Complementarities and Games: New Developmentsblog.iese.edu/xvives/files/2011/09/111.pdfComplementarities and Games: New Developments X AVIER V IVES 437 ... gic situations is the theory

Journal of Economic LiteratureVol XLIII (June 2005) pp 437ndash479

Complementarities and Games New Developments

XAVIER VIVESlowast

437

lowast Vives INSEAD and ICREA-UPF I am grateful toTim Van Zandt two anonymous referees and the editorfor very helpful comments and to INSEAD for financialsupport

1 Introduction

Complementarities are pervasive in eco-nomics ranging from coordination

problems in macroeconomics and financeto pricing and product selection issues inindustrial organization At the heart ofcomplementarity is the notion due toEdgeworth that the marginal value of anaction or variable increases in the level ofanother action or variable

Complementarities have been a recurrentand somewhat contentious topic of study foreconomic analysis Indeed while Paul ASamuelson (1947) in his Foundations statedthat ldquoIn my opinion the problem of comple-mentarity has received more attention thanis merited by its intrinsic importancerdquo (at thestart of the section on complementarity p183 1979 edition) he later corrected himselfin this very journal in 1974 on the occasionof the fortieth anniversary of the HicksndashAllenrevolution in demand theory when he statedat the very beginning of his paper that

The time is ripe for a fresh modern look at theconcept of complementarity Whatever the

intrinsic merits of the concept forty years ago ithelped motivate Hicks and Allen to performtheir classical ldquoreconsiderationrdquo of ordinaldemand theory And as I hope to show the lastword has not yet been said on this ancient pre-occupation of literary and mathematical econo-mists The simplest things are often the mostcomplicated to understand fully

Complementarities have a deep connec-tion with strategic situations and the conceptof strategic complementarity is at the centerstage of game-theoretic analyses Examplesabound An arms race If the Soviet Unionincreased its spending on nuclear weapons itwould pay the United States to respond byincreasing its own spending A bank run Ifother customers of the bank that holds yoursavings withdraw their money it might payyou to also take your money out A currencycrisis If other investors attack the currency itmight pay me to do so as well An RampD raceIf my main rivals in the pharmaceuticalindustry increase their RampD spending doesit pay me to increase it also Technologyadoption My neighbors have introduced anew type of crop does it pay me to followLocation If a new store locates in the shop-ping mall does the mall become a moreattractive location for other stores

The modeling of complementaritiesincluding strategic situations has proved

jn05_Article 3 62205 100 PM Page 437

438 Journal of Economic Literature Vol XLIII (June 2005)

challenging The reason is that the tools athand were not attuned to deal with environ-ments where indivisibilities nonsmooth pay-offs and complex strategy spaces werenaturally the norm rather than the exceptionTo make matters worse multiple equilibriaare typical in the presence of complementar-ities and policy analysis is left orphanInstances of coordination failure with multi-ple equilibria abound bank runs debt runson a country low employmentactivity equi-libria revolutions and development trapsprovide some examples In all these situa-tions there are multiple self-fulfilling expec-tations of agents in the economy A key issueis how to build coherent models that are use-ful for policy analysis A challenging aspectof any crisis situation is to disentangle self-fulfilling from fundamentals-driven explana-tions that help answer questions such asWhat is the effect of an increase in theamount of central bank reserves on the prob-ability of a run on the currency What is theimpact of an increase in the solvency ratio onthe probability of failure of a bank What isthe effect of a change in foreign short-termdebt exposure on the probability of defaultof a small open economy

The appropriate toolbox to deal with com-plementarities and in particular with strate-gic situations is the theory of monotonecomparative statics and supermodulargames This theory provides the method formaking the analysis of complementarity sim-ple The basic idea of the theory is to exploitfully both order and monotonicity proper-ties The achievements of this approach areas follows First it provides a framework forthinking rigorously about complementari-ties identifying key parameters in the envi-ronment to look at (eg what are the criticalproperties of the payoffs and action spaces)Second it simplifies the analysis clarifyingthe drivers of the results (eg is the regular-ity condition really needed to obtain thedesired comparative static result) Third itencompasses the analysis of multiple equi-libria situations by ranking equilibria and

helping understand how potential equilibriamove with the parameters of interestFinally it easily incorporates complex strate-gy spaces such as those arising in dynamicgames and games of incomplete information

The paper provides an introduction to thetools of supermodular games and a range ofapplications to industrial organization(Cournot and Bertrand markets monopolis-tic competition RampD races multimarketoligopoly switching costs dynamic invest-ment games) macroeconomic models(menu costs search aggregate demandexternalities technology adoption adjust-ment costs) and finance (currency crisisbank runs) The next section presents theapproach in somewhat more detail as well asthe plan of the paper

2 Tools Results and Plan of the Paper

The basic theory was developed byDonald M Topkis (1978 1979) and furtherdeveloped and applied to economics byXavier Vives (1985a 1990a) and PaulMilgrom and John Roberts (1990a) Yet thetheory continues to be extended at the fron-tier of researchmdashfor example to dynamicgames and games of incomplete informa-tion The beauty of the approach is not itscomplexity but rather how much it simplifiesthe analysis and clarifies results In fact eventhe basic tools are not fully exploited byeconomists in current research

The theory of supermodular games andmonotone comparative statics based on lat-tice-theoretic methods has provided a pow-erful toolbox for analyzing the consequencesof complementarities in economicsMonotone comparative statics analysis pro-vides conditions under which optimal solu-tions to optimization problems movemonotonically with a parameter In this paperI provide an introduction to this methodologyand then apply it to the study of strategicinteraction in the presence of complementar-ities This approach exploits order and monot-onicity properties in contrast to classical


Vives Complementarities and Games New Developments 439

convex analysis The central piece of attentionwill be games of strategic complementaritieswhere the best response of a player to theactions of rivals is increasing in their level

The purpose of this paper is to bring for-ward some recent applications of the lattice-theoretic methodology and at the same timeprovide an introduction to the toolbox Ishall demonstrate the usefulness of theapproach for

bull providing a common analytical frame tostudy complementarities

bull deriving new results andbull casting new light on old results by simpli-

fying proofs and discarding unnecessaryassumptions

Modeling strategic interaction presentsformidable problems A Nash equilibriummay not exist at least in pure strategies Orinstead there may be multiple equilibriaHow do players coordinate on one of themHow can the policymaker be sure thatchanging a parameter will have the desiredeffect Classical comparative statics analysisprovides ambiguous results in the presenceof multiple equilibria and imposes strongregularity conditions These regularity con-ditions become particularly strong whenapplied to games with complex functionalstrategy spaces such as dynamic or Bayesiangames We will see how complementaritiesare intimately linked to multiple equilibriaand how supermodular methods provide anatural tool for characterizing them

The class of games with strategic comple-mentarities and the tools to analyze themhave very nice properties

bull They allow very general strategy spacesincluding indivisibilities and functionalspaces such as those arising in dynamicor Bayesian games

bull They ensure the existence of equilibriumin pure strategies (without requiring quasiconcavity of payoffs smoothnessassumptions or interior solutions)

bull They allow a global analysis of the equi-librium set which has an order structurewith largest and smallest elements

bull Equilibria have useful stability proper-ties and there is an algorithm to computeextremal equilibria

bull Monotone comparative statics results areobtained with minimal assumptions byeither focusing on extremal equilibria orconsidering best-response dynamicsafter the perturbation

Furthermore as we shall see results canbe extended beyond the class of games withstrategic complementarities

Let me highlight here some examples ofhow the lattice-theoretic approach eitherobtains new results that are hard or impossi-ble to derive using the classical approach orimproves upon results already obtained bygetting rid of unnecessary assumptions orsimplifies and deepens our understanding ofthe proof of known results

bull Consider an RampD race where each firminvests continuously to obtain a break-through and where we want to knowwhat the effect is of increasing the num-ber of participants n in the race (TomLee and Louis L Wilde 1980) Undervery weak assumptions this game is oneof strategic complementarities and it willhave multiple equilibria The problem ofusing the classical approach is thatincreasing n may make some equilibriadisappear while others may appearClassical analysis will not help here butwith the lattice approach we obtain anunambiguous comparative statics resultincreasing n will necessarily increaseRampD effort provided that out-of-equi-librium adjustment dynamics are of ageneral adaptive form

bull Consider a dynamic monopolistic com-petition model with menu costs wherefirms interact repeatedly over an infinitehorizon and each firm receives an idio-syncratic demand or cost shock everyperiod Under what conditions doesthere exist a Markov perfect equilibri-um When is the current price of a firmincreasing in its past price and the dis-tribution of prices in the market The



lattice approach provides the keyassumptions needed to answer thesequestions (Christopher Sleet 2001 andByoung Jun and Vives 2004)

bull Masahiro Okuno-Fujiwara AndrewPostlewaite and Kotaro Suzumura(1990) provided conditions under whichfully revealing equilibria obtain in duop-oly games of voluntary disclosure ofinformation when information is verifi-able The conditions involve restrictiveregularity assumptions such as concavityof payoffs interiority of equilibria andindependent types for the players Ourapproach allows us to omit these unnec-essary regularity assumptions highlightthe crucial ones (those related to monot-onicity conditions) and extend the resultto n-player games

bull Global games (Hans Carlsson and Ericvan Damme 1993) are games of incom-plete information with types determinedby each player observing a noisy signal ofthe state They are proving to be a popu-lar methodology for equilibrium selec-tion using iterated elimination ofdominated strategies and have wideapplications to currency and bankingcrises and macroeconomics (StephenMorris and Hyun Song Shin 2002)Global games are Bayesian games andthe lattice approach is particularly suitedto analyze them For example recentmajor advances in the difficult problem ofshowing the existence of Bayesian equi-librium in pure strategies have beenmade using the lattice-theoretic method-ology Furthermore by realizing thatglobal games are typically games of strate-gic complementarities we understandwhy and how iterated elimination of dom-inated strategies works and why andunder what conditions equilibrium selec-tion is successful Indeed we will see howequilibrium is unique precisely whenstrategic complementarities are weak andthat comparative statics results can bederived even for multiple equilibria

The methodology of supermodular gamesprovides the tools and an appropriate frame-work for satisfactorily confronting multipleequilibria and comparative statics Howeverwe should be aware also that the lattice-the-oretic approach is not a panacea and cannotbe applied to everything (Indeed theapproach builds on a set of assumptions) Togive an obvious example the approach can-not make equilibria appear in a game thathas no equilibrium to start with

We begin by introducing a simple class ofgames in section 3 where many of theimportant issues are highlighted The classincludes monopolistic competition searchand adoption games Section 4 provides anintroduction to the theory and basic resultsSection 5 provides applications to oligopolyand comparative statics in the context ofCournot Bertrand and RampD games includ-ing multimarket oligopoly competitionSection 6 deals with dynamic games therewe examine when increasing or decreasingdominance will obtain in investment gamesand we also characterize strategic incentivesin Markov games Applications to menuswitching and adjustment costs are providedas well Section 7 studies Bayesian gamescharacterizing equilibria in pure strategiesand comparative statics properties withapplications (among others) to games of vol-untary disclosure and global games includ-ing currency and banking crises Theappendix provides a brief recollection of themost important definitions and results of thelattice-theoretic method

3 A Simple Framework

Games of strategic complementarities arethose in which players respond to anincrease in the strategies of the rivals with anincrease in their own strategy This sectionpresents an example suggesting the flavor ofmany of the results that can be obtained withthe approach

Consider a game with a continuum ofplayers in which the payoff to a player is



1 The analysis with n players is similar

π(aisimai) Here ai is the action of the playerlying in a (normalized) compact interval[01]sima is the average or aggregate actionand i is a (possibly idiosyncratic) payoff-relevant parameter1 I consider first the casewith homogeneous players and then laterthe case with heterogeneous players

31 Homogeneous Players

Consider the symmetric case where thepayoff to a player is given by π(aisima)Suppose that π is smooth in all argumentsand strictly concave in ai and let r be thebest response of an individual player toaggregate action sima In this framework equi-libria will be symmetric because given anyaggregate action sima there is a unique bestresponse r (sima) For interior solutions wewill have that

(r (sima)sima) = 0

If part2π(partai)2 lt 0 then r is continuously differ-

entiable and

Therefore sign r(~a) = signpart2πpartaipart~a andbest replies are increasing if part2πpartaipart~a ge 0 A symmetric equilibrium is characterized by r(a) = a Suppose also that part2πpartaipart ge 0so that an increase in increases the mar-ginal profit of the action of a player and hisbest response r

Two examples of the game are monopolis-tic competition and search In monopolisticcompetition (see Vives 1999 section 66)the action would be the price of a firm with~a the average price in the market and ademand or cost parameter We have

π(ai~a) = (ai minus )D(ai~a)

with D the demand function and the(common) marginal cost As we will see insection 4 for many demand systemspart2logDpartaipart~a gt 0 (meaning that the elasticity

prime( ) = minuspart part part

part part( )r a

a a

ai

i

~ ~

2

2 2

π

π

partpart

πai

of demand for product i is decreasing in theaverage price) and therefore part2logπpartaipart~a =part2logDpartaipart~a gt 0 Under this condition wewill have that r(~a) gt 0 because best repliesare invariant to an increasing transformationof the payoffs such as the logarithm In thesearch model (Peter A Diamond 1981) theaction ai is the effort of trader i in looking fora partner The benefit (probability of findinga partner) is proportional to own effort and isincreasing in the aggregate effort ~a of others

π(ai~a) = aig(~a) minus C(ai)

where gt 0 is the efficiency of the searchtechnology and where g and the cost ofeffort C are increasing functions In thiscase part2πpartaipart~a = g(~a) ge 0

In these examples it is easy to generatemultiple equilibria For instance in thesearch model let g(~a) equiv ~a and let C beincreasing with C(0) = 0 then ai = 0 for all iis always an equilibrium If C is smooth andstrictly convex with C(0) = 0 then there aretwo equilibria ai = 0 and ai = acirc gt 0 with acirc = C(acirc) for all i The latter equilibriumincreases strictly with and is Pareto supe-rior to the no-effort equilibrium Anotherpossibility is when g has an S-shaped func-tion and C(a) equiv a then there will be threeequilibria They will be the solutions (iea acirc and ndasha) to g(a) = a as depicted in figure 1 (lower branch) In this exampler(~a) = g(~a) Obviously equilibria are thesolution to r (a) = a and r(~a) = g(~a)

Several properties of the equilibria in thesearch example are worth noticing1 A sufficient condition to have multiple

equilibria is that strategic complementar-ities be sufficiently strongmdashnamely thatr(a) gt 1 for some candidate equilibriumr (a) = a (such as point acirc in figure 1)

2 The symmetric equilibria are orderedThere exits a largest (ndasha) and a smallest (ndasha)equilibrium (this follows trivially heregiven that actions are one-dimensional)and equilibria can be Pareto ranked Thisis a general property whenever π isincreasing in ~a (positive externalities)



Figure 1 Best Response (with Homogenoues Players) and Multiple Equilibria

acirc a1

1

a a a

3 Extremal equilibria ndasha and ndasha are stablewith respect to the usual best-replydynamics Indeed it is immediate thatbest response dynamics starting at a = 0(resp a = 1) will converge to ndasha (resp ndasha)See figure 1

4 Iterated elimination of strictly dominatedstrategies defines two sequences that con-verge respectively to ndasha and ndasha For exam-ple let ndasha

0 = 0 Players will never use astrategy a lt r(0) because it is strictly dom-inated by ndasha

1 = r(0) Now knowing that noone will use a strategy in [0r(0)) theregion [0r(r(0))) will also be strictlydominated Let ndasha

2 = r(ndasha1) and define ndasha

k

recursively The sequence ndashak is increasing

and converges to ndasha (indeed it coincideswith best-reply dynamics starting at a =0) (See figure 1) This means that ratio-nalizable strategies will lie in the interval[ndashandasha] and if the equilibrium is uniquethen the game will be dominance solv-able That is the final outcome of theprocess of iterated elimination of strictly

2 Indeed at a stable equilibrium r lt 1 (or part2πi(partai)2 +part2πipartaipart~a lt 0) At equilibrium r(a) = a and therefore inthe vicinity of da d = (partrpart)(1 minus r) gt partrpart providedthat r gt 0 (or part2π partaipart~a gt 0)

dominated strategies is both unique andan equilibrium

5 An increase in the parameter will leadto an increased action in equilibriumgiven out-of-equilibrium best-responsedynamics and this increase will be overand above the direct effect of the increasein the parameter Indeed increasing will move r upward (as in figure 1) andthe equilibrium level of a will increaseStarting at a = ndasha the direct effect will leadus to r(ndasha) gt ndasha and the full equilibriumimpact to ndasha gt ndasha2 The consequences of a common shock (or for that matter an idiosyncratic shock) are amplifiedBecause of strategic complementaritiesthere is a multiplier effect Indeed thedirect effect of an increase in in theaction of an agent taking as given the



3 Roger Guesnerie (1992) has shown this in a version ofthe model

4 See Satyajit Chatterjee and Cooper (1989) MarcoPagano (1989) or Philip H Dybvig and Chester S Spatt(1983) for related examples

average action is amplified by theincrease in the average action This hap-pens whether we focus on at extremal (orstable) equilibria or rather consider best-response dynamics after the perturbationEven starting at an unstable equilibriumor at an equilibrium that disappears once increases an increase in will result inan increase in a over and above the directeffect In figure 1 the unstable equilibri-um acirc disappears with the increase in moving r upward and best-replydynamics lead to the new equilibrium ndashaWith strategic substitutability among

strategies part2πpartaipart~a lt 0 there cannot bemultiple symmetric equilibria In this case itis immediate that there is a unique symmetricequilibrium (because

part2πi part2πi partπindashndashndashndash + ndashndashndash lt 0 and ndashndashndash (aa) = 0(partai)

2 partaipart~a partaiwill have a unique solution) It is easy to seethat when 0 gt r gt minus 1 (or |r| lt 1) thegame is dominance solvable3 This corre-sponds to the case where the symmetricequilibrium is stable according to the usualcobweb dynamics Equivalently in terms ofiterated elimination of strictly dominatedstrategies agents recognize that no one willtake an action larger than r(0) this starts theprocess of elimination of strategies nowwith alternating regions on both sides of thecandidate equilibrium

Models of aggregate demand externalitiesand models of Keynesian effects have a sim-ilar flavor to our simple model (see RussellCooper and Andrew John 1988) Themonopolistic competition model has beenused extensively in the growth developmentregional and international trade literaturesto generate complementarities and multipli-er processes (see Kiminori Matsuyama 1995for a survey) In all those instances the pres-ence of multiple Pareto rankable equilibria

multiplier effects and cumulative self-reinforcing processes is central to the analysis

32 Heterogeneous Players

A variation of the search example encom-passes heterogeneous agents4 Suppose anagent must decide whether or not to adopt anew technology (or whether to ldquoinvestrdquoldquoactrdquo or ldquoparticipaterdquo) His action is ai = 0 ifthere is no adoption and is ai = 1 if there isadoption The cost of adoption is idiosyn-cratic and follows a distribution function Fon the interval [ndashndash ] The cost i for agent i isan independent draw from F The benefit ofadoption is g(~a) where ~a is the total massadopting (which will be between 0 and 1)and no adoption yields no benefit Therefore

π(ai~ai) = ai(g(~a) minus i)

The game is one of strict strategic comple-mentarities if g gt 0 It is worth noticingthat because of independence the adoptingmass ~a will be nonrandom Player i willadopt if g(~a) minus i ge 0 An equilibrium will begiven by an adoption threshold and anadopting mass ~a = F() such that g(~a) = and agent i will adopt if i le The aggregatebest reply to the adopting mass ~a is justF(g(~a)) or equivalently the best reply to athreshold used by other players is g(F())The equilibria can be depicted as in figure 1where on the horizontal axis we have ~a or and along the vertical axis the best-reply r(~a) = F(g(~a))or r () = g(F()) For examplelet g(~a) = ~a ndash lt 0 and ndash gt 1 Then for gt 0to adopt is a dominant strategy (ie it pays toadopt even if no one else adopts) for gt 1not to adopt is a dominant strategy (ie itdoes not pay to adopt even if everyone elseadopts) In equilibrium ~a = F() =

The equilibrium threshold solvesg(F()) minus = 0 The solution will be uniqueif gF minus 1 = gƒ minus 1 lt 0 where f is the den-sity of F It is thus immediate that if g lt 0(strategic substitutability) then the equilib-rium is unique The question is When dowe have a unique equilibrium with strategiccomplementarities



5 Recall that if x~N(microσ 2) then f(micro) = (σradic2π)1 wheref is the density of x

It is instructive to think of the case wherei follows a normal distribution with mean micro

and variance σ 2 and where the costs of adop-

tion i and j (j ne i) are potentially correlat-ed with covariance σ 2

for isin[01) The case = 0 corresponds to the independent caseconsidered previously Suppose that playersadopt strategies with adoption threshold (In section 7 we will see that equilibriummust be of this form) From the point ofview of player i and given i the adoptingmass will be given by

where is the cumulative distributionfunction of the standard normal The agentwill adopt if and only if g(~ai) minus i ge 0 andthe equilibrium threshold will satisfy

where

The solution will be unique if (1 minus )(1 + )g minus 1 lt 0 where φ = is the density of the standard normal It isthen immediate that the equilibrium will beunique when (1 minus )(1 + )g2π lt 1where g equiv sup a[01] g(a)5 There will be aunique equilibrium when the degree ofstrategic complementarity is not too strongThis may happen either because payoff com-plementarities are weak (ndashg low) or becauseeach player ex ante faces a large cost uncer-tainty (σ high) or because the correlation ofthe costs is high ( close but not equal to 1)All three factors tend to lessen the strengthof strategic complementarities

Let g(~a) = ~a in order to illustrate theeffect of uncertainty If costs are perfectlycorrelated then there are multiple equilibria

Pr ˆ ˆˆ

θ θ θ θ ρρ

θ microσ

θ

θj ile = = minus

+minus

11

g j iPr ˆ ˆ ˆθ θ θ θ θle = ( ) minus = 0

îθ ρθ ρ micro

σ ρθ

θ

=minus + minus( )( )

minus

1

1 2

~ Pr âi j i θ θ θle

6 What the two cases have in common is that the playerputs very little weight on prior information when σ 2

is verylarge because the prior is flat when is close to 1 becausethe type of the player predicts (almost) perfectly the typesof others

for isin(01) In this case there is completeinformation because a playermdashby knowinghis own costmdashknows the costs of any otherplayer However a little bit of imperfect costcorrelation ( close to 1) will yield a uniqueequilibrium Note for example that((1 minus )(1 + )( minus micro))) tends to 12either when rarrinfin or as rarr1 yielding theunique solution = 12 In figure 2 the case micro = 12 is displayed and = micro = 12is the equilibrium threshold Then if (1 minus )(1 + )2π gt 1 two more equi-libria appear

Either with a diffuse prior or when thecost of a player gives very precise informa-tion about the costs of others the (strategic)uncertainty of player i is maximal withrespect to the behavior of others Thisinduces a best response for the player whichis quite ldquoflatrdquo that is not very sensitive to thethreshold used by others6

33 How General Are the Results

The question arises of how far the niceresultsmdashabout existence and characteriza-tion of equilibria and comparative staticpropertiesmdashin our simple game of section31 extend to different specifications (what ifpayoffs are not concave and best responsesare not unique) or more general games withstrategic complementarities or evenbeyond As we will see in the next sectionmost of the properties generalize to multidi-mensional strategy spaces discrete or con-tinuous and even functional spaces as wellas to nonsmooth and nonconcave payoffsThe basic insight of the next section will bethat to obtain the desired results only themonotonicity properties of incremental pay-offs and the order properties of strategiesmatter Most of the regularity conditions typ-ically assumed will not be crucial In section7 we will study Bayesian games and see how



Figure 2 Equilibrium Thresholds (with Heterogeneous Agents) and Multiple Equilibria

1

12

microθ = 12 1 θ

our intuitions concerning the game with het-erogeneous players of section 32 generalizeto equilibrium selection in global games Wewill see how the consideration of thresholdstrategies is in fact without loss of generalityand how the key to uniqueness will be aninformation structure (diffuse prior or thesignal of a player giving very precise infor-mation about the signals of others) thatweakens strategic complementarities Thegeneral principle is that in order to obtainuniqueness of equilibrium in the presence ofcomplementarities the degree of strategiccomplementarities must be weak

4 An Introduction to Games with Strategic Complementarities

In this section I provide first a briefintroduction to the tools and main resultsof the theory and then comment on thetheoryrsquos scope

41 Modeling Complementarities andResults

Complementarities are dealt with usingtools provided by the theory of monotonecomparative statics and supermodulargames Those tools are based on lattice-the-oretic results that exploit order and monoto-nicity properties of action sets and payoffsThe basis of the approach are monotonecomparative statics results developed byTopkis (1978) and the application of AlfredTarskirsquos (1955) fixed point theorem toincreasing functions In a game situationminimal assumptions are put on strategy setsand payoffs so that best responses areincreasing and move monotonically with theparameters of interest Then Tarskirsquos fixedpoint theorem delivers existence of equilib-ria as well as order properties of the equilib-rium set and comparative statics resultsfollow naturally



7 The reader is referred to chapter 2 of Vives (1999) fora more thorough and general treatment of the theory aswell as proofs further references and applications

This approach provides a powerful analyt-ical tool that confronts the usual obstacleswhen analyzing a game existence of pure-strategy equilibria comparison of equilibriaand comparative statics In particular ingames of strategic complementarities thepresence of multiple equilibria need not bean obstacle to performing comparative stat-ics analysis

The emphasis of the exposition will be onintuition and not the technical details Thissection will provide the minimal backgroundnecessary for a reader to follow the rest ofthe paper and the appendix contains techni-cal definitions and intermediate resultsExamples will be developed to illustrate themethodology7

We will use the intuitive concept of agame of strategic complementarities (GSC)whenever the best responses of the playersin the game are increasing in the actions ofrivals The technical concept of supermodu-lar game (to be shortly defined below)defined provides sufficient conditions forbest responses to be increasing

I will provide a definition of a supermod-ular game in a smooth context in order tokeep the mathematical apparatus to a mini-mum but this is by no means the most gen-eral way to define it Consider the game (Aiπi i isin N) where N is the set of players i = 1hellip n Ai is the strategy set a compactcube in Euclidean space and πi the payoff ofplayer i isin N (defined on the cross productof the strategy spaces of the players A) Letai isin Ai and ai isin ΠjiAj (ie we denote byai the strategy profile (a1hellip an) exceptingthe ith element) Let aih denote the hth component of the strategy ai of player i

We will say that the game (Aiπi i isin N) issmooth supermodular if for all i

bull Ai is a compact cube in Euclidean spacebull πi(aiai) is twice continuously differen-

tiable

8 Continuity is needed to ensure the existence of bestreplies and the continuity requirement can also be weak-ened See the appendix for the general definitions of lat-tices supermodularity increasing differences andsupermodular game

9 Supermodularity and increasing differences can evenbe weakened to define an ldquoordinal supermodularrdquo gamerelaxing supermodularity to the weaker concept of quasi-supermodularity and increasing differences to a single-crossing property (see Milgrom and Chris Shannon 1994)However such properties (unlike supermodularity andincreasing differences) have no differential characteriza-tion and need not be preserved under addition or partialmaximization operations

1 supermodular in ai for fixed ai orpart2πipartaihpartaik ge 0 for all k ne h and

2 with increasing differences in (aiai)or part2πipartaihpartajk ge 0 for all j ne 1 and forall h and k

The game is smooth strictly supermodularif the inequality in (2) is strict

Condition (1) is the complementarityproperty (supermodularity) in own strate-gies It means that the marginal payoff to anystrategy of player i is increasing in the otherstrategies of the player Condition (2) is thestrategic complementarity property in rivalsrsquostrategies ai It means that the marginalpayoff to any strategy of player i is increasingin any strategy of any rival player This prop-erty of πi is termed increasing differences in(aiai) In the general formulation of asupermodular game strategy spaces needonly be ldquocomplete latticesrdquo only continuity(not differentiability) of payoffs is neededand properties (1) and (2) are stated in termsof increments8

In a supermodular game very generalstrategy spaces can be allowed These includeindivisibilities as well as functional strategyspaces such as those arising in dynamic orBayesian games (as we will see in section 62and section 7) Regularity conditions such asconcavity and interior solutions can be dis-pensed with The complementarity propertiesare robust in the sense that they are preservedunder addition or integration pointwise lim-its and maximization (with respect to a subsetof variables preserving supermodularity forthe remaining variables)9



10 See chapter 6 of Vives (1999) However this is not auniversal result as we will see in section 42

11 See lemma 1 in the appendix for a precise statementof the result

Two leading oligopoly models fit in manyspecifications the complementarity assump-tions made A first example is a Cournot oli-gopoly with complementary products In thiscase the strategy sets are compact intervalsof quantities and the complementarityassumptions are natural (what MichaelSpence (1976) called ldquostrong comple-mentsrdquo) A second example is a Bertrand oli-gopoly with differentiated substitutableproducts where each firm produces a differ-ent variety The demand for variety i is givenby Di(pipi) where pi is the price of firm iand pi denotes the vector of the pricescharged by the other firms A linear demandsystem will satisfy the complementarityassumptions

The application of the theory can beextended by considering increasing transfor-mations of the payoff (which does notchange the equilibrium set of the game) Wesay that the game is log-supermodular if πi isnonnegative and if πi fulfills conditions (1)and (2) In the Bertrand oligopoly examplethe profit function πi = (pi minus ci)Di(pipi) offirm i where ci is the constant marginal costis log-supermodular in (pipi) wheneverpart2logDipartpipartpj ge 0 For firm i this holdswhenever the own-price elasticity ofdemand ηi is decreasing in pi as with con-stant elasticity logit or constant expendituredemand systems10

The key results of the theory are obtainedby a combination of monotone comparativestatics results due to Topkis and Tarskirsquos fixedpoint theorem The results by Topkis (1978)deliver monotone increasing best responseseven when πi is not quasi-concave in ai

The basic monotone comparative staticsresult states that the set of optimizers of afunction u(x t) that is parameterized by tsupermodular in x and with increasing dif-ferences in x and t has a largest and a smallestelement and that both are increasing in t11

12 This definition was used in Vives (1985a) who con-centrated attention on monotone increasing best respons-es as the defining characteristic of games with strategiccomplementarities See the appendix for a more formaldefinition along those lines

In a supermodular game this means thateach player i has a largest ndashψi(ai) = supψi(ai)as well as a smallest ndashψi(ai) = inf ψi(ai) bestreply and that they are increasing in thestrategies of the other players If the game isstrictly supermodular then any selectionfrom the best-reply correspondence isincreasing

Some intuition can be gained from thesimple framework of section 31 Weobserved that if part2πi(partai)

2 lt 0 then r(~a) = sign(part2πipartaipart~a) Now even when πi

is not quasi-concave the monotone com-parative statics result implies that if part2πipartaipart~a gt 0 then any selection from thebest-reply correspondence of player i (whichmay have jumps) is increasing in the averageaction

We could define also the (weaker) conceptof a game of strategic complementarities(GSC) under our maintained assumptionsas a game where (a) strategy sets are com-pact cubes (or ldquocomplete latticesrdquo) (b) thebest reply of any player has extremal (largestand smallest) elements and (c) those ele-ments are increasing in the strategies ofrivals Similarly we could define a game ofstrict strategic complementarities if in addi-tion any selection from the best reply of anyplayer is increasing in the strategies of therivals12 All the results stated hereafter willthen hold replacing (strictly) supermodulargame by GSC (game of strict SC)

The following results hold in a supermodu-lar game Let ndashψ = (ndashψ1hellipndashψn) and ndashψ =(ndashψ1hellipndashψn) denote the extremal best-replymaps

Result 1 Existence and order struc-ture (Topkis 1979) In a supermodulargame there always exist extremal equilibriaa largest element ndasha = supaisinA ndashψ(a) ge a anda smallest element ndasha = infaisinA ndashψ(a) le aof the equilibrium set



13 The equilibrium set has additional order properties(see Vives 1985a Vives 1990a problem 25 in Vives(1999and Lin Zhou(1994)

14 Indeed if -a1 ne -a2 then because the game is symmet-ric (-a2 -a1 -a3hellip -an) will also be an equilibrium and there-fore because (--a1 -a2 -a3hellip -an) is the largest equilibrium -a1 ge -a2 ge -a1 and -a1 = -a2

The result is shown by applying Tarskirsquosfixed point theorem (which implies that anincreasing function from a compact cubeinto itself has a largest and a smallest fixedpoint see appendix) to the extremal selec-tions of the best-reply map ndashψ and ndashψ whichare monotone by the strategic complemen-tarity assumptions There is no reliance onquasi-concave payoffs and convex strategysets to deliver convex-valued best replies asis required when showing existence usingKakutanirsquos fixed point theorem13

In the Bertrand oligopoly for examplewhen the payoff is supermodular or log-supermodular then it follows that extremalprice equilibria do exist The results can beextended to convex costs and multiproductfirms and so provide a large class of Bertrandoligopoly cases for which the classical nonex-istence of equilibrium problem encounteredby Roberts and Hugo Sonnenschein (1977)does not arise

Result 2 Symmetric games For a sym-metric supermodular game (exchangeableagainst permutations of the players) the fol-lowing statements hold

bull Symmetric equilibria exist because theextremal equilibria ndasha and ndasha are symmet-ric14 Hence if there is a unique sym-metric equilibrium then the equilibriumis unique (since ndasha = ndasha) This result provesto be a very useful tool for showinguniqueness in symmetric supermodulargames For example in a symmetric ver-sion of a Bertrand oligopoly system withconstant elasticity of demand and con-stant marginal costs it is easy to see thatthere exists a unique symmetric equilib-rium Since the game is strictly log-supermodular we can conclude that theequilibrium is unique

15 See footnote 23 in Vives (1999) for a proof of thestatement

16 See section 231 in Vives (1999)17 Milgrom and Roberts (1994) also state and prove the

theorem with S = [01]18 The argument is simple Consider a symmetric game

with compact intervals as strategy spaces and let πi(aiai)= π(aiΣjneiaj) as in a Cournot game with homogeneousproduct and identical cost functions (or as in the game ofsection 3 with a continuum of players) Existence of sym-metric equilibria follows then from the stated result if thebest-reply Ψi of a player (identical for all i due to symme-try) has no jumps down This is in fact true if costs are con-vex in the Cournot game Symmetric equilibria are givenby the intersection of the graph of ai = Ψi(Σjneiaj) with theline ai = (Σjneiaj)(n minus 1)

bull If the strategy spaces of the players areone-dimensional (or more generallycompletely ordered) then a symmetricstrictly supermodular game has onlysymmetric equilibria15

For one-dimensional strategy spaces exis-tence of symmetric equilibria can beobtained by relaxing the monotonicityrequirement of best responses It is enoughthen that all jumps in the best reply of a play-er be up Existence follows from Tarskirsquosintersection point theorem16 The result iseasy to grasp considering a functionƒ [01]rarr[01] which when discontinuousjumps up but not down The function mustthen cross the 45˚ line at some point Indeedsuppose that it starts above the 45˚ line (oth-erwise 0 is a fixed point) then it either staysabove it (and then 1 is a fixed point) or itcrosses the 45˚ line Versions of this fixedpoint theorem have been derived by MMcManus (1962 1964) and Roberts andSonnenschein (1976) to show existence ofequilibria in symmetric Cournot games withconvex costs17 More generally In a sym-metric game where the strategy space ofeach player is a compact interval and thepayoff to a player depends only on her ownstrategy and the aggregate strategy of rivalsif the best reply of a player has no jumpsdown then symmetric equilibria exist Thisimplies in particular that for the game insection 31 under very weak assumptions asymmetric equilibrium will always exist18



Figure 3 Cournot Tatocircnnement in a Supermodular Game with Best Reply Functions r1 r2

a2

A+

A+

Aminus

Aminus

a

a

a1

r2()

r1()

Result 3 Welfare (Milgrom andRoberts 1990a Vives 1990a) In a super-modular game if the payoff to a player isincreasing in the strategies of the other play-ers (positive externalities) then the largest(resp smallest) equilibrium point is thePareto best (resp worst) equilibrium Thisis a very simple result that is at the base offinding equilibria that can be Pareto rankedin many games with strategic complemen-tarities For example in the Bertrand oli-gopoly example the profits associated withthe largest price equilibrium are also thehighest for every firm

Result 4 Stability and rationalizabili-ty Consider a supermodular game with con-tinuous payoffs Then we have the following1 Simultaneous response best-reply

dynamics (Vives 1990a)bull Approach the ldquoboxrdquo [ndashandasha ] defined by

the smallest and the largest equilib-rium points of the game Hence ifthe equilibrium is unique then it isglobally stable

bull Converge monotonically downward(upward) to an equilibrium startingat any point in the intersection of

the upper (lower) contour sets of the largest (smallest) best replies ofthe players A+ equiv a isin A ndashψ(a) le a(A- equiv a isin A ndashψ(a) ge a) See figure 3

2 The extremal equilibria ndasha and ndasha corre-spond to the largest and smallest seriallyundominated strategies Therefore if theequilibrium is unique then the game isdominance solvable (Milgrom andRoberts 1990a)

This result implies that all relevant strate-gic action is happening in the box [ndashandasha ]defined by the smallest and largest equilib-rium points For example rationalizableoutcomes (B Douglas Bernheim 1984David G Pearce 1984) and supports ofmixed-strategy and correlated equilibriamust lie in the box [ndashandasha ] The argument forthe second part of result 41 is quite simplebecause for example starting at any pointin A+ (see figure 3) best-reply dynamicsdefine a monotone decreasing sequencethat converges to a point that must ( by con-tinuity of payoffs) be an equilibriumResults in 41 extend to a large class of adap-tive dynamics of which best-reply dynamicsare a particular case



In fact starting at the largest (smallest)point of the strategy space Amdashrecall it is acubemdashbest-reply dynamics with the largest(smallest) best-response map ndashψ(ndashψ) will leadto the largest (smallest) equilibrium ndasha(ndasha )(Topkis 1979) For example starting atinf A (see figure 3) best-reply dynamicswith the smallest best reply map ndashψ definea monotone increasing sequence thatconverges to a point y that by continuityof payoffs must be an equilibriumFurthermore this must be the smallestequilibrium y = ndasha For any other equilibri-um x x ge inf A and iterating the best replymap ndashψ on both sides of the inequality yieldsx ge ndasha because ndashψ is increasing

Starting at an arbitrary point we cannotensure convergence because for instance acycle is possible For example in figure 3starting at a0 = (ndasha1ndasha 2) the simultaneousresponse best-reply dynamics cycle between(ndasha1ndasha 2) and (ndasha 1ndasha2)

In the Bertrand oligopoly example withlinear constant elasticity or logit demandsthe equilibrium is unique and so the game isdominance solvable and globally stable

Another interesting result is that properlymixed equilibria (ie Nash equilibria forwhich at least two playersrsquo strategies are notpure strategies) in strictly supermodulargames are unstable with respect to best-reply or more general adaptive dynamics(Federico Echenique and Aaron S Edlin2004) An example is the mixed-strategyequilibrium in the battle of the sexes

Result 5 Comparative staticsConsider a n-player supermodular gamewith payoff for firm i πi(aiai t) par-ameterized by a vector t = (t1hellip tn) If πi has increasing differences in (ai t)(part

2πipartaihparttj ge 0) for all h and j) thenwith an increase in t(i) the largest and smallest equilibrium

points increase and(ii) starting from any equilibrium best-

reply dynamics lead to a (weakly) largerequilibrium following the parameterchange

Furthermore the latter result can beextended to a class of adaptive dynamicsincluding fictitious play and gradient dynam-ics Continuous equilibrium selections thatdo not increase monotonically with t predictunstable equilibria (Echenique 2002) Thecomparative statics result is generalized inMilgrom and Shannon (1994)

An heuristic argument for the result is asfollows The largest best reply of player i isincreasing in t and from this it followsthat the largest equilibrium point (asdetermined by the largest best replies)also increases with t Indeed ndasha = supa isin A ndashψ(a t) ge a and ndashψ(a t) is increasingin t Obviously for an increase in the equi-librium to take place we need only forexample that the payoff to firm i be affect-ed by ti and not by any other tj An increasein t leaves the old equilibrium in A- (seefigure 3) and thus sets in motion viabest reply (or more generally via adap-tive dynamics) a monotone increasingsequence that converges to a larger equi-librium Increasing actions by one playerreinforce the desire of all other players toincrease their actions and the increasesare mutually reinforcing (ie they exhibitpositive feedback)

Another way to look at the feedback loopis to think in terms of multiplier effects Asstated in section 3 a multiplier effect inthe parameter tj obtains if the equilibriumreaction of each player to a change in theparameter is strictly larger than the reac-tion of the player keeping the strategies ofthe other players constant This will hap-pen for example in a smooth strictlysupermodular game with one-dimensionalstrategy spaces for which part2πipartaiparttj ge 0with strict inequality for at least one firm ifeither considering extremal equilibria orfollowing best-reply adjustment dynamicsafter a parameter change In figure 4where there is a unique equilibrium theeffect of an increase in t1 is to move out-ward the best reply of player 1 If player 2were to stay put at a0

2 then the best



19 Martin Peitz (2000) gives sufficient conditions for aprice game to display multiplier effects

Figure 4 Effect of an Increase in Parameter t1

a2

a1acirc1 a1

r1 r2

a20

response of player 1 would be acirc1 but inequilibrium a1 gt acirc1

19

In games with strategic complementari-ties unambiguous monotone comparativestatics obtain if we concentrate on stableequilibria This is a multidimensional glob-al version of Samuelsonrsquos (1979) correspon-dence principle which links unambiguouscomparative statics with stable equilibriaand is obtained with standard calculusmethods applied to interior and stable one-dimensional models

As an example consider the (supermodu-lar or log-supermodular) Bertrand oligop-oly There extremal equilibrium pricevectors are increasing in an excise tax tIndeed we have that πi = (pi minus t minus ci)Di(p)and part2πipartpipartt = minuspartDipartpi gt 0

Result 6 Duopoly with strategic sub-stitutability (Vives 1990a) If n = 2 andthere is (a) strategic complementarity inown strategies with πi supermodular in ai

or part2πipartaihpartaik ge 0 for all k ne h and (b)strategic substitutability in rivalsrsquo strategieswith πi with decreasing differences in (aiaj)or part2πipartaihpartajk le 0 for all j ne i and for all hand k then the transformed game with newstrategies s1 = a1 and s2 = minusa2 is smoothsupermodular (See figure 5 and note thatthis is the mirror image of figure 3 withrespect to the ordinate axis) Therefore allthe results stated previously apply to thisduopoly game as well Unfortunately thetrick does not work for n gt 2 and the exten-sion to the strategic substitutability case forn players does not hold

A typical example of duopoly with strate-gic substitutability is a Cournot marketwhere usually best replies are decreasing Inthis case the welfare result is as follows If



Figure 5 A Duopoly Game with Decreasing Best Replies

a2

r1()

r2()

a1

20 See however the modified Hotelling game in Thisseand Vives (1992) where best responses may be discontinuousbut are increasing

for some players payoffs are increasing in thestrategies of rivals and for other players theyare decreasing then the largest equilibriumis best for the former and worst for the lat-ter This is the case in the Cournot duopolywith the strategy transformation yielding asupermodular game The preferred equilib-rium for a firm is the one in which its outputis largest and the output of the rival lowest

42 The Scope of the Theory Is Anything aGSC

If not everything is a game of strategiccomplementarities where are the bounds ofthe theory

First of all if we take the view that the orderof the strategy spaces is part of the descriptionof the game or that there is a ldquonaturalrdquo orderin the strategy spaces then there are manygames that are not of strategic complementar-ities For example not all Bertrand gameswith product differentiation are supermodulargames Roberts and Sonnenschein (1977)James Friedman (1983) and Vives (1999 sec-tion 62) provide examples including games

with avoidable fixed costs and the classicalHotelling model where firms are located closeto each other In those cases at some pointbest replies may jump down and a price equi-librium (in pure strategies) may fail to exist20

Indeed with goods that are gross substitutesprices may be strategic substitutes becausethe own-price elasticity of demand need notdecrease in the prices charged by rivals Aprice increase by rival j may lead to an increasein the own-price elasticity of demand for firmi because it makes consumers of brand i whodo not have a strong preference for any prod-uctmdashthat is who are more price sensitivemore willing to switch brands It may then payfor firm i to cut the price to gain these con-sumers Steven Berry James Levinsohn andAriel Pakes (1999) find some empirical sup-port for this in certain markets Anotherinstance of strategic price substitutabilityamong prices may come from the presence of



21 In a GSC as defined in this section there may be noequilibrium in pure strategies because strategy sets neednot be complete lattices

strong network externalities For example inthe logit model with network externalities(Simon Anderson Andre de Palma andJacques Franccedilois Thisse 1992 chapter 7)increasing the price set by a rival raises thevalue for consumers of the network of firm iso it may pay this firm to cut prices in order toenlarge this lead (although this will not hap-pen for small network externalities) To put itanother way in many games best responsesare simply nonmonotone For example theyare increasing in some portion of the strategyspace and decreasing in another

However we could also take the view thatthe order of the strategy sets of the players isa modeling choice at the convenience of theresearcher This is what we have done toextend the reach of the theory to duopolieswith strategic substitutes Then if we allowalso the construction of this order ex postwith knowledge of the equilibria of thegame then the answer to the question of thebounds of the theory is that most games areof strategic complementarities This meansthat complementarities alone in the weaksense stated do not have much predictivepower unless coupled with additional struc-ture (Echenique 2004a) Indeed define agame with strategic complementarities asone in which there is a partial order onstrategies (that can be chosen by the model-er) so that best responses are monotoneincreasing (and with strategy sets having alattice structure)21 Then (i) a game with aunique pure strategy equilibrium is a GSC ifand only if Cournot best-response dynamics(with unique or finite-valued best replies)have no cycles except for the equilibriumand (ii) a game with multiple pure strategyequilibria is always a GSC As a corollary (iii)2 times 2 games generically are either GSC orhave no pure strategy equilibria (like match-ing pennies) Result (i) in particular meansthat a game with a unique and globally stable 22 If a correspondence φ from X to X has two fixed

points a and b then define an order ge on X as follows Lety ge x if and only if any of the following is true x = y x = aor y = b Then φ is actually weakly increasing in the sensethat if y ge x then there is a z in φ and a z in φ(y) with z ge z (In fact (X ge ) is a complete lattice see appendix)

equilibrium is a GSC according to the defi-nition given An example is the strategic sub-stitutes case in the continuum of agentsmodel of section 2 when r gt minus 1 Note thatin this case the game is dominance solvable

Result (ii) is shown by taking one equilib-rium to be the largest and another thesmallest strategy profiles in a way that bestresponses are increasing22 Indeed a gamewith multiple equilibria always involves acoordination problem (ie coordinating onone equilibrium) We can then find an orderon strategies that makes the game one ofstrategic complementarities However notethat this is done with a priori knowledge ofthe equilibria and the defined orderindeed it may not be ldquonaturalrdquo at all

5 Oligopoly and Comparative Statics

This section reviews some of the basicapplications to oligopoly surveys very recentones and provides some new ones It devel-ops comparative statics results in Cournotmarkets (including entry) patent races andmultidimensional competition

The analysis illustrates several points thepotential pitfalls of classical analysis theextension of the methods to games that neednot display complementarities globally andthe isolation of the crucial assumptions driv-ing the results As an example of the firstissue classical analysismdashwhen studying theeffects of increasing the number of firms ninto a Cournot marketmdashignores that someequilibria may disappear (or appear) whenchanging n making any ldquolocalrdquo study mean-ingless (see Rabah Amir and Val E Lambson2000) An analysis of a multimarket oligopolycoming from two-sided competition willexemplify the second issue Using lattice-the-oretic methods conditions for ldquoperverserdquocomparative statics will be derived in a



context where the underlying game is notsupermodular (Luis Cabral and Lluis Villas-Boas 2004) Finally an examination of a patentrace will isolate the crucial assumptionsbehind the comparative statics of RampD effortwith respect to the number of participants inthe race (Vives 1999) We deal in turn withcomparative statics in Cournot markets patentraces and multidimensional competition

51 Comparative Statics in Cournot Markets

The standard Cournot game displaysstrategic substitutability therefore thegame is supermodular only in the duopolycase (by changing the sign of the strategyspace of one player) as discussed in result 6However the lattice-theoretic approachdelivers results also with n firms I considerhere a symmetric market (see Amir 1996aand Vives 1999 for other results)

Consider a n-firm symmetric Cournotmarket in which the profit function of firm iis given by

πi = P(Q)qi minus C(qi)

Here P is the smooth inverse demandwith P lt 0Q is total output C is the costfunction of the firm and qi its output levelWe parameterize the cost function of firm iby and let C(qi) be smooth withpart2Cpartpartqi le 0

In the standard approach (Jesuacutes K Seade1980a 1980b Avinash K Dixit 1986) it isassumed that payoffs are quasi-concave andconditions

(n + 1)P(nq) + nP(nq)q lt 0 and C(q) minus P(nq) gt 0

are imposed so that there is a unique andlocally stable symmetric equilibrium qlowastThen standard calculus techniques showthat an increase in increases qlowast and thattotal output increases and profits per firmdecrease as n increases The comparativestatics of output per firm with respect to thenumber of firms are ambiguous

The classical approach has several prob-lems First of all it is silent about the

23 See Vives (1999 pp 42ndash43 93ndash96 section 431) fordetails

potential existence of asymmetric equilib-ria Second it is restrictive and may be mis-leading For example if the uniquenesscondition for symmetric equilibria does nothold and there are multiple symmetricequilibria changing n may either cause theequilibrium considered to disappear orintroduce more equilibria (as in figure 1)

In the lattice-theoretic approach (Amirand Lambson 2000 Vives 1999) it isassumed only that P lt 0 and C minus P gt 0Then it can be shown that a symmetricequilibrium (and no asymmetric equilibri-um) exists23 Furthermore at extremalCournot equilibria (or following out-of-equilibrium best-reply dynamics) individ-ual outputs are increasing in total outputis increasing in n and profits per firmdecrease with n Furthermore it can beshown that individual outputs decrease(increase) with n if demand is log-concave(log-convex and costs are zero) Thisapproach does away with the unnecessaryassumptions of the standard approach andderives new results

The approach here also delivers the con-ditions (in a differentiated product environ-ment) for Bertrand prices to be lower thanCournot prices as a corollary of the fact thatCournot prices must lie in region A+(figure3) when actions are prices (Vives 1985b1990a see also Vives 1999 section 63)

52 Patent Races

Suppose that n firms are engaged in amemoryless patent race and have access tothe same RampD technology (Lee and Wilde1980) An innovating firm obtains the prizeV and losers obtain nothing If a firm spendsx continuously then the (instantaneous)probability of innovating is given by h(x)where h is a smooth concave function withh(0) = 0 and h gt 0 limxrarrinfinh(x) = 0 h(0) = infin(a region of increasing returns for small x



may be allowed) In the absence of innovat-ing the normalized profit of firms is zeroUnder these conditions the expected dis-counted profits (at rate r) of firm i investingx if rival j invests xj is given by

(see Lee and Wilde 1980 and JenniferReinganum 1989) Denote the bestresponse of a firm by xi = R(Σjih(xj) + r)this is well defined under the assump-tions Lee and Wilde (1980) restrict atten-tion to symmetric Nash equilibria of thegame and show that under a stability condition at a symmetric equilib-rium xlowast R((n minus 1)h(xlowast))(n minus 1)h(xlowast) lt 1 xlowast

increases with nHowever this approach suffers from the

same problems as the comparative statics ofentry in Cournot markets It requiresassumptions to ensure a unique and stablesymmetric equilibrium and cannot rule outthe existence of asymmetric equilibriaNonetheless the following mild assump-tions ensure that the game is strictly log-supermodular h(0) = 0 and h is strictlyincreasing in [0ndashx] with h(x)V minus x lt 0 for x ge ndashx gt 0 It follows then from result 2 thatequilibria exist and all are symmetric

Let xi = x and xj = y for j ne i Then logπi

has (strictly) increasing differences in (xn)for all y(y gt 0) and at extremal equilibriathe expenditure intensity xlowast is increasing inn Furthermore if h is smooth with h gt 0and h(0) = infin then partlogπi partxi is strictlyincreasing in n and (at extremal equilibria)xlowast is strictly increasing in n This followsbecause under our assumptions equilibriaare interior and must fulfill the first-orderconditions

As before starting at any equilibrium anincrease in n will raise the research intensitywith out-of-equilibrium adjustment accord-ing to best-reply dynamics This will be soeven if some equilibria disappear or newones appear as a result of increasing n

π ii i

i j i j

h x V xh x h x r

=( ) minus

( ) + ( ) +ne

53 Multidimensional Competition

Multidimensional competition providesanother fertile ground for application of theapproach which can readily handle multidi-mensional strategy spaces We will considerfirst an example with advertising and priceas strategies and then examine multimarketoligopoly situations

531 Advertising and Prices

Consider our Bertrand oligopoly examplewhere the demand Di(p ti) of firm i dependson advertising effort ti with partDipartti gt 0Suppose that goods are gross substitutespartDipartpj ge 0 for j ne i and that demand isdownward sloping partDipartpi lt 0 Let

πi = (pi minus ci)Di(p ti) minus Fi(ti)

here Fi is the cost of advertising with Fi gt 0The action of the firm is ai = (pi ti) with nat-ural upper bounds for pi and ti Profits πi arestrictly supermodular in ai = (pi t) if

A sufficient condition for the condition tohold is that part2Dipartpipartti ge 0 This amounts torequiring that advertising increases the cus-tomers willingness to pay Furthermore πi

has increasing differences in ((pi ti)(pi ti))if part2Dipartpipartpj ge 0 for j ne i (given thatpartDipartpiparttj = 0 j ne i) Under these assump-tions the game is supermodular and thelargest (smallest) equilibrium has the featureof having high (low) prices and advertisinglevels Multiple equilibria obtain with a sym-metric linear demand system where ti

increases the demand intercept if F is con-cave enough We have thus found conditionsunder which high prices are associated withhigh advertising levels

532 Multimarket Oligopoly

The previous example can be extendedreadily to multiproduct firms and even to

partpart part

= minus( ) partpart part

+partpart

gt2 2

0π i

i ii i

i

i i

i

ip tp c

Dp t

Dt



24 See Mark Armstrong (2002) for a survey of two-sidedcompetition

pricing games which are neither supermod-ular nor log-supermodular For example inthe multiproduct logit model of Richard HSpady (1984) best responses are increasingand there is a unique Bertrand equilibriumdespite the fact that payoffs are single-peaked (not quasi-concave) and neithersupermodular or log-supermodular in ownactions or prices Even so strategic com-plementarity across prices of differentfirms holds

A multimarket mixed oligopoly featuringproducts demand complements within thefirm as well as substitutes across firms pro-vides another example This situation is typ-ical of two-sided markets where two groupsof market participants benefit from interac-tion via a platform or intermediaryIntermediaries compete for business fromboth groups and set prices Examples arenumerous and include readersviewers andadvertisers in media markets cardholdersconsumers and merchantsretailers in pay-ment systems such as credit cards con-sumers and shops in shopping malls authorsand readers in academic journals borrowersand depositors in banking ldquosubscription to anetworkrdquo and ldquonumber of calls made to anetworkrdquo in telecom markets and in generalbuyers and sellers put together with the helpof intermediaries (in real estate financialproducts or auction markets) The interac-tion between the two sides gives rise to com-plementarities or externalities betweengroups that are not internalized by endusers For example a consumer who uses acredit card does not internalize the benefitthat it confers to the other side of the market(the merchants)

Consider a situation of two-sided exclusiveintermediation with two groups of partici-pants (say columnists and readers datingbars workers and firms in a single regionconsumers and shops in a mall) where eachparticipant joins one of the two existingintermediaries and where the utility derivedby a member of a group from joining a par-ticular intermediary is increasing in the

number of members of the other group join-ing the same intermediary24 When lineardemands arise from Hotelling-type prefer-ences for the intermediaries the result isthat prices charged by intermediaries arestrategic complements across firms butstrategic substitutes within the firm Themultimarket oligopoly game is therefore nota supermodular game as defined in section4 However best replies will be increasing aslong as the demand complementarity amongthe products of the same firmintermediaryis not very strong In the context of the lineardemand game with small and symmetric net-work effects best replies are increasing andthere is a unique symmetric equilibrium

An interesting result in the Hotellinggame where total demand is inelastic is thatan increase in the cross-group network effect(ie an increase in the degree of demandcomplementarity of the ldquoproductsrdquo of theintermediary) reduces equilibrium profitsAn increase in the impact of the benefits thatone side of the market confers on the otherwhen they go to an intermediary has in facta detrimental equilibrium effect on profitsThe reason is that the externality increasehas no positive direct impact on demand at asymmetric equilibrium in which the wholemarket is covered and it motivates eachintermediary to cut prices Since totaldemand stays constant (because it is priceinelastic) equilibrium profits decrease Thisresult can be generalized whenever (a) thedirect effect of the externality on demand atsymmetric equilibria is small so that profitsfor any intermediary have decreasing differ-ences in the price charged to a group (and inconsequence the externality parameter andbest replies shift inward as the externalityparameter increases) and (b) total demand isfixed (or it is quite price inelastic) so that theequilibrium price decrease translates into aprofit decrease In those circumstances thestrategic pricing effect dominates the direct



25 Peitz (forthcoming) uses supermodular methods tostudy the effects of asymmetric access price regulation intelecom markets

effect (Cabral and Villas-Boas 2004)Economies of scope have a similar effectthan demand externalities25

6 Dynamic Games

This section will take a look at dynamicissues building on comparative statics resultsfor supermodular games (like result 5) thatpredict movements of equilibrium variableswhen a parameter changes

I examine first the conditions under whichincreasing or decreasing dominance occursin oligopolymdashthat is whether leaders or lag-gards have more incentives to invest This isparticularly relevant in situations whereinvestment today which could be a largerfirm size if there are learning effects andoradjustment costs affects competitive condi-tions tomorrow This application will illus-trate the power of the approach to isolate thedrivers of results and extend them beyondGSC (Susan Athey and Armin Schmutzler2001) I deal afterwards with full-blowndynamic Markov games and Markov perfectequilibria (MPE) First I tackle how staticcomplementarities translate into dynamiccomplementarities and use the methodologyto characterize MPE Conditions are giventhat enable contemporaneous (intraperiod)strategic complementarity (SC) andintertemporal (interperiod) SC to obtainThe relationships between static and dynam-ic strategic substitutability and complemen-tarity are studied in alternating move games(Eric Maskin and Jean Tirole 1987 1988ab)and in games with adjustment costs (Jun andVives 2004) Finally the problem of exis-tence and characterization of Markov per-fect equilibria is addressed (Laurent OCurtat 1996 Sleet 2001)

The outcome of the analysis are newresults uncovered (characterization ofdynamic strategic complementarity linkage

between static and dynamic complementari-ty concepts existence of MPE) and isolationof crucial assumptions in known results(increasing dominance monotonicity ofdynamic reaction functions in alternatingmove games)

61 Increasing or Decreasing Dominance

Suppose that the payoff to player i is givenby πi(aiai t) with t = (t1hellip tn) The param-eter ti is to be interpreted as the state vari-able or initial conditions of player i in thegame Let both actions and state variables beone-dimensional We would like to find con-ditions under which if two firms differ onlyin their state variables if and ti gt tj then atany equilibrium we have ai(t) ge aj(t) withthe interpretation being that an initial domi-nance is reinforced by actions of the firmsFor example in the presence of a learningcurve the firm that has accumulated moreoutput has incentive to produce more

Suppose that πi(aiai t) has increasingdifferences in (aiaj) for i ne j (strategic com-plementarity) and increasing differences in(ai(ti minusti)) Assume also that all the playershave the same strategy set and that the pay-offs are exchangeable (players do not careabout the identity of their opponents onlyabout their actions and payoff-relevantparameters or state variables) This meansthat the payoffs of two players are the sameif actions and state variables are exchangedamong them Suppose also that payoffs arestrictly quasi-concave so there is a uniquebest-response function for any player andthat we have an equilibrium for which(without loss of generality) a1 lt a2 with t1 gt t2 Fix the actions of the players n = 3hellip n at their equilibrium levelsBecause of strict quasi-concavity andexchangeability we can write the bestresponse of firm 1 as r(a2 t1 t2) and that offirm 2 as r(a1 t2 t1) Because of strategiccomplementarity and a1 lt a2 we have a1 = r(a2 t1 t2) ge r(a1 t2 t1) Since t1 gt t2 andsince πi(aiai t) has increasing differ-ences in (ai(ti minusti)) it follows that



Figure 6 Effects of Going from t1 = t2 to t1t2

a2

r2()

r1()

45deg

a1

r(a1 t1 t2) ge r(a1 t2 t1) = a2 contradictingthe supposition that a1 lt a2 We conclude as desired that a1 ge a2 if t1 gt t2 (See Atheyand Schmutzler 2001)26

To help the intuition just think of the casen = 2 in figure 6 starting from a symmetricequilibrium at t1 = t2 and increasing t1 Wesee how the best reply of firm 1 shifts out-ward while the best reply of player 2 shiftsinward and the equilibrium moves to aregion with a1 ge a2

An example is provided by the Bertrandoligopoly model via product differentiationwith learning by doing or alternatively withproduction adjustment costs or even withswitching costs With learning by doing theprofit function of firm i is

where ti is the accumulated output of the firmLetting ai = minuspi we have that part2πi partaipartti gt 0and part2πi partaiparttj lt 0 With production adjust-ment costs the profit function is

where ti is the price of the firm in the previous period and F is the increasing and convex production adjustment costwith F(0) = 0 Then part2πi partpipartti gt 0 andpart2πi partpiparttj lt 0 In both cases the firm start-ing with a higher output level (lower price)has an incentive to set lower prices in equi-librium However this does not mean thatthere is increasing dominance Even thoughin any period the larger firm sets a lowerprice it may well be that the price differ-ence between the firms disappears overtime In fact this is exactly what happens atthe MPE of an infinite-horizon version ofthe model (Jun and Vives 2004)

π i i i i ip c D p F D p D t= minus( ) ( ) minus ( ) minus ( )( )

π i i i ip c f t D p= minus minus ( )(( ) ( )

26 Without requiring quasi-concavity we could makethe same argument with the extremal best replies Theresult would then be true for extremal equilibria



27 This is so in the Avner Shaked and John Sutton(1982) model of vertical quality differentiation when themarket is covered However in the classical linearBertrand duopoly with product differentiation invest-ments in quality that raise the intercept of demand for theown product (Vives 1985a) or that increase the willingnessto pay by lowering the absolute value of the slope ofdemand |partDipartp|(Vives 1990b) are strategic complements

In the switching costs model (Alan WBeggs and Paul Klemperer 1992) firmscompete in prices and ti is the loyal customerbase of firm i In this case we have thatpart2πi partpipartti gt 0 because lowering prices ismore costly to a firm with a larger customerbase and part2πi partpiparttj lt 0 It then follows thata firm with a larger customer base will besofter in pricing This is to be interpreted asdecreasing dominance (and indeed theauthors show that at an MPE of the full-blown dynamic game initial asymmetries inmarket shares are eroded) However thereader is warned that in a dynamic gamefirms are forward looking and the continua-tion payoffs need not look like the static pay-offs Therefore the static dominance neednot translate in dominance in the dynamicgame We will show in the next section therelationships between static and dynamicproperties of payoffs

The result can be extended to the strategic substitutes case (where πi(aiai t)has decreasing differences in (aiaj) i ne j)with the restriction that minuspart2πi (partai)

2 gtpart2πi partaipartaj| i ne j (this implies that the profitfunction of any player is concave and that aduopoly game would have a unique equilib-rium) The conditions in the result for strate-gic substitutes are typically met when actionsare investments in cost reduction and also insome models of quality enhancement27

Then profits at the market stage as a functionof those investments display strategic substi-tutability both in Cournot and Bertrandmodels The result can also be used to showthat learning by doing in a Cournot marketleads to increasing dominance That is thefirm that is ahead of the learning curveremains ahead because it has incentives to

28 A similar example with product differentiation andnetwork demand externalities (Michael L Katz and CarlShapiro 1986) would have πi = (Pi(q) minus (c minus ƒ(ti))qi whereq is the vector of output levels of the firms and ti is the accumulated sales of product i

produce more Actions are current rates ofoutput and state variables are the inheritedaccumulated production of each firm Letthe profit function of firm i be given by

Here P is the inverse demand Q is totaloutput ƒ is the learning curve a differ-entiable and concave function of totalaccumulated output of the firm ti with ƒ gt0 and qi is its current output level Ifinverse demand is log-concave then bestreplies are downward sloping (strategicsubstitutes) Furthermore part2πi partqipartti =ƒ gt 0 and part2πi partqiparttj = 0 As a conse-quence ti gt tj implies that at the (unique)Cournot equilibrium qi(t) ge qj(t)

28

62 Markov Games

An important issue is how static comple-mentarities translate (or not) into dynamiccomplementarities In this section we willexplore the issue in the context of discretetime Markov games A Markov strategydepends only on (state) variables denoted ythat condense the direct effect of the past onthe current payoff Let the current payoff ofplayer i be πi(xy) where x is the currentaction profile vector and y is the state evolv-ing according to y = ƒ(xmacrymacr) where xmacr and ymacrare (respectively) the lagged action profilevector and the lagged state A Markov per-fect equilibrium (MPE) is a subgame-perfect equilibrium in Markov strategiesThat is an MPE is a set of strategies optimalfor any firm and for any state of systemgiven the strategies of rivals

What do we mean by dynamic strategiccomplementarity (SC) or dynamic strategicsubstitutability (SS) We can think of ldquocon-temporaneousrdquo SC when the value function atan MPE Vi(y) displays SC (Vi has increasingdifferences in (yiyndashi)) We can think of

π i i iP Q c f t q= ( ) minus minus ( )( )(



29 This section draws on Vives (2004)

ldquointertemporalrdquo SC when dynamic bestreplies or the policy function at an MPE aremonotone There is intertemporal SC (SS)when a player raising his state variable todayincreases (decreases) the state variable of hisrival tomorrow I will investigate these prop-erties for a class of simple dynamic Markovgames that admits two-stage games simulta-neous move games with adjustment costs andalternating moves games

The class of simple dynamic Markov gamesis defined as follows Consider the n-playergame in which the actions of player i in anyperiod lie in Ai a compact cube of Euclideanspace here πi(xy) is the current payoff forplayer i with y isin A the action profile in theprevious period (state variables) and x isin A thecurrent action profile This simple class ofgames encompasses two-stage games andinfinite-horizon games of simultaneous moveswith adjustment costs or of alternating movesIn a two-stage game y isin A is the action pro-file in the first stage x isin A the action profilein the second stage and πi(xy) the payoff forplayer i Consider now an infinite-horizondiscrete time game with discount factor 13With simultaneous moves and adjustmentcosts the payoff to player i is given by

where ui(x) is the current profit in the periodand Fi(x y) is the adjustment cost in goingfrom past actions (y) to current actions (x)Assume that Fi(x y) = 0 i = 1 2 that is whenactions are not changed there is no adjustmentcost With alternating moves in a duopoly x isthe action of the player moving now and y isthe action of the player who moved last period

We take in turn the issues of contempora-neous SC in two-stage games and intertem-poral SC or SS in infinite-horizon games Weend the section with some remarks on theexistence of MPE

621 Contemporaneous SC in Two-StageGames

The contemporaneous SC property obtains(a) if at the second stage for any actions y in

π i i ix y u x F x y ( ) = ( ) + ( )

30 However the result cannot be extended to the casewhere each pay off function πi(xy) fulfills the ordinalcomplementarity conditions (or single-crossing propertySCP) in any pair of variables Indeed it is easy to con-struct examples where each payoff fulfills the SCP for allpairs of variables while the property is not preserved inthe reduced form first-period payoffs (Echenique 2004b)Neither supermodularity nor increasing differences canbe weakened to the ordinal SCP even though the simul-taneous move (ldquoopen looprdquo) game would be an ordinalGSC and even though the second-period equilibrium ismonotone in first-period choices

the first stage payoffs πi(xy) display SC and(b) if the SC property is preserved when pay-offs are folded back at the first stage in asubgame-perfect equilibrium29

Suppose that πi(xy) displays increasingdifferences(or is supermodular) in any pairof variables Let Vi(y) equiv πi(x

lowast(y)y) wherexlowast(y) is an extremal equilibrium in the sec-ond stage Extremal equilibria exist at thesecond-stage for any y because the secondstage game is supermodular A particularcase is when contingent on y a uniqueNash equilibrium xlowast(y) obtains at the secondstage Vi(y) is thus the first-period reducedform payoff for player i I claim that Vi(y) issupermodular in y

The argument is simple We have

Note that xlowastj(y) increases in y because πi

has increasing differences in (xiy) It fol-lows that Vi(y) is supermodular in ybecause (i) πi is supermodular in all argu-ments (ii) xlowast

j(y) is increasing in y (iii)supermodularity is preserved by increasingtransformations of the variables and (iv)supermodularity is preserved under themaximization operation

The result can be readily generalized tofinite-horizon multistage games with observ-able actions (as defined eg by DrewFudenberg and Tirole 1991)where the payoffto each player displays increasing differencesin any two variables30

V y x y y x x y yi i x i i ii

( ) ( )( ) = ( )( )lowastminuslowast π π max



32 It is worth noting that with high enough spilloversfirmsrsquo RampD cost-reduction investments are SC in thetwo-stage game (Claude drsquoAspremont and AlexisJacquemin 1988)

An example of the result is provided bythe Bertrand oligopoly with advertisingUnder the assumptions made (section 531)

is supermodular in any pair of argumentsand the first-stage value function atextremal equilibria is supermodular Thatis advertising expenditures are strategiccomplements The assumptions are ful-filled in the classical linear differentiatedproduct Bertrand competition model withconstant marginal costs when either adver-tising or investment in product quality rais-es the demand intercept of the firmexerting the effort (Vives 1985a) or increas-es the willingness to pay for the product ofthe firm by lowering the absolute value ofthe slope of demand |partDi partpi| (Vives1990b) In this case for a given advertisingeffort there is a unique price equilibrium atthe second stage31

The result can be extended easily to aduopoly case in which for all i πi(x y) hasincreasing differences in (xi minusxj)(yi minusyj)and (xi(yi minusyj)) j ne i An example is provid-ed by a Cournot duopoly in which outputsare strategic substitutes and yi is the cost-reduction effort by firm i Let

with part2Ci partxipartyi le 0 Then the assumptionsare fulfilled because part2πi partxipartyi ge 0 andpart2πi partxipartyj = part2πi partzipartyj = 0 for j ne i Wethen have that cost reduction investmentsare strategic substitutes at the first stageWith linear demand there is a unique equi-librium at the second stage (see Vives 1990bfor a computed example where investmentreduces the slope of marginal costs and for a

π i i i i iP x x x C x y= ( ) minus ( )1 2

π i i i i i i ip c D p z F z= minus( ) ( ) minus ( )

reinterpretation in terms of firms that investin expanding their own market)32

622 Intertemporal StrategicComplementarity

Consider a stationary MPE of an infinite-horizon simultaneous move game with dis-count factor 13 and let Vi(y) be the valuefunction associated to player i at the MPEPlayer i solves

Let xlowast(y) be the (assumed unique) contem-poraneous Nash equilibrium given y andthe MPE policy functions for the playersFrom result 5 we have that for all i if

1 πi(xy) + 13Vi(x) has increasing differ-ences in (xixi) and

2 πi has increasing differences in (xiy)then xlowast(y) is increasing in y (ie we haveintertemporal SC xlowast

i increases with yj for j nei) In order for (1) to hold it is suffi-cient that both πi and Vi have increasing differences in (xixi)

Likewise we have the corresponding resultfor a duopoly with strategic substitutabilityFor all i if

1 πi(xy) + 13Vi(x) has increasing differ-ences in (xi minusxj) j ne i and

2 πi has increasing differences in (xi(yi minusyj))

then xlowasti increases in (yi minusyj) (ie we have

intertemporal SS xlowasti decreases with yj for j ne i)

The question is When will the assump-tions be fulfilled We will consider in turn theadjustment cost model and the alternatingmove duopoly

Simultaneous moves with adjustmentcosts Consider the adjustment cost modeland interpret actions as either prices orquantities Let production or price bear theconvex adjustment cost F Models with price

max x i i

i

x y V xπ δ( ) + ( )

31 However if firms invest in cost reduction the sec-ond-stage SC is transformed into a first-stage SS Thesame happens with product enhancement investments inthe Shaked and Sutton (1982) model of vertical qualitydifferentiation when the market is covered



adjustment costs or ldquomenu costsrdquo are com-monly used in macroeconomics It is easy tosee that with price competition (with staticSC) and menu costs the marginal profit forfirm i is increasing in the price yi charged bythe firm in the previous period and decreas-ing in the price yj charged by the rival in theprevious period This case falls in thedomain of the foregoing general result pro-vided that the value function V displays SC(this is true in the linear-quadratic specifica-tion) With quantity competition (static SS)and production adjustment costs the mar-ginal profit for firm i is increasing in theproduction yi of the firm in the previousperiod and decreasing in the production yj ofthe rival in the previous period This casefalls in the domain of the duopoly result withSS provided the value function displays SS(as in the linear-quadratic specification)

In these two cases static SC or SS is trans-formed into intertemporal SC or SSHowever this need not be always so Jun andVives (2004) have fully characterized the lin-ear and stable MPE in a symmetric differen-tiated duopoly model with quadratic payoffsand adjustment costs in a continuous timeinfinite-horizon differential game by build-ing on the work of Stanley Reynolds (1987)and Robert A Driskill and StephenMcCafferty (1989) Jun and Vives (2004)consider both SC (Bertrand) and SS(Cournot) competition with production orprice (menu) adjustment costs It is foundthat contemporaneous (dynamic) SC or SSare inherited from static SC or SS IndeedVi displays increasing (decreasing) differ-ences in (yiyj) when there is static SC (SS)Intertemporal SC or SS then obtainsdepending on what variable bears the adjust-ment cost We know already from the previ-ous paragraph that under price competitionwith menu costs (quantity competition withproduction adjustment costs) static SC (SS)is transformed into intertemporal SC (SS)

In contrast for the mixed case of pricecompetition with production adjustmentcosts Jun and Vives show that the static SC

is transformed into intertemporal SS Thenwe have that the marginal profit for firm i isincreasing in the price yi of the firm in theprevious period and decreasing in the priceof the rival yj in the previous period Thereasonmdashmuch as in the learning curvemodel with price competitionmdashis that a firmwants to make the rival small today in orderto induce it to price softly tomorrow Indeeda smaller rival will face a stiff cost of increas-ing its output A cut in price today will there-fore bring a price increase by the rivaltomorrow The result is that if production iscostly to adjust then intertemporal SSobtains whereas if price is costly to adjustthen intertemporal SC obtains

Having intertemporal SC or SS mattersbecause it governs strategic incentives atthe MPE with respect to innocent behaviorat the open-loop equilibrium Indeed Junand Vives show that with intertemporal SC(SS) steady-state prices at the MPE will beabove (below) the stationary open-loopequilibrium prices which coincide with thestatic equilibrium prices with no adjustmentcosts In fact this provides a generalizationof the taxonomy of strategic behavior intwo-stage games of Drew Fudenberg andTirole (1984) to the full-blown infinite-horizon game

Alternating move duopoly Consider aduopoly game in which the payoff to firmi(i = 1 2) is πi(a1a2) and the action setavailable to the firm is a compact intervalTwo players in a duopoly interact repeated-ly with player 1 moving in odd periods t = 1 3hellip and player 2 in even periods t = 0 2hellip (Richard M Cyert and MorrisH DeGroot 1970 Maskin and Tirole 19871988a 1988b) The action (eg price orquantity) of player i is fixed for one periodDenote by x the action of the player mov-ing now and by y the action of the playerwho moved last period The state variablefor firm i is therefore the action taken inthe previous period by firm j A (pure)Markov strategy for firm i is a function Rithat maps the past action of firm j into an



action for firm i This is truly a dynamicreaction function in contrast with the best-response functions derived in the staticgames considered in section 3 (in whichbest-response functions are useful in find-ing equilibria and characterizing stabilityproperties)

A Markov perfect equilibrium (MPE) is apair of dynamic reaction functions (R1 R2) such that for any state a firmmaximizes its present discounted profitsgiven the strategy of the rival The pair(R1 R2) is an MPE if and only if thereexist value functions (V1 V2 ) such that player 1 solves R1(y) isinarg maxx

π1(x y) + 13~V1(x) where

and similarly for player 2 That is given thestate variable y (the current action of firm2) for firm 1 V1(y) gives the present dis-counted profits when it is firm 1rsquos to moveand both firms use the dynamic reactionfunctions (R1 R2)

Suppose that MPE dynamic reaction func-tions exist Then according to our resultthey will be monotone increasing (decreas-ing) if the underlying one-shot simultaneousmove game is strictly supermodular (super-modular in (xminusy)) that is if πi(xy) hasstrictly increasing differences in (xy) (in(xminusy) Then any selection R1 from the setof maximizers of π1(xy) + 13

~V1(x) is increasing(decreasing) No other property is needed

Existence of an MPE can be establishedeasily with quadratic payoff functionsCournot with homogeneous products(Maskin and Tirole 1987) and Bertrand withdifferentiated products (Vives 1999 Ex912) It can be shown that for any 13 thereis a unique linear MPE that is symmetricand (globally) stable and that the steady-state action is increasing in 13 and equals thestatic Nash equilibrium when 13 = 0

The strategic incentives for a firm in theCournot case are to increase its output in

V y x y V xx1 1 1( ) = ( ) + ( )max ~π δ

~V x x R x V R x1 1 2 1 2( ) = ( )( ) + ( )( )[ ]π δ

33 See Amir (1996b) for another application

order to reduce the output of the rival and inthe Bertrand case (with product differentia-tion) to increase price in order to induce therival to be softer in pricing Thus in theCournot (Bertrand) case the static strategicsubstitutability (complementarity) translatesinto intertemporal strategic substitutability(complementarity) In other words in theCournot (Bertrand) case both static anddynamic reaction functions are downward(upward) sloping An increase in the weightfirms that put into the future (a larger 13)increases the strategic incentives with theresult of a higher output (price) in theCournot (Bertrand) market In any case theequilibrium action is larger than the staticequilibrium when 13 = 0

With homogenous products and pricecompetition dynamic reaction functions areno longer monotone This is so because witha homogeneous product the marginal profitof a firm is not monotone in the pricecharged by the rival For example if the rivalsets a (strictly) lower price than firm i thenfirm irsquos marginal profit of changing its price iszero and if the rival sets a (strictly) largerprice then firm irsquos marginal profit is positiveprovided that its price is below the monopolyprice However if the prices of both firms areequal the marginal profit is negative A con-sequence of this lack of monotonicity is thefinding of multiple equilibria (including equi-libria of the ldquokinked demand curverdquo type andprice cycles see Maskin and Tirole 1988b)

623 Existence of MPE

Until now we have not dealt with the exis-tence problem for MPE only with the char-acterization of equilibria Lattice-theoreticmethods can be used when there is enoughmonotonicity in the problem under study

Laurent Curtat (1996) shows existence ofMPE of stochastic games with complemen-tarities in discrete time and infinite hori-zon33 He considers multidimensional actionspaces and a multidimensional state evolving



according to a transition probability as afunction of the current state and action pro-file Payoffs are smooth and display per-peri-od complementarities and positiveexternalities or spillovers (the payoff to aplayer is increasing in the actions of rivalsand the state) the transition distributionfunction is smooth displays complementari-ties and is stochastically increasing in actionsand states Furthermore the payoff to aplayer as well as the transition distributionfunction fulfill a strict dominant diagonalcondition These strong assumptions allowus to collapse the multiperiod problem to areduced form static game (with continuationvalue functions increasing in the state vari-able) which is shown to be supermodular Anequilibrium can then be found with valuefunctions increasing in the state

Examples of games fulfilling the assump-tions are a dynamic version of the searchgame considered in section 3 (where theparameter evolves stochastically in a mon-otone increasing way with the average searcheffort of the population the higher the aver-age effort ~at in period t the higher the t + 1

in expected terms) and a dynamic version ofa Cournot oligopoly with complementaryproducts and learning by doing where ahigh level of accumulated output by onefirm yields stochastically higher levels ofcumulated experience and lower productioncosts to the firm (learning by doing) and tothe rivals (spillovers)

Another successful application of thetechniques used to prove the existence of anMPE is provided by Sleet (2001) Sleet con-siders a version of the adjustment cost modelof the previous section in an infinite-horizondiscrete game with a continuum of playersand symmetric payoffs Firms set prices andprices are costly to adjust The payoff to aplayer in any period is given by π(xyG) = R(xG) minus F(xy) where x isthe current price of the firm y the price inthe previous period G the distribution ofprices chosen by other firms a firm-specif-ic shock (iid across firms and dates) R the

34 See Olivier J Blanchard and Stanley Fischer (1989section 81) and Julio Rotemberg (1982)

net revenue function and F the adjustmentcost The payoff is increasing in G and hasincreasing differences in (x(G)) andminus F(xy) is supermodular Under some fur-ther technical restrictions the existence of asymmetric monotone MPE is shown inwhich each firm uses the same increasingMPE policy function this yields an action inthe current period that is contingent on lastperiodrsquos action last periodrsquos distribution ofactions and the playerrsquos specific shock This is done by showing the existence of afixed point of an increasing function thatmaps (increasing) policy functions ontothemselves The problem is simplifiedbecause with a continuum of firms no firmcan influence any aggregate and each firmfaces a dynamic programming problemFurthermore and as usual with the lattice-theoretic approach an algorithm to computethe largest or the smallest equilibrium policyfunctions can be provided

The model corresponds to a monopolisticcompetition model where no firm influencesthe market aggregates yet each retains somemarket power the demand or technologyfirms are subject to a period-specific shockand prices are subject to continuous adjust-ment costs For example the demand for theproduct of a firm may depend on the averageprice charged in the market or on a priceindex The assumptions are fulfilled with lin-ear or constant elasticity demands and quad-ratic or constant elasticity production costs(subject to a multiplicative shock) and withquadratic costs of price adjustment34

7 Bayesian Games

Bayesian games provide a fertile groundfor applications of the lattice-theoreticalapproach The reason is that it allows forgeneral strategy spaces and payoff functionsIn section 71 I present the setup of theBayesian game and basic approaches to thedifficult issue of existence of equilibrium in



pure strategiesmdashtogether with some exam-ples and applications to oligopoly teams andgames of voluntary disclosure Most recentadvances are based on the lattice-theoreticapproach be it with supermodular games(Vives 1990a) single-crossing properties(Athey 2001) or ldquomonotone supermodularrdquogames (Timothy Van Zandt and Vives 2004)The last two approaches deliver conditionsfor equilibria to be monotone in type adesirable property in auctions and globalgames for example In fact global gamestypically belong to the monotone supermod-ular class Section 72 deals with globalgames and applications to currency andbanking crisis It makes clear that domi-nance solvability in the standard global game(Morris and Shin 2002) obtains because theunderlying game is one of strategic comple-mentarities and the key to uniqueness isprecisely that the strength of the strategiccomplementarities is not too large

71 Bayesian Nash Equilibrium Existenceand Characterization

In a Bayesian game the type of a playerembodies all the decision-relevant privateinformation Let Ti be a subset of Euclideanspace and the set of possible types ti of play-er i The types of the players are drawn froma common prior distribution on

where T0 is residual uncertainty notobserved by any player The action space ofplayer i is a compact cube of Euclideanspace Ai and his payoff is given by the(measurable and bounded) function πi A times Trarr where

The (ex post) payoff to player i when thevector of actions is a = (a1hellipan) and whenthe realized types t = (t1hellip tn) is thus πi(a t)Action spaces payoff functions type setsand the prior distribution are commonknowledge The Bayesian game is then fullydescribed by (AiTiπi i isin N)

A Ai==prod i

n

1

T Ti==prod i

n

0

A (pure) strategy for player i is a (measur-able) function σiTirarrAi that assigns anaction to every possible type of the playerLet Σi denote the strategy space of player iand identify strategies σi and i if they areequal with probability 1 Let σ = (σ1hellipσn)Denote the expected payoff to player iwhen agent j uses strategy σj by Ui(σ) =Eπi(σ1(t1)hellipσn(tn) t)

A Bayesian Nash equilibrium is a Nashequilibrium of the game (ΣiUi iisinN) wherethe strategy space and payoff function of play-er i are denoted Σi and Ui respectivelyDenote by i ΣirarrΣi player irsquos best-replycorrespondence in terms of strategies Then aBayesian Nash equilibrium is a strategy profileσ such that σi isin i(σi) for i isin N We candefine a natural order in the strategy spaceΣiσi le σ i if σi(ti) le σ i(ti) in the usual compo-nentwise order with probability 1 on Ti

This formulation of a Bayesian game isgeneral and encompasses common and pri-vate values as well as perfect or imperfectsignals With ldquopurerdquo private values allowingfor correlated types we have πi(a ti) Forexample types are private cost parametersof firms A ldquocommon valuerdquo case is πi(a t) = vi(aΣiti) For instance there is acommon demand shock in an oligopoly andfirm i observes component ti only As anexample of imperfect signals suppose firmsobserve with noise their cost parametersThen t0 could represent the n-vector offirmsrsquo cost parameters and ti the private costestimate of firm i The various cost parame-ters as well as the error terms in the privatesignals may be correlated

711 Equilibrium Existence in PureStrategies

Existence of pure-strategy Bayesianequilibria in games with a continuum oftypes andor actions has proved to be a dif-ficult issue Typical sufficient conditions forexistence of pure-strategy Bayesian equi-libria include conditionally independenttypes finite action spaces and atomlessdistributions for types (see Roy Radner and



Richard Rosenthal 1982 and Milgrom andRobert Weber 1985)35 Under theseassumptions the authors show first theexistence of mixed strategy equilibria andthen obtain a purification result In orderfor this approach to work independence(or at least conditional independence) ofthe distribution of types is needed

The lattice-theoretic method has providedthree types of results

1 for supermodular games with generalaction and type spaces (Vives 1990a)

2 for games in which each player uses astrategy increasing in type in responseto increasing strategies of rivals (Athey2001) and

3 for ldquomonotonerdquo supermodular gameswith general action and type spaces(Van Zandt and Vives 2004)

Supermodular games In the firstapproach (Vives 1990a and Vives 1999 sec-tion 273) existence of pure-strategyBayesian equilibria follows from supermod-ularity of the underlying family of gamesdefined with the ex post payoffs for givenrealizations of the types of the players A keyobservation is that supermodularity of thisunderlying family of games is inherited bythe Bayesian game

Let πi be supermodular in ai and haveincreasing differences in (ai ai) ThenUi() is supermodular in i and has increas-ing differences in (ii) because super-modularity and increasing differences arepreserved by integration Furthermorestrategy spaces in the Bayesian game Σi canbe shown to have the appropriate orderstructure (ie they are complete lattices)Then the game (ΣiUi i isin N) is a GSC andfor all iisin Σ ii(i) contains extremalelements i(i) and

i(i) Existence of

extremal pure strategy Bayesian equilibriathen follows from the general versions of

the results in section 2 (see also Vives1990a 1999 section 273) This existenceresult holds for multidimensional actionspaces and requires no distributionalrestrictions The driving assumption isstrategic complementarities

Applications of this approach can befound in oligopoly games and team theory(as we shall see below) Diamondrsquos (1982)search model natural resource explorationgames with private information (seeKenneth Hendricks and Dan Kovenock(1989) and Milgrom and Roberts 1990a)and global games (see section 73)

Single-crossing properties In the sec-ond approach (Athey 2001) conditions areimposed so that an equilibrium in mono-tone increasing strategies (in types) can befound Suppose that both action Ai andtypes sets Ti for any player i are compactsubsets of the real line and that types havea joint density that is bounded atomlessand log-supermodular (ie types are affili-ated) Suppose also that πi(a t) is continu-ous and supermodular in ai and hasincreasing differences in (aiai) and (ai t)or alternatively that πi(a t) is nonnegativeand log-supermodular in (a t) Then theBayesian game has a pure-strategy equilib-rium in increasing strategies Note that inthe first case the first approach outlinedalready delivers existence of a pure-strategyequilibrium

The proof of these results relies on thestandard Kakutani fixed point theoremwhich relies on convex-valued correspon-dences It turns out that with discrete actionspaces and under the prevailing assump-tions best-response correspondences areconvex valued A key step in the proof is toshow that under our assumptions if therivals of player i use increasing strategiesthen the payoff to player i is log-supermodu-lar or has increasing differences (or in gen-eral fulfills an appropriate single-crossingproperty) in action and type This ensuresthat a player uses a strategy that is increasingin his type as a best response to increasing

35 M Ali Khan and Yeneng Sun (1995) show existenceof pure-strategy equilibria when types are independentpayoffs continuous and action sets countable



38 The assumptions on type and action spaces can beweakened considerably (out of the realm of Euclideanspaces) and there is no need to assume a common prior(see Van Zandt and Vives 2004)

37 The result is shown using an intermediate resultallowing multidimensional types and multidimensional(Euclidean) action spaces that puts assumptions on non-primitives The conditions are atomless types and interim(conditional on type) expected payoff of each player(quasi)-supermodular in his action and with single-cross-ing in own action and type given that other players usestrategies increasing in types

strategies of rivals36 The existence result fordiscrete action spaces can then be used toshow existence with a continuum of actionsvia a purification argument

An example of the result is our differenti-ated Bertrand oligopoly in which firm i hasrandom marginal cost ti Then it is immedi-ate that E(Di(pipi(ti)|ti) is log-supermod-ular in (pi ti) if both Di(pipi) and the jointdensity of (t1hellip tn) are log-supermodularand if the strategies of rivals pj j ne i areincreasing in types It follows that

E(πi|ti) = (pi minus ti)E(Di(pipi(ti)|ti)

is log-supermodular in (pi ti) and that thebest-reply map of player i is increasing in ti

The approach can be used also in gamesthat are not of SC and with discontinuous pay-offs For example in auctions the existence ofmonotone equilibria in pure strategies can beshown for

bull first-price auctions with heterogeneous(weakly) risk-averse bidders character-ized by private affiliated values or com-mon value and conditionally independentsignals (Athey 2001) and

bull uniform price auctions featuring multiu-nit demand with nonprivate values andindependent types (David McAdams2003)37

Monotone supermodular gamesCombining both approaches Van Zandtand Vives (2004) show a stronger result forldquomonotonerdquo supermodular games with

multidimensional action spaces and typespaces Let (Ti) be the set of probabilitydistributions on Ti and let player irsquos pos-teriors be given by the (measurable) func-tion pi Tirarr(Ti) consistent with theprior where pi(middot|ti) isin (Ti) denotes irsquosposteriors on Ti conditional on ti

38 Amonotone supermodular game is definedby the following properties1 Supermodularity and complementarity

between action and type πi supermodu-lar in ai and with increasing differencesin (aiai) and in (ai t)

2 Monotone posteriors pi Tirarr(Ti)increasing with respect to the partialorder on (Ti) of first-order stochasticdominance (a sufficient but not necessarycondition is that be affiliated)

Under these conditions there is a largestand a smallest Bayesian equilibrium and eachone is in monotone strategies There mightbe other equilibria that are in nonmonotonestrategies but if so they will be ldquosand-wichedrdquo between the largest and the smallestone which are monotone in type Theassumptions on action and type spaces can beconsiderably weakened but the result cannotbe extended to log-supermodular payoffs

The argument for the result is powerfulyet simple First the Bayesian game is ofstrategic complementarities as in the firstapproach to existence This means that theextremal best-reply maps are well-definedfor each player and are increasing in thestrategies of rivals Second the extremal bestreplies to monotone (in type) strategies aremonotone (in type) This follows because πi

is supermodular in ai and has increasing dif-ferences in (ai(ai t)) and because posteri-ors are monotone The result is that a highertype for i chooses a higher action because ashift in beliefs (the posterior pi is increasingand higher types believe that other players

36 The result follows directly from the assumptions bynoting that (a) if ƒnrarr is supermodular (or log-super-modular) then so is the function ƒ(h1(x1)hellip hn(xn)) provid-ed the hi functions are increasing and (b) that if g(x t) issupermodular (log-supermodular) in (x t) then Eg(x t)|tiis supermodular (log-supermodular) in (x ti) provided that(the random vector) t is affiliated See Vives (1999 pp 69229ndash30) and Athey (2001) for details



are more likely to be of higher types as well)and also because the induced expected pay-off has increasing differences in (ai ti)

39

Third if the largest best-reply map i(i) isincreasing the largest best reply to mono-tone strategies is monotone and payoffs arecontinuous then there is a largest equilibri-um and it is in monotone strategies This fol-lows by starting a Cournot tacirctonnement withstrategies for each player i equal for anytype to the largest element in the action setAi (it exists because Ai is a cube in Euclideanspace) Then the Cournot tacirctonnementdefines a decreasing sequence of monotonestrategies its limit must be an equilibriumowing to the continuity of payoffs and thelimit is also in monotone strategiesFurthermore it is easy to see that the limitmust be the largest equilibrium

712 Monotone Supermodular GamesApplications

Monotone supermodular games fit a vari-ety of problems Van Zandt and Vives (2004)present an application to the discrete setupof an adoption game on a graph with localnetwork effects Section 72 provides anapplication to global games I provide hereillustrative examples of multimarket oligop-oly and a team problem as well as a com-parative statics result that can be applied toextending the results of games of voluntarydisclosure

Multimarket oligopoly Consider aBertrand multimarket oligopoly where firm i = 1hellip n produces potentially H varieties h = 1hellip H and has profits

where h is a random demand shock for

π θih

H

ih ih ih i i hp c D p p= sum minus( ) ( )= minus

1

market h Let the type of firm i be ti = (cisi)where si is a multidimensional signal aboutthe random vector and ci- = (ci1hellipciH) (Inthe notation used the vector is part of T0) The payoff ui is supermodular in theprices and has increasing differences in pi

and (ci) if for example Dih is linear andincreasing in h and if all the goods are grosssubstitutes (both across markets and acrossbrands) For instance if and (cisi)iN areaffiliated then the increasing posteriors condition is satisfied Extremal equilibriawill then be monotone so prices at extremalequilibria will increase in cost and demandsignals

Teams Consider a team problem(Radner 1962) in which the common func-tion to be optimized is supermodular thereare increasing differences between actionsand types and the distribution of typesyields monotone posteriors Each memberof the team chooses a decision rule or strat-egy that is contingent on his private informa-tion (type) in order to maximize the commonobjective We know that the team optimumwill be a Bayesian equilibrium of the gameamong team members (Radner 1962)Suppose there is a unique equilibrium Wethen conclude that there is a team optimumand at the optimum players use decisionrules that are monotone in type For exam-ple in a multidivisional firm in which thetotal profit of the firm has been internalizedby the divisionrsquos managers aj could be thevector of actions or ldquoeffortsrdquo under the con-trol of manager j and sj his private informa-tion relating to cost and demand conditionsfor division j

Comparative statics and strategicinformation revelation Monotone super-modular Bayesian games have a useful com-parative statics property Extremal equilibriaare increasing in posteriors A consequenceis that if payoffs display positive externalities(πi is increasing in ai) then increasing pos-teriors increases the equilibrium expectedpayoffs Therefore in a game with positiveexternalities the expected payoff of each

39 Let Vi(ai tiPi)equivTiπi(aii(ti) ti ti)dPi(ti) Then

Vi has increasing differences in (ai ti) and in (ai Pi)because πi(aii(ti) ti ti) has increasing differences in(ai t) (since πi has increasing differences in (ai(ai t)) andi is increasing) and because of the monotone posteriorscondition Furthermore Vi is supermodular in ai becauseπi is supermodular in ai



40 With partuj partajh strictly increasing in tj and ti for all hand with partuj partajh strictly increasing in aih for all h

player in an extremal equilibrium is increas-ing in the posteriors of the other players(ordered by first-order stochastic domi-nance) This result can easily be strength-ened to ldquostrictly increasingrdquo under certainregularity assumptions (including somesmooth strict complementary conditions andrequiring πi to be strictly increasing in aj)

40

This comparative statics result has a readyapplication to games of voluntary disclosureOkuno-Fujiwara Postlewaite andSuzumura (1990) have provided conditionsunder which fully revealing equilibria obtainin duopoly games of voluntary disclosure ofinformation when information is verifiableThe conditions involve restrictive regularityassumptions such as one-dimensionalactions concavity of payoffs uniqueness andinteriority of equilibrium and independenttypes for the players The basic intuition ofthe result is that in equilibrium inferencesare skeptical if a player reports a set of typesothers believe the worst (ie others believethat the player is of the most unfavorabletype in the reported set) This unravels theinformation For example consider aCournot duopoly in which types are the(constant marginal) costs of firms which canbe high or low Then a firm reporting noth-ing (the full set) will be assumed to havehigh costs because if the firm had low costsit would have said so

Noting that the authors work in the con-text of a monotone supermodular game ourapproach dispenses with these unnecessaryregularity assumptions and highlights thecrucial ones those related precisely to the complementarity and monotonicityassumptions of a strict version of the mono-tone supermodular game that the marginalpayoff of an action of a player is strictlyincreasing in the actions of rivals and in thetypes of players The outcome is an exten-sion of the result to n-player GSC games orto a duopoly with strategic substitutability

multidimensional actions affiliated typesand possibly multiple noninterior equilibria(provided they are extremal) (See VanZandt and Vives 2004 for the details)

72 Global Games

Global games were introduced byCarlsson and van Damme (1993) as games ofincomplete information with types deter-mined by each player observing a noisy sig-nal of the underlying state The aim is toselect an equilibrium with a perturbation ofa complete information game The basicidea is that when analyzing a completeinformation game with potentially multipleequilibria players must entertain the ldquoglobalpicturerdquo of slightly different possible gamesbeing played Each player has a noisy esti-mate of the game being played and knowsthat the other players are also receiving noisyestimates

Carlsson and van Damme (1993) showthat in 2 times 2 games if each player observes anoisy signal of the true payoffs and if ex antefeasible payoffs include payoffs that makeeach action strictly dominant then as noisebecomes small an iterative strict dominanceselects one equilibrium The equilibriumselected is the John Harsanyi and ReinhardSelten (1988) risk-dominant one if there aretwo equilibria in the complete informationgame Carlsson and van Damme do notexplicitly consider supermodular games butin the interesting case of two equilibria in acomplete information game the game is oneof strategic complementarities

I will analyze a standard symmetric binaryaction global game with the tools of super-modular games provide some applicationsto currency and financial crises and con-clude with some robustness considerations

721 A Binary Action Game of StrategicComplementarities

Consider a version of the game with a con-tinuum of players in the simple frameworkof section 3 The action set of player i is Ai equiv 01 with ai = 1 interpreted as ldquoactingrdquo



Figure 7 Critical Fraction of Players Above Which It Pays to Act

0

1

h(θ)

θ θ

41 In the notation of section 71 we have t0 = and ti = + εi

and ai = 0 ldquonot actingrdquo (and let ai = 1 beldquolargerrdquo than ai = 0) To act may be to investadopt a technology or standard revolt attacka currency or run on a bank The fraction ofpeople acting is ~a and the state of the worldis There is a critical fraction of people h()above which it pays to act with h strictlyincreasing and crossing 0 at =

and 1 at

= See figure 7

Let π1 = π(ai = 1~a) and π0 = π(ai = 0~a)The differential payoff to acting is given by thefollowing chart

For any given state of the world thisdefines a supermodular game in whichπ1 minus π0 is increasing in ~a and minus or equiva-lently π(ai

~a) has increasing differences in(ai(

~aminus)) It is immediate that if le then it is a dominant strategy to act if ge then it is a dominant strategy not to act andfor isin( ) there are multiple equilibriaeither everyone acting or no one acting Weknow also according to result 5 in section 3

that extremal equilibrium strategies will bemonotone (decreasing) in Indeed thelargest equilibrium is ai = 1 for all i if le and ai = 0 for all i if gt and it is(weakly)decreasing in

Consider now the incomplete informationgame where players have a normal prior onthe state of the world ~N(microτ

1) and whereplayer i observes a private signal si = + εi

with normally distributed noise εi~N(0τε1)

iid across players41 Morris and Shin (2002)show that iterated elimination of dominatedstrategies then leads to a unique outcomeprovided that τ τε is small Thus we have aunique Bayesian equilibrium We show herehow the tools of supermodular games can beused to conclude that the game is dominancesolvable without actually having to gothrough the elaborate process of iteratedelimination of dominated strategiesFurthermore we shall see in a very transpar-ent way how the approach brings intuitionbehind the uniqueness result

~a ge h() ~a lt h()

π1π0 B gt 0 C lt 0



Note first that the game is monotonesupermodular since π(ai

~a) has increasingdifferences in (ai(

~aminus)) and since signalsare affiliated This means that extremal equi-libria exist are symmetric (because the gameis symmetric) and are in monotone (decreas-ing) strategies (according to the results insection 711) Because of binary action thestrategies must then be of the thresholdform ai = 1 if and only if si lt s Thereforethe extremal equilibrium thresholds ndashs and ndashsbound the set of rationalizable strategies

Now an equilibrium will be characterizedby two thresholds (slowastlowast) with slowast yielding theacting signal threshold and lowast the state-of-the-world threshold below which the actingmass is successful and an acting playerobtains the payoff B minus C gt 0 (the currencyfalls the bank fails or the revolt succeeds)The critical thresholds must fulfill twoequations

1

2

The first equation states that at the criticalstate of the world in equilibrium the frac-tion of acting players must equal the criticalfraction above which it pays to act The sec-ond equation states that at the critical signalthreshold the expected payoff of acting andnot acting is the same

Equations (1) and (2) may have multiplesolutions However it can be shown that if(and only if) ττε is small enough the solu-tion is unique in which case the equilibriumis unique and the game is dominance solvablebecause then ndashs = ndashs

If τ τε is not small enough then typical-ly there are three equilibria The main rea-son why the equilibrium is unique with smallnoise in the signals is that decreasing theamount of noise decreases the strength of

= le( ) + gt( ) minus( ) =lowast lowast lowast lowastPr Prθ θ θ θs B s C 0

E a a s sπ θ θ π θ θ1 0~ ~ ( ) )( minus ( ) )( = lowast

where γ +( ) ltC B C 1

Pr θ θ γle( ) =lowast lowasts

~ a s s s hθ θ θlowast lowast lowast lowast lowast( ) = le( ) = ( )Pr

42 We have that ( s) is the solution in of Pr(s le s|)( = (τε(s minus ))) = h() From this equation we can solve for the inverse function and obtain s() = + (1τε)1(h()) with derivative s = 1 + (1τε)h()[φ(1(h()))]1 where φ is the density of the stan-dard normal Since φ is bounded above by 12π it followsthat s is bounded below s() ge 1 + 2πτε h where

h = min[]h() gt 0 Hence ( s) le [1 + 2πτε h]1

(with strict inequality except when h() = 12 because then1(12) = 0 and φ attains its maximum φ() = 12π)

the strategic complementarity among theactions of the players Indeed multipleequilibria come about when the strategiccomplementarity is strong enough

It is instructive to sketch the proof ofuniqueness in order to bring forward theintuition Suppose that h is contin-uously differentiable with h gt 0 Let _h equiv min[]h() gt 0 Let P(s s) be theprobability that the acting players succeed if they use a threshold s and the playerreceives a signal s That is

Here (s) is the critical below whichthere is success when players use a strategywith threshold s and is the cumulativedistribution of the standard normal randomvariable N(01)

It is immediate that P is strictly decreasingin s partPparts lt 0 and nondecreasing in s partPpart s ge 0 Given that other players use astrategy with threshold s the best responseof a player is to use a strategy with threshold~s where P(~s s) = act if and only if P(~s s) gt or equivalently if and only if s lt ~s Thisdefines a best-response function

The game is of strategic complementaritiesand we have that r = minus(partPpart s)(partPparts) ge 0a higher threshold s by others induces aplayer to use also a higher thresholdFurthermore it is easily checked that(s) le [1 + 2πτεh]142 As a consequence

r s sˆ ˆ ˆ( ) =+ ( )minus minus + ( )minusτ ττ

θττ

micro τ ττ

γθ ε

ε

θ

εθ

θ ε

ε

1

sˆ ˆ+ ( ) minus+

τ τ θτ micro τ

θ εθ θ εε

θ ετ τs

+

P s s s s ˆ ˆ ˆ( ) lt ( ) = Pr θ θ



with equality only when h() = 12 Thisensures that r(s) crosses the 45deg line onlyonce and that the equilibrium is unique Ifh() = then -h = 1 and if radicε gt radic2πthen r( s) gt 1 for h() = = 12 Thus forexample for such that lowast = 12 there arethree equilibria (similarly as in figure 2 insection 32)43

With small noise the strategic comple-mentarity is lessened and r le 1 becausethen a player faces greater uncertainty aboutthe behavior of others Indeed consider thelimit cases εrarr + infin (or equivalently a dif-fuse prior = 0) Then it is not hard to seethat the distribution of the proportion of act-ing players ~a(slowast) is uniformly distributedover [01] conditional on si = slowast This meansthat players face maximal strategic uncer-tainty and cannot coordinate on differentequilibria44 In contrast with completeinformation there are multiple equilibriawhen isin(ndashndash) Indeed at any of the equilib-ria players face no strategic uncertainty Forexample in the equilibrium in which every-one acts a player has a point belief that allother players will act

The intuition for the uniqueness resultshould be familiar to the reader from oursimple framework with heterogeneousagents (section 32 and figure 2) There thecost of adoption (ai = 1) for player i is i andfollows a normal distribution with mean microvariance σ 2

and covariance with the adop-tion cost for j ne i j σ 2

with isin[01) Thebenefit of adoption is g(~a) where ~a is thetotal mass adopting and no adoption yieldsno benefit The payoff is therefore π (ai~ai) = ai(g(~a) minus i) Now if g gt 0 thenthe game is monotone supermodularbecause π (ai~ai) has increasing differences

ττ

πτ τ

τθθ

ε

θ ε

ε

le rArr ( ) =+ ( ) le2 1h r s s ˆ ˆ ˆ in (ai(~aminus)) and types are affiliated This

means that extremal equilibria exist aresymmetric (because the game is symmetric)and are in monotone (decreasing) strategiesof the form ai = 1 if and only if i le Thisprovides the rationale for concentrating onthreshold strategies (something that wasassumed in section 32) From section 32 weknow that the equilibrium will be unique if

where -g equiv supaisin[01]g(a) This may happenbecause of weak payoff complementarities (-g low) because of a diffuse prior ( low) orbecause the correlation of the costs is high (and the signal of each player is very precise)As before in all these situations the degreeof strategic complementarity is not toostrong and we are in the ldquoflatrdquo best-responsecase of figure 2

In summary by using the theory of super-modular games we bring the intuition for theuniqueness result clarify the role of theassumptions and obviate the necessity ofsolving for iterated elimination of dominatedstrategies We can start by noting that thegame is monotone supermodular Thismeans that extremal equilibria exist and arein monotone (threshold) strategies Thoseextremal equilibria can be found starting atextremal points of the strategy sets of players(-s = infin and -s = minusinfin) and iterating using bestresponses (Vives 1990a) We must make surethat the process is not stuck at extremalpoints of strategy space (eg the boundaryassumptions on h guarantee this since if 1 gt h() gt 0 or ndash = infin and ndash = minusinfin then bothto act and not act coexist as equilibria no mat-ter what signal is realized) The extremalequilibrium thresholds -s and -s bound the setof rationalizable strategies and if the equilib-rium is unique then the game is dominancesolvable The condition for equilibriumuniqueness is precisely that strategic comple-mentarities are not too strong and this holdswhen the signals are precise enough or if theprior is diffuse In this situation each player

11 2

1minus+

prime ltρρ

τπθ g

43 As ranges from 0 to 1 lowast goes from = 1 to = 044 It is worth noting that the uniqueness argument made

is robust to general distributions for the uncertainty as longas the noise in the signals is small Indeed with very precisesignals all priors ldquolook uniformrdquo (Morris and Shin 2002)



45 This follows immediately because (lowastslowast) solves thesystem (τε(s minus )) = h() and

from this it follows that lowast solves () equiv τ( minus ) minusτε

1(h()) minus τ + τε1() = 0 and lt 0 when ττε

is small enough Furthermore lowast and slowast move together

τ τ θτ micro τ

τ τθ εθ θ ε

θ ε

s+ minus

++

= γ

46 Michael Suk-Young Chwe (1998) provides evidenceof the per-viewer price of advertising in TV it is higher forbig sports events consistent with the multiplier effect ofpublic information

47 An early model with incomplete information thatobtains a unique Bayesian equilibrium with a positiveprobability of a crisis is Postlewaite and Vives (1987)

faces a lot of (strategic) uncertainty about theaggregate action of the other players

722 Applications

It is well known that multiple equilibriamake comparative statics and policy analysisdifficult The uniqueness of an equilibriumdelivered by the global game approachcomes to the rescue In the region where theequilibrium is unique we can obtain severaluseful results as follows

bull When lt lowast the acting mass of playerssucceeds In the range [lowastndash) there iscoordination failure from the point ofview of players because if all them wereto act then they would succeed

bull Both lowast and slowast (and the probability thatthe acting mass succeeds) is decreasingin the relative cost of failure equiv C(B + C) and in the expectedvalue of the state of the world micro

45

bull There is a multiplier effect of publicinformation An increase in micro will havean effect on the equilibrium thresholdslowast over and above the direct impact on the best response of a player partrpartmicro = minusε Indeed the prior mean of can be understood as a publicsignal of precision and exactly as inthe simple framework of section 31

whenever the uniqueness condition (r lt 1) ismet and the game is a GSC (r gt 0) The mul-tiplier is largest when r is close to 1mdashthat iswhen we approach the multiplicity of equilib-ria region The multiplier effect of publicinformation is emphasized by Morris andShin (2002) who interpret it in terms of thecoordinating potential of public information

dsd

rr

rlowast

=part part

minus primegt part

partmicromicro

microθ

θ

θ

1

beyond its strict information content Thereason is that public information becomescommon knowledge and affects the equilib-rium outcome Every player knows that anincrease in micro will shift downward the bestreplies of the rest of the players thereaftereveryone will be more cautious in actingThis phenomenon for example may bebehind the apparent overreaction of financialmarkets to Fed announcements46

The approach is useful for policy analysisbecause it links the probability of occurrenceof a ldquocrisisrdquo (successful mass action) at theunique equilibrium with the state of theworld Pr( le lowast) This is in contrast with thecomplete information model where multipleself-fulfilling equilibria arise in the range(ndash ndash ) Hence the theory builds a bridgebetween the self-fulfilling theory of crisis (egDouglas W Diamond and Dybvig 1983) andthe theory that links crisis to the fundamentals(eg Gary Gorton 1985 1988)47

The uniqueness property is nice in agame but we can still perform comparativestatics analysis in a GSC even if there aremultiple equilibria Suppose we are in themultiple equilibrium region and that micro

increases The comparative statics result thatthe critical thresholds lowast and slowast decrease stillholds for extremal equilibria or for reason-able out-of-equilibrium dynamics that elimi-nate the middle ldquounstablerdquo equilibriumIndeed we know that extremal equilibria ofmonotone supermodular games are increas-ing in the posteriors of the players A suffi-cient statistic for the posterior of a playerunder normality is E(|s) = micro + ε s + εwhich is increasing in micro It follows then thatextremal equilibrium thresholds (minuslowastminusslowast)increase with micro The out-of-equilibriumadjustment can take the form of best-reply



dynamics where at any stage after the per-turbation a new state of the world is drawnindependently and a player best-responds tothe strategy threshold used by other playersat the previous stage

We present here two applications thatillustrate the power of the approach Supposein all of them that the uniqueness conditionis fulfilled (ie radicε is small enough)48

The first application is a modified versionof the currency attacks model of Morris andShin (1998) This is a highly streamlinedmodel of currency attacks Let be thereserves of the central bank (with le 0meaning that reserves are exhausted)There is a continuum of speculators andspeculator i has one unit of resources toattack the currency (ai = 1) at a cost C afterreceiving a signal about the level ofresources of the central bank Let h() = and let the attack succeed if ~a ge The(capital) gain if there is a depreciation is B(and it is fixed) Let B = B minus C The result isthat the probability of a currency crisis isdecreasing in CB and in the expected valueof the reserves of the central bank In theregion [lowastndash ) if speculators were to coordi-nate their attack then they would succeedbut in fact the currency holds

The second application is an instance ofcoordination failure in the interbank marketproviding a rationale for a Lender of LastResort (LLR) intervention (Jean CharlesRochet and Vives 2004) Consider a marketwith three dates t = 0 1 2 At date t = 0the bank possesses own funds E and collectsuninsured wholesale deposits (CDs forexample) for some amount D0 equiv 1 Thesefunds are used to finance some investment Iin risky assets (loans) the rest being held incash reserves M Under normal circum-stances the returns I on these assets arecollected at date t = 2 the CDs are repaid attheir face value D and the stockholders ofthe bank get the difference (when it is posi-tive) However early withdrawals may occur

at an interim date t = 1 following the obser-vation of private signals on the future real-ization of If the proportion ~a of thesewithdrawals exceeds the cash reserves M ofthe bank then the bank is forced to sellsome of its assets A continuum of fund man-agers make investment decisions in theinterbank market At t = 1 each fund manag-er after receiving a private signal about decides whether to cancel (ai = 1) or renewhis CD (ai = 0) Let m equiv MD be the liquid-ity ratio ndash equiv D ndash MI the solvency thresh-old of the bank gt 0 the fire sales premiumof early sales of bank assets and ndash

equiv (1 +)ndash the ldquosupersolvencyrdquo point where a bankdoes not fail even if no fund manager renewshis CDs The bank fails if ~a ge h() where

for isin[ndash ndash ] and h() = 0 for le ndash A fund

manager is rewarded for making the rightdecision The equilibrium failure threshold ofthe bank is lowast isin [ndash ndash

] and in the range [ndashlowast)the bank is solvent but illiquid This providesa rationale for a LLR intervention with thediscount window Comparative statics resultsare also easily obtained The critical lowast (andprobability of failure) is a decreasing functionof the liquidity ratio m and the solvency (EI)of the bank of the critical withdrawal proba-bility and of the expected return on thebankrsquos assets micro it is an increasing function ofthe fire-sale premium and of the face valueof debt D

723 Robustness and Extensions

David M Frankel Morris and AdyPauzner (2003) obtain a generalization ofthe limit uniqueness result to games ofstrategic complementarities The authorsconsider a Bayesian game (AiTiπi) for i isinNwhere Ai is a compact interval and whereπi(aiai) is continuous and has increasingdifferences in (ai(ai)) The state isdrawn from a continuous density with con-nected support and player i receives a pri-vate signal si = + εi with gt 0 where εi is

h mmθ

λθθ

( ) + minus minus

11

48 See Morris and Shin (2002) for other applications



drawn from an atomless density with com-pact support (and the error terms are iidacross players) The authors also assumethat for extreme values of extreme actionsin Ai are strictly dominant (this is the equiv-alent of the assumption that h crosses 0and 1 at finite values) and make the techni-cal assumption that πi(a) has sensitivity toactions with a Lipschitz bound The result isthat if is uniformly distributed over a largeinterval or for tending to 0 there is a(essentially) unique Bayesian equilibrium inpure strategies (and it is increasing in type)Under the given assumptions we are in theframe of monotone supermodular games(section 71) and hence the existence ofextremal equilibria monotone in type isguaranteed

The framework can be extendedbull To include large players (see Giancarlo

Corsetti Amil Dasgupta Morris andShin 2004 on currency attacks as well asCorsetti Bernardo Guimaraes andNouriel Roubini 2003 and Morris andShin 2002 on the impact of the IMF asprovider of ldquocatalytic financerdquo)

bull To relax the strategic complementaritycondition of actions to a single-crossingcondition and obtain a uniquenessresult in switching strategies assumingthat signals fulfill the monotone likeli-hood ratio property However then itcannot be guaranteed that there are noother equilibria in nonmonotone strate-gies (see Athey 2001) Itay Goldsteinand Ady Pauzner (2003) apply a similarstrategy to model bank runs when thedepositorrsquos game is not one of strategiccomplementarities

bull To consider dynamic settings model-ingmdashfor example contagion (Dasgupta2003) and dynamic speculative attacks(Christophe Chamley 2003)

8 Concluding RemarksIn the paper I have surveyed the theory

and several applications of the lattice-theoret-ic approach in the study of complementarities

in games The survey has been by no meansexhaustive Indeed the method as has beenmade clear in the text can be applied fruit-fully to comparative statics analysis and so isuseful in practically all domains of economictheory For example it has been applied todemand analysis the theory of the firm andorganizations and dynamic optimizationproblems (see eg the applications inMilgrom and Roberts 1990b and Milgromand Shannon 1994) cooperative games (seeTopkis 1998 for a survey and Robert Shimerand Lones Smith 2000 for a model of assorta-tive matching with frictions) and evolution-ary games (see eg Carlos Aloacutes-Ferrer andAna Ania 2002) Despite these applications itis safe to say that the approach promises todeliver much more when the tools becomepart of the standard methods in economicsand when empirical analysis develops andinteracts with model building The empiricalanalysis of complementarities with implica-tions for the new methods is taking off in thestudy of innovation (Eugenio Miravete andJoseacute Perniacuteas 2004 and Pierre Mohnen andLars-Hendrik Roumlller forthcoming) and ofmarkets with potentially multiple equilibria(Andrew Sweeting 2004 on the timing ofradio commercials Federico Ciliberto andElie Tamer 2004 on airline markets)

The strength of the approach is its simplic-ity its capacity to generate new results andthe power it has to make results transparentThe challenges are multiple

bull continue pushing the frontier of the the-ory with a view toward applications indynamic games and games of incompleteinformation

bull incorporate the methodology fully inthe standard toolbox of economists and

bull develop the empirical analysis

9 Appendix Summary of Lattice-Theoretic Methods

For the convenience of the reader I includea few definitions and results of lattice meth-ods More complete treatments can be foundin Vives (1999 chapter 2) and Topkis (1998)



A binary relation ge on a nonempty set X isa partial order if ge is reflexive transitiveand antisymmetric An upper bound on asubset AX is z isinX such that z ge x for allx isinA A greatest element of A is an elementof A that is also an upper bound on ALower bounds and least elements aredefined analogously The greatest and leastelements of A when they exist are denotedmax A and min A respectively A supre-mum (resp infimum) of A is a least upperbound (resp greatest lower bound) it isdenoted supA (resp inf A)

A lattice is a partially ordered set (X ge )in which any two elements have a supre-mum and an infimum A lattice (X ge ) iscomplete if every nonempty subset has asupremum and an infimum A subset L ofthe lattice X is a sublattice of X if the supre-mum and infimum of any two elements of Lbelong also to L

Let (X ge ) and (T ge ) be partially orderedsets A function ƒ XrarrT is increasing if forx y in X x ge y implies that ƒ(x) ge ƒ(y)

A function g Xrarr on a lattice X is supermodular if all x y in X g(inf(xy)) +g(sup(x y)) ge g(x) + g(y) It is strictly supermodular if the inequality is strict forall pairs xy in X that cannot be comparedwith respect to ge (ie neither x ge y nor y ge x holds) A function f is (strictly) sub-modular if ndash f is (strictly) supermodular afunction f is (strictly) log-supermodular if fis (strictly) supermodular

Let X be a lattice and T a partiallyordered set The function g X times TrarrR has(strictly) increasing differences in (x t) ifg(x t) minus g(x t) is (strictly) increasing in t forx gt x or equivalently if g(x t) minus g(x t) is(strictly) increasing in x for t gt tDecreasing differences are defined analo-gously If X is a convex subset of n and if g XrarrR is twice-continuously differentiablethen g has increasing differences in (xixj) ifand only if part2g(x)partxipartxj ge 0 for all x and i ne j

Supermodularity is a stronger propertythan increasing differences if T is also a lat-tice and if g is (strictly) supermodular on

X times T then g has (strictly) increasing differ-ences in (x t) The two concepts coincide onthe product of linearly ordered sets if X issuch a lattice then a function g Xrarr issupermodular if and only if it has increasingdifferences in any pair of variables

The main comparative statics tool for ourpurposes is the following

Lemma 1 Let X be a compact lattice andlet T be a partially ordered set Let u X times Trarr be a function that (a) is super-modular and continuous on the lattice X foreach t isin T and (b) has increasing differencesin (x t) Let ϕ(t) = arg maxxXu(x t) Then

1 ϕ(t) is a nonempty compact sublatticefor all t

2 ϕ is increasing in the sense that for t gt tand for xisinϕ(t) and x isinϕ(t) we havesup(xx) isinϕ (t) and inf(xx) isinϕ (t)and

3 tmaxφ(t) and tminφ(t) are well-defined increasing functions

Remark If u has strictly increasing dif-ferences in (x t) then all selections of ϕ areincreasing

Remark If Xm solutions are interi-or and partupartxi is strictly increasing in t forsome i then all selections of ϕ are strictlyincreasing (Edlin and Chris Shannon1998)

The basic fixed point theorem in the lattice-theoretic approach is Tarski (1955)

Theorem 2 (Tarski 1955) Let A be acomplete lattice (eg a compact cube in m)Then an increasing function f ArarrA has alargest supa isinA f(a) ge a and a smallestinfa isinA a ge f(a) fixed point

Supermodular game The game (Aiπii isinN) is supermodular if for all i the following statements hold

bull Ai is a compact latticebull πi(aiai) is continuous

1 is supermodular in ai and2 has increasing differences in (aiai)

Game of strategic complementaritiesGiven a set of players N strategy spaces Ai and (nonempty) best-reply maps ψi i = 1hellipn we define a game of strategic



complementarities (GSC) as one in whichfor each i Ai is a complete lattice and ψi isincreasing and has well-defined extremalelements

REFERENCES

Aloacutes-Ferrer Carlos and Ana Ania 2002 ldquoTheEvolutionary Logic of Feeling Smallrdquo University ofVienna Working Paper No 0216

Amir Rabah 1996a ldquoContinuous Stochastic Games ofCapital Accumulation with Convex TransitionsrdquoGames and Economic Behavior 15(2) 111ndash31

Amir Rabah 1996b ldquoCournot Oligopoly and theTheory of Supermodular Gamesrdquo Games andEconomic Behavior 15(2) 132ndash48

Amir Rabah and Val E Lambson 2000 ldquoOn theEffects of Entry in Cournot Marketsrdquo Review ofEconomic Studies 67(2) 235ndash54

Anderson Simon Andre de Palma and JacquesFranccedilois Thisse 1992 Discrete Choice Theory ofProduct Differentiation Cambridge MIT Press

Armstrong Mark 2002 ldquoCompetition in Two-SidedMarketsrdquo Mimeo

Athey Susan 2001 ldquoSingle Crossing Properties andthe Existence of Pure Strategy Equilibria in Gamesof Incomplete Informationrdquo Econometrica 69(4)861ndash89

Athey Susan and Armin Schmutzler 2001ldquoInvestment and Market Dominancerdquo RANDJournal of Economics 32(1) 1ndash26

Beggs Alan W and Paul Klemperer 1992 ldquoMulti-period Competition with Switching CostsrdquoEconometrica 60(3) 651ndash66

Bernheim B Douglas 1984 ldquoRationalizable StrategicBehaviorrdquo Econometrica 52(4) 1007ndash28

Berry Steven James Levinsohn and Ariel Pakes 1999ldquoVoluntary Export Restraints on AutomobilesEvaluating a Trade Policyrdquo American EconomicReview 89(3) 400ndash430

Blanchard Olivier J and Stanley Fischer 1989Lectures on Macroeconomics Cambridge MIT Press

Cabral Luis and Luis Villas-Boas 2004 ldquoBertrandSupertrapsrdquo Mimeo

Carlsson Hans and Eric van Damme 1993 ldquoGlobalGames and Equilibrium Selectionrdquo Econometrica61(5) 989ndash1018

Chamley Christophe 2003 ldquoDynamic SpeculativeAttacksrdquo American Economic Review 93(3) 603ndash21

Chatterjee Satyajit and Russell Cooper 1989ldquoMultiplicity of Equilibria and Fluctuations inDynamic Imperfectly Competitive EconomiesrdquoAmerican Economic Review 79(2) 353ndash57

Chwe Michael Suk-Young 1998 ldquoBelieve the HypeSolving Coordination Problems with TelevisionAdvertisingrdquo Mimeo

Ciliberto Federico and Elie Tamer 2004 ldquoMarketStructure and Multiple Equilibria in AirlineMarketsrdquo Mimeo

Cooper Russell and Andrew John 1988 ldquoCoordinatingCoordination Failures in Keynesian ModelsrdquoQuarterly Journal of Economics 103(3) 441ndash63

Corsetti Giancarlo Amil Dasgupta Stephen Morrisand Hyun Shin 2004 ldquoDoes One Soros Make aDifference A Theory of Currency Crises with Largeand Small Tradersrdquo Review of Economic Studies71(1) 87ndash113

Corsetti Giancarlo Bernardo Guimaraes and NourielRoubini 2003 ldquoInternational Lending of LastResort and Moral Hazard A Model of IMFrsquosCatalytic Financerdquo Mimeo

Curtat Laurent O 1996 ldquoMarkov Equilibria ofStochastic Games with Complementaritiesrdquo Gamesand Economic Behavior 17(2) 177ndash99

Cyert Richard M and M H DeGroot 1970ldquoMultiperiod Decision Models with AlternatingChoice as a Solution to the Duopoly ProblemrdquoQuarterly Journal of Economics 84(3) 410ndash29

drsquoAspremont Claude and Alexis Jacquemin 1988ldquoCooperative and Noncooperative RampD in Duopolywith Spilloversrdquo American Economic Review 78(5)1133ndash37

Dasgupta Amil 2003 ldquoFinancial Contagion throughCapital Connections A Model of the Origin andSpread of Bank Panicsrdquo Mimeo

Diamond Douglas W and Philip H Dybvig 1983ldquoBank Runs Deposit Insurance and LiquidityrdquoJournal of Political Economy 91(3) 401ndash19

Diamond Peter A 1981 ldquoMobility Costs FrictionalUnemployment and Efficiencyrdquo Journal of PoliticalEconomy 89(4) 798ndash812

Diamond Peter A 1982 ldquoAggregate DemandManagement in Search Equilibriumrdquo Journal ofPolitical Economy 90(5) 881ndash94

Dixit Avinash K 1986 ldquoComparative Statics forOligopolyrdquo International Economic Review 27(1)107ndash22

Driskill Robert A and Stephen McCafferty 1989ldquoDynamic Duopoly with Adjustment Costs ADifferential Game Approachrdquo Journal of EconomicTheory 49(2) 324ndash38

Dybvig Philip H and Chester S Spatt 1983ldquoAdoption Externalities as Public Goodsrdquo Journal ofPublic Economics 20(2) 231ndash47

Echenique Federico 2002 ldquoComparative Statics byAdaptive Dynamics and the CorrespondencePrinciplerdquo Econometrica 70(2) 833ndash44

Echenique Federico 2004a ldquoA Characterization ofStrategic Complementaritiesrdquo Games and EconomicBehavior 46(2) 325ndash47

Echenique Federico 2004b ldquoExtensive-Form Gamesand Strategic Complementaritiesrdquo Games andEconomic Behavior 46(2) 348ndash64

Echenique Federico and Aaron Edlin 2004 ldquoMixedEquilibria Are Unstable in Games of StrategicComplementsrdquo Journal of Economic Theory 118(1)61ndash79

Edlin Aaron S and Chris Shannon 1998 ldquoStrictMonotonicity in Comparative Staticsrdquo Journal ofEconomic Theory 81(1) 201ndash19

Frankel David M Stephen Morris and Ady Pauzner2003 ldquoEquilibrium Selection in Global Games withStrategic Complementaritiesrdquo Journal of EconomicTheory 108(1) 1ndash44

Friedman James 1983 Oligopoly Theory Cambridge



Cambridge University PressFudenberg Drew and Jean Tirole 1984 ldquoThe Fat-Cat

Effect the Puppy-Dog Ploy and the Lean andHungry Lookrdquo American Economic Review 74(2)361ndash66

Fudenberg Drew and Jean Tirole 1991 GameTheory Cambridge MIT Press

Goldstein Itay and Ady Pauzner 2003 ldquoDemandDeposit Contracts and the Probability of BankRunsrdquo Mimeo

Gorton Gary 1985 ldquoBank Suspension ofConvertibilityrdquo Journal of Monetary Economics15(2) 177ndash93

Gorton Gary 1988 ldquoBanking Panics and BusinessCyclesrdquo Oxford Economic Papers 40(4) 751ndash81

Guesnerie Roger 1992 ldquoAn Exploration of theEductive Justifications of the Rational-ExpectationsHypothesisrdquo American Economic Review 82(5)1254-78

Harsanyi John and Reinhard Selten 1988 A GeneralTheory of Equilibrium Selection in GamesCambridge MIT Press

Hendricks Kenneth and Dan Kovenock 1989ldquoAsymmetric Information Information Externalitiesand Efficiency The Case of Oil Explorationrdquo RANDJournal of Economics 20(2) 164ndash82

Jun Byoung and Xavier Vives 2004 ldquoStrategicIncentives in Dynamic Duopolyrdquo Journal ofEconomic Theory 116(2) 249ndash81

Katz Michael L and Carl Shapiro 1986 ldquoTechnologyAdoption in the Presence of Network ExternalitiesrdquoJournal of Political Economy 94(4) 822ndash41

Khan M Ali and Yeneng Sun 1995 ldquoPure Strategiesin Games with Private Informationrdquo Journal ofMathematical Economics 24(7) 633ndash53

Lee Tom and Louis L Wilde 1980 ldquoMarketStructure and Innovation A ReformulationrdquoQuarterly Journal of Economics 94(2) 429ndash36

Maskin Eric and Jean Tirole 1987 ldquoA Theory ofDynamic Oligopoly III Cournot CompetitionrdquoEuropean Economic Review 31(4) 947ndash68

Maskin Eric and Jean Tirole 1988a ldquoA Theory ofDynamic Oligopoly I Overview and QuantityCompetition with Large Fixed CostsrdquoEconometrica 56(3) 549ndash69

Maskin Eric and Jean Tirole 1988b ldquoA Theory ofDynamic Oligopoly II Price Competition KinkedDemand Curves and Edgeworth CyclesrdquoEconometrica 56(3) 571ndash99

Matsuyama Kiminori 1995 ldquoComplementarities andCumulative Processes in Models of MonopolisticCompetitionrdquo Journal of Economic Literature33(2) 701ndash29

McAdams David 2003 ldquoIsotone Equilibrium inGames of Incomplete Informationrdquo Econometrica71(4) 1191ndash1214

McManus M 1962 ldquoNumbers and Size in CournotOligopolyrdquo Yorkshire Bulletin of Social andEconomic Research 14

McManus M 1964 ldquoEquilibrium Numbers and Sizein Cournot Oligopolyrdquo Yorkshire Bulletin of Socialand Economic Research 16 68ndash75

Milgrom Paul and John Roberts 1990

ldquoRationalizability Learning and Equilibrium inGames with Strategic ComplementaritiesrdquoEconometrica 58(6) 1255ndash77

Milgrom Paul and John Roberts 1990b ldquoTheEconomics of Modern Manufacturing TechnologyStrategy and Organizationrdquo American EconomicReview 80(3) 511ndash28

Milgrom Paul and John Roberts 1994 ldquoComparingEquilibriardquo American Economic Review 84(3)441ndash59

Milgrom Paul and Chris Shannon 1994 ldquoMonotoneComparative Staticsrdquo Econometrica 62(1) 157ndash80

Milgrom Paul and Robert Weber 1985 ldquoDistributionalStrategies for Games with Incomplete InformationrdquoMathematics of Operations Research 10(3) 619ndash32

Miravete Eugenio and Jose C Perniacuteas 2004ldquoInnovation Complementarity and Scale ofProductionrdquo CEPR Discussion Paper 4483

Mohnen Pierre and Lars-Hendrick RoumlllerForthcoming ldquoComplementarities in InnovationPolicyrdquo European Economic Review

Morris Stephen and Hyun Song Shin 1998 ldquoUniqueEquilibrium in a Model of Self-Fulfilling CurrencyAttacksrdquo American Economic Review 88(3) 587ndash97

Morris Stephen and Hyun Song Shin 2002 ldquoGlobalGames Theory and Applicationsrdquo in Advances inEconomics and Econometrics Proceedings of theEighth World Congress of the Econometric SocietyM Dewatripont L Hansen and S Turnovsky edsCambridge Cambridge University Press

Okuno-Fujiwara Masahiro Andrew Postlewaite andKotaro Suzumura 1990 ldquoStrategic InformationRevelationrdquo Review of Economic Studies 57(1)25ndash47

Pagano Marco 1989 ldquoTrading Volume and AssetLiquidityrdquo Quarterly Journal of Economics 104(2)255ndash74

Pearce David G 1984 ldquoRationalizable StrategicBehavior and the Problem of PerfectionrdquoEconometrica 52(4) 1029ndash50

Peitz Martin 2000 ldquoAggregation in a Model of PriceCompetitionrdquo Journal of Economic Theory 90(1)1ndash38

Peitz Martin Forthcoming ldquoAsymmetric Access PriceRegulation in Telecommunications MarketsrdquoEuropean Economic Review

Postlewaite Andrew and Xavier Vives 1987 ldquoBankRuns as an Equilibrium Phenomenonrdquo Journal ofPolitical Economy 95(3) 485ndash91

Radner Roy 1962 ldquoTeam Decision Problemsrdquo Annalsof Mathematical Statistics 33(3) 857ndash81

Radner Roy and Robert Rosenthal 1982 ldquoPrivateInformation and Pure-Strategy EquilibriardquoMathematical Operations Research 7 401ndash09

Reinganum Jennifer 1989 ldquoThe Timing of InnovationResearch Development and Diffusionrdquo inHandbook of Industrial Organization RSchmalensee and R Willig eds Amsterdam NorthHolland 849ndash908

Reynolds Stanley S 1987 ldquoCapacity InvestmentPreemption and Commitment in an Infinite HorizonModelrdquo International Economic Review 28(1)69ndash88



Roberts John and Hugo Sonnenschein 1976 ldquoOn theExistence of Cournot Equilibrium without ConcaveProfit Functionsrdquo Journal of Economic Theory13(1) 112ndash17

Roberts John and Hugo Sonnenschein 1977 ldquoOn theFoundations of the Theory of MonopolisticCompetitionrdquo Econometrica 45(1) 101ndash13

Rochet Jean Charles and Xavier Vives 2004ldquoCoordination Failures and the Lender of LastResort Was Bagehot Right After Allrdquo Journal of theEuropean Economic Association 2(6) 1116ndash47

Rotemberg Julio J 1982 ldquoMonopolistic PriceAdjustment and Aggregate Outputrdquo Review ofEconomic Studies 49(4) 517ndash31

Samuelson Paul A 1979 Foundations of EconomicAnalysis New York Atheneum

Samuelson Paul A 1974 ldquoComplementarity-An Essayon the 40th Anniversary of the HicksndashAllenRevolution in Demand Theoryrdquo Journal of EconomicLiterature 12(4) 1255ndash89

Seade Jesus K 1980a ldquoThe Stability of CournotRevisitedrdquo Journal of Economic Theory 23(1) 15ndash27

Seade Jesus K 1980b ldquoOn the Effects of EntryrdquoEconometrica 48(2) 479ndash89

Shaked Avner and John Sutton 1982 ldquoRelaxing PriceCompetition through Product DifferentiationrdquoReview of Economic Studies 49(1) 3ndash13

Shimer Robert and Lones Smith 2000 ldquoAssortativeMatching and Searchrdquo Econometrica 68(2) 343ndash69

Sleet Christopher 2001 ldquoMarkov Perfect Equilibria inIndustries with Complementaritiesrdquo EconomicTheory 17(2) 371ndash97

Spady Richard H 1984 ldquoNon-Cooperative Price-Setting by Asymmetric Multiproduct Firmsrdquo BellLaboratories Mimeo

Spence Michael 1976 ldquoProduct Differentiation andWelfarerdquo American Economic Review 66(2) 407ndash14

Sweeting Andrew 2004 ldquoCoordination GamesMultiple Equilibria and the Timing of Radio

Commercialsrdquo MimeoTarski Alfred 1955 ldquoA Lattice-Theoretical Fixpoint

Theorem and its Applicationsrdquo Pacific Journal ofMathematics 5(1) 285ndash308

Thisse Jacques-Francois and Xavier Vives 1992ldquoBasing Point Pricing Competition versusCollusionrdquo Journal of Industrial Economics 40(3)249ndash60

Topkis Donald M 1978 ldquoMinimizing a SubmodularFunction on a Latticerdquo Operations Research 26(2)305ndash21

Topkis Donald M 1979 ldquoEquilibrium Points inNonzero-Sum N-Person Submodular Gamesrdquo SIAMJournal of Control and Optimization 17(6) 773ndash87

Topkis Donald M 1998 Supermodularity andComplementarity Princeton Princeton UniversityPress

Van Zandt Timothy and Xavier Vives 2004ldquoMonotone Equilibria in Bayesian Games of StrategicComplementaritiesrdquo INSEAD Working Paper

Vives Xavier 1985a ldquoNash Equilibrium in OligopolyGames with Monotone Best Responsesrdquo CARESSWorking Paper 85-10

Vives Xavier 1985b ldquoOn the Efficiency of Bertrand andCournot Equilibria with Product DifferentiationrdquoJournal of Economic Theory 36(1) 166ndash75

Vives Xavier 1990a ldquoNash Equilibrium with StrategicComplementaritiesrdquo Journal of MathematicalEconomics 19(3) 305ndash21

Vives Xavier 1990b ldquoInformation and CompetitiveAdvantagerdquo International Journal of IndustrialOrganization 8(1) 17ndash35

Vives Xavier 1999 Oligopoly Pricing Old Ideas andNew Tools Cambridge MIT Press

Vives Xavier 2004 ldquoStrategic Complementarity inMulti-Stage Gamesrdquo Mimeo

Zhou Lin 1994 ldquoThe Set of Nash Equilibria of aSupermodular Game Is a Complete Latticerdquo Gamesand Economic Behavior 7(2) 295ndash300



challenging The reason is that the tools athand were not attuned to deal with environ-ments where indivisibilities nonsmooth pay-offs and complex strategy spaces werenaturally the norm rather than the exceptionTo make matters worse multiple equilibriaare typical in the presence of complementar-ities and policy analysis is left orphanInstances of coordination failure with multi-ple equilibria abound bank runs debt runson a country low employmentactivity equi-libria revolutions and development trapsprovide some examples In all these situa-tions there are multiple self-fulfilling expec-tations of agents in the economy A key issueis how to build coherent models that are use-ful for policy analysis A challenging aspectof any crisis situation is to disentangle self-fulfilling from fundamentals-driven explana-tions that help answer questions such asWhat is the effect of an increase in theamount of central bank reserves on the prob-ability of a run on the currency What is theimpact of an increase in the solvency ratio onthe probability of failure of a bank What isthe effect of a change in foreign short-termdebt exposure on the probability of defaultof a small open economy

The appropriate toolbox to deal with com-plementarities and in particular with strate-gic situations is the theory of monotonecomparative statics and supermodulargames This theory provides the method formaking the analysis of complementarity sim-ple The basic idea of the theory is to exploitfully both order and monotonicity proper-ties The achievements of this approach areas follows First it provides a framework forthinking rigorously about complementari-ties identifying key parameters in the envi-ronment to look at (eg what are the criticalproperties of the payoffs and action spaces)Second it simplifies the analysis clarifyingthe drivers of the results (eg is the regular-ity condition really needed to obtain thedesired comparative static result) Third itencompasses the analysis of multiple equi-libria situations by ranking equilibria and

helping understand how potential equilibriamove with the parameters of interestFinally it easily incorporates complex strate-gy spaces such as those arising in dynamicgames and games of incomplete information

The paper provides an introduction to thetools of supermodular games and a range ofapplications to industrial organization(Cournot and Bertrand markets monopolis-tic competition RampD races multimarketoligopoly switching costs dynamic invest-ment games) macroeconomic models(menu costs search aggregate demandexternalities technology adoption adjust-ment costs) and finance (currency crisisbank runs) The next section presents theapproach in somewhat more detail as well asthe plan of the paper

2 Tools Results and Plan of the Paper

The basic theory was developed byDonald M Topkis (1978 1979) and furtherdeveloped and applied to economics byXavier Vives (1985a 1990a) and PaulMilgrom and John Roberts (1990a) Yet thetheory continues to be extended at the fron-tier of researchmdashfor example to dynamicgames and games of incomplete informa-tion The beauty of the approach is not itscomplexity but rather how much it simplifiesthe analysis and clarifies results In fact eventhe basic tools are not fully exploited byeconomists in current research

The theory of supermodular games andmonotone comparative statics based on lat-tice-theoretic methods has provided a pow-erful toolbox for analyzing the consequencesof complementarities in economicsMonotone comparative statics analysis pro-vides conditions under which optimal solu-tions to optimization problems movemonotonically with a parameter In this paperI provide an introduction to this methodologyand then apply it to the study of strategicinteraction in the presence of complementar-ities This approach exploits order and monot-onicity properties in contrast to classical



































(r (sima)sima) = 0


entiable and






part part( )r a

a a

ai

i

~ ~

2

2 2

π

π

partpart

πai











acirc a1

1

a a a






k





























where



Pr ˆ ˆˆ

θ θ θ θ ρρ

θ microσ

θ

θj ile = = minus

+minus

11


îθ ρθ ρ micro

σ ρθ

θ


minus

1

1 2











1

12

microθ = 12 1 θ















tiable









































a2

A+

A+

Aminus

Aminus

a

a

a1

r2()

r1()






















2 then the best





a2

a1acirc1 a1

r1 r2

a20


19









a2

r1()

r2()

a1



































52 Patent Races











π ii i

i j i j

h x V xh x h x r

=( ) minus

( ) + ( ) +ne












partpart part


+partpart

gt2 2

0π i

i ii i

i

i i

i

ip tp c

Dp t

Dt













6 Dynamic Games











a2

r2()

r1()

45deg

a1



















28

62 Markov Games













π i i ix y u x F x y ( ) = ( ) + ( )



































max x i i

i

x y V xπ δ( ) + ( )




















V y x y V xx1 1 1( ) = ( ) + ( )max ~π δ

~V x x R x V R x1 1 2 1 2( ) = ( )( ) + ( )( )[ ]π δ
















7 Bayesian Games









A Ai==prod i

n

1

T Ti==prod i

n

0















i(i) Existence of






























39






π θih

H


1











40




72 Global Games









0

1

h(θ)

θ θ



and 1 at

= See figure 7










~a ge h() ~a lt h()








1

2


















θττ

micro τ ττ

γθ ε

ε

θ

εθ

θ ε

ε

1

sˆ ˆ+ ( ) minus+

τ τ θτ micro τ

θ εθ θ εε

θ ετ τs

+









ττ

πτ τ

τθθ

ε

θ ε

ε





11 2

1minus+

prime ltρρ

τπθ g









τ τ θτ micro τ

τ τθ εθ θ ε

θ ε

s+ minus

++

= γ




722 Applications




45



dsd

rr

rlowast

=part part

minus primegt part

partmicromicro

microθ

θ

θ

1


















h mmθ

λθθ

( ) + minus minus

11











































REFERENCES

















































































































































(r (sima)sima) = 0


entiable and






part part( )r a

a a

ai

i

~ ~

2

2 2

π

π

partpart

πai











acirc a1

1

a a a






k





























where



Pr ˆ ˆˆ

θ θ θ θ ρρ

θ microσ

θ

θj ile = = minus

+minus

11


îθ ρθ ρ micro

σ ρθ

θ


minus

1

1 2











1

12

microθ = 12 1 θ















tiable









































a2

A+

A+

Aminus

Aminus

a

a

a1

r2()

r1()






















2 then the best





a2

a1acirc1 a1

r1 r2

a20


19









a2

r1()

r2()

a1



































52 Patent Races











π ii i

i j i j

h x V xh x h x r

=( ) minus

( ) + ( ) +ne












partpart part


+partpart

gt2 2

0π i

i ii i

i

i i

i

ip tp c

Dp t

Dt













6 Dynamic Games











a2

r2()

r1()

45deg

a1



















28

62 Markov Games













π i i ix y u x F x y ( ) = ( ) + ( )



































max x i i

i

x y V xπ δ( ) + ( )




















V y x y V xx1 1 1( ) = ( ) + ( )max ~π δ

~V x x R x V R x1 1 2 1 2( ) = ( )( ) + ( )( )[ ]π δ
















7 Bayesian Games









A Ai==prod i

n

1

T Ti==prod i

n

0















i(i) Existence of






























39






π θih

H


1











40




72 Global Games









0

1

h(θ)

θ θ



and 1 at

= See figure 7










~a ge h() ~a lt h()








1

2


















θττ

micro τ ττ

γθ ε

ε

θ

εθ

θ ε

ε

1

sˆ ˆ+ ( ) minus+

τ τ θτ micro τ

θ εθ θ εε

θ ετ τs

+









ττ

πτ τ

τθθ

ε

θ ε

ε





11 2

1minus+

prime ltρρ

τπθ g









τ τ θτ micro τ

τ τθ εθ θ ε

θ ε

s+ minus

++

= γ




722 Applications




45



dsd

rr

rlowast

=part part

minus primegt part

partmicromicro

microθ

θ

θ

1


















h mmθ

λθθ

( ) + minus minus

11











































REFERENCES































































































































(r (sima)sima) = 0


entiable and






part part( )r a

a a

ai

i

~ ~

2

2 2

π

π

partpart

πai











acirc a1

1

a a a






k





























where



Pr ˆ ˆˆ

θ θ θ θ ρρ

θ microσ

θ

θj ile = = minus

+minus

11


îθ ρθ ρ micro

σ ρθ

θ


minus

1

1 2











1

12

microθ = 12 1 θ















tiable









































a2

A+

A+

Aminus

Aminus

a

a

a1

r2()

r1()






















2 then the best





a2

a1acirc1 a1

r1 r2

a20


19









a2

r1()

r2()

a1



































52 Patent Races











π ii i

i j i j

h x V xh x h x r

=( ) minus

( ) + ( ) +ne












partpart part


+partpart

gt2 2

0π i

i ii i

i

i i

i

ip tp c

Dp t

Dt













6 Dynamic Games











a2

r2()

r1()

45deg

a1



















28

62 Markov Games













π i i ix y u x F x y ( ) = ( ) + ( )



































max x i i

i

x y V xπ δ( ) + ( )




















V y x y V xx1 1 1( ) = ( ) + ( )max ~π δ

~V x x R x V R x1 1 2 1 2( ) = ( )( ) + ( )( )[ ]π δ
















7 Bayesian Games









A Ai==prod i

n

1

T Ti==prod i

n

0















i(i) Existence of






























39






π θih

H


1











40




72 Global Games









0

1

h(θ)

θ θ



and 1 at

= See figure 7










~a ge h() ~a lt h()








1

2


















θττ

micro τ ττ

γθ ε

ε

θ

εθ

θ ε

ε

1

sˆ ˆ+ ( ) minus+

τ τ θτ micro τ

θ εθ θ εε

θ ετ τs

+









ττ

πτ τ

τθθ

ε

θ ε

ε





11 2

1minus+

prime ltρρ

τπθ g









τ τ θτ micro τ

τ τθ εθ θ ε

θ ε

s+ minus

++

= γ




722 Applications




45



dsd

rr

rlowast

=part part

minus primegt part

partmicromicro

microθ

θ

θ

1


















h mmθ

λθθ

( ) + minus minus

11











































REFERENCES





















































































































(r (sima)sima) = 0


entiable and






part part( )r a

a a

ai

i

~ ~

2

2 2

π

π

partpart

πai











acirc a1

1

a a a






k





























where



Pr ˆ ˆˆ

θ θ θ θ ρρ

θ microσ

θ

θj ile = = minus

+minus

11


îθ ρθ ρ micro

σ ρθ

θ


minus

1

1 2











1

12

microθ = 12 1 θ















tiable









































a2

A+

A+

Aminus

Aminus

a

a

a1

r2()

r1()






















2 then the best





a2

a1acirc1 a1

r1 r2

a20


19









a2

r1()

r2()

a1



































52 Patent Races











π ii i

i j i j

h x V xh x h x r

=( ) minus

( ) + ( ) +ne












partpart part


+partpart

gt2 2

0π i

i ii i

i

i i

i

ip tp c

Dp t

Dt













6 Dynamic Games











a2

r2()

r1()

45deg

a1



















28

62 Markov Games













π i i ix y u x F x y ( ) = ( ) + ( )



































max x i i

i

x y V xπ δ( ) + ( )




















V y x y V xx1 1 1( ) = ( ) + ( )max ~π δ

~V x x R x V R x1 1 2 1 2( ) = ( )( ) + ( )( )[ ]π δ
















7 Bayesian Games









A Ai==prod i

n

1

T Ti==prod i

n

0















i(i) Existence of






























39






π θih

H


1











40




72 Global Games









0

1

h(θ)

θ θ



and 1 at

= See figure 7










~a ge h() ~a lt h()








1

2


















θττ

micro τ ττ

γθ ε

ε

θ

εθ

θ ε

ε

1

sˆ ˆ+ ( ) minus+

τ τ θτ micro τ

θ εθ θ εε

θ ετ τs

+









ττ

πτ τ

τθθ

ε

θ ε

ε





11 2

1minus+

prime ltρρ

τπθ g









τ τ θτ micro τ

τ τθ εθ θ ε

θ ε

s+ minus

++

= γ




722 Applications




45



dsd

rr

rlowast

=part part

minus primegt part

partmicromicro

microθ

θ

θ

1


















h mmθ

λθθ

( ) + minus minus

11











































REFERENCES


















































































































acirc a1

1

a a a






k





























where



Pr ˆ ˆˆ

θ θ θ θ ρρ

θ microσ

θ

θj ile = = minus

+minus

11


îθ ρθ ρ micro

σ ρθ

θ


minus

1

1 2











1

12

microθ = 12 1 θ















tiable









































a2

A+

A+

Aminus

Aminus

a

a

a1

r2()

r1()






















2 then the best





a2

a1acirc1 a1

r1 r2

a20


19









a2

r1()

r2()

a1



































52 Patent Races











π ii i

i j i j

h x V xh x h x r

=( ) minus

( ) + ( ) +ne












partpart part


+partpart

gt2 2

0π i

i ii i

i

i i

i

ip tp c

Dp t

Dt













6 Dynamic Games











a2

r2()

r1()

45deg

a1



















28

62 Markov Games













π i i ix y u x F x y ( ) = ( ) + ( )



































max x i i

i

x y V xπ δ( ) + ( )




















V y x y V xx1 1 1( ) = ( ) + ( )max ~π δ

~V x x R x V R x1 1 2 1 2( ) = ( )( ) + ( )( )[ ]π δ
















7 Bayesian Games









A Ai==prod i

n

1

T Ti==prod i

n

0















i(i) Existence of






























39






π θih

H


1











40




72 Global Games









0

1

h(θ)

θ θ



and 1 at

= See figure 7










~a ge h() ~a lt h()








1

2


















θττ

micro τ ττ

γθ ε

ε

θ

εθ

θ ε

ε

1

sˆ ˆ+ ( ) minus+

τ τ θτ micro τ

θ εθ θ εε

θ ετ τs

+









ττ

πτ τ

τθθ

ε

θ ε

ε





11 2

1minus+

prime ltρρ

τπθ g









τ τ θτ micro τ

τ τθ εθ θ ε

θ ε

s+ minus

++

= γ




722 Applications




45



dsd

rr

rlowast

=part part

minus primegt part

partmicromicro

microθ

θ

θ

1


















h mmθ

λθθ

( ) + minus minus

11











































REFERENCES






































































































































where



Pr ˆ ˆˆ

θ θ θ θ ρρ

θ microσ

θ

θj ile = = minus

+minus

11


îθ ρθ ρ micro

σ ρθ

θ


minus

1

1 2











1

12

microθ = 12 1 θ















tiable









































a2

A+

A+

Aminus

Aminus

a

a

a1

r2()

r1()






















2 then the best





a2

a1acirc1 a1

r1 r2

a20


19









a2

r1()

r2()

a1



































52 Patent Races











π ii i

i j i j

h x V xh x h x r

=( ) minus

( ) + ( ) +ne












partpart part


+partpart

gt2 2

0π i

i ii i

i

i i

i

ip tp c

Dp t

Dt













6 Dynamic Games











a2

r2()

r1()

45deg

a1



















28

62 Markov Games













π i i ix y u x F x y ( ) = ( ) + ( )



































max x i i

i

x y V xπ δ( ) + ( )




















V y x y V xx1 1 1( ) = ( ) + ( )max ~π δ

~V x x R x V R x1 1 2 1 2( ) = ( )( ) + ( )( )[ ]π δ
















7 Bayesian Games









A Ai==prod i

n

1

T Ti==prod i

n

0















i(i) Existence of






























39






π θih

H


1











40




72 Global Games









0

1

h(θ)

θ θ



and 1 at

= See figure 7










~a ge h() ~a lt h()








1

2


















θττ

micro τ ττ

γθ ε

ε

θ

εθ

θ ε

ε

1

sˆ ˆ+ ( ) minus+

τ τ θτ micro τ

θ εθ θ εε

θ ετ τs

+









ττ

πτ τ

τθθ

ε

θ ε

ε





11 2

1minus+

prime ltρρ

τπθ g









τ τ θτ micro τ

τ τθ εθ θ ε

θ ε

s+ minus

++

= γ




722 Applications




45



dsd

rr

rlowast

=part part

minus primegt part

partmicromicro

microθ

θ

θ

1


















h mmθ

λθθ

( ) + minus minus

11











































REFERENCES























































































































where



Pr ˆ ˆˆ

θ θ θ θ ρρ

θ microσ

θ

θj ile = = minus

+minus

11


îθ ρθ ρ micro

σ ρθ

θ


minus

1

1 2











1

12

microθ = 12 1 θ















tiable









































a2

A+

A+

Aminus

Aminus

a

a

a1

r2()

r1()






















2 then the best





a2

a1acirc1 a1

r1 r2

a20


19









a2

r1()

r2()

a1



































52 Patent Races











π ii i

i j i j

h x V xh x h x r

=( ) minus

( ) + ( ) +ne












partpart part


+partpart

gt2 2

0π i

i ii i

i

i i

i

ip tp c

Dp t

Dt













6 Dynamic Games











a2

r2()

r1()

45deg

a1



















28

62 Markov Games













π i i ix y u x F x y ( ) = ( ) + ( )



































max x i i

i

x y V xπ δ( ) + ( )




















V y x y V xx1 1 1( ) = ( ) + ( )max ~π δ

~V x x R x V R x1 1 2 1 2( ) = ( )( ) + ( )( )[ ]π δ
















7 Bayesian Games









A Ai==prod i

n

1

T Ti==prod i

n

0















i(i) Existence of






























39






π θih

H


1











40




72 Global Games









0

1

h(θ)

θ θ



and 1 at

= See figure 7










~a ge h() ~a lt h()








1

2


















θττ

micro τ ττ

γθ ε

ε

θ

εθ

θ ε

ε

1

sˆ ˆ+ ( ) minus+

τ τ θτ micro τ

θ εθ θ εε

θ ετ τs

+









ττ

πτ τ

τθθ

ε

θ ε

ε





11 2

1minus+

prime ltρρ

τπθ g









τ τ θτ micro τ

τ τθ εθ θ ε

θ ε

s+ minus

++

= γ




722 Applications




45



dsd

rr

rlowast

=part part

minus primegt part

partmicromicro

microθ

θ

θ

1


















h mmθ

λθθ

( ) + minus minus

11











































REFERENCES


















































































































1

12

microθ = 12 1 θ















tiable









































a2

A+

A+

Aminus

Aminus

a

a

a1

r2()

r1()






















2 then the best





a2

a1acirc1 a1

r1 r2

a20


19









a2

r1()

r2()

a1



































52 Patent Races











π ii i

i j i j

h x V xh x h x r

=( ) minus

( ) + ( ) +ne












partpart part


+partpart

gt2 2

0π i

i ii i

i

i i

i

ip tp c

Dp t

Dt













6 Dynamic Games











a2

r2()

r1()

45deg

a1



















28

62 Markov Games













π i i ix y u x F x y ( ) = ( ) + ( )



































max x i i

i

x y V xπ δ( ) + ( )




















V y x y V xx1 1 1( ) = ( ) + ( )max ~π δ

~V x x R x V R x1 1 2 1 2( ) = ( )( ) + ( )( )[ ]π δ
















7 Bayesian Games









A Ai==prod i

n

1

T Ti==prod i

n

0















i(i) Existence of






























39






π θih

H


1











40




72 Global Games









0

1

h(θ)

θ θ



and 1 at

= See figure 7










~a ge h() ~a lt h()








1

2


















θττ

micro τ ττ

γθ ε

ε

θ

εθ

θ ε

ε

1

sˆ ˆ+ ( ) minus+

τ τ θτ micro τ

θ εθ θ εε

θ ετ τs

+









ττ

πτ τ

τθθ

ε

θ ε

ε





11 2

1minus+

prime ltρρ

τπθ g









τ τ θτ micro τ

τ τθ εθ θ ε

θ ε

s+ minus

++

= γ




722 Applications




45



dsd

rr

rlowast

=part part

minus primegt part

partmicromicro

microθ

θ

θ

1


















h mmθ

λθθ

( ) + minus minus

11











































REFERENCES
























































































































tiable









































a2

A+

A+

Aminus

Aminus

a

a

a1

r2()

r1()






















2 then the best





a2

a1acirc1 a1

r1 r2

a20


19









a2

r1()

r2()

a1



































52 Patent Races











π ii i

i j i j

h x V xh x h x r

=( ) minus

( ) + ( ) +ne












partpart part


+partpart

gt2 2

0π i

i ii i

i

i i

i

ip tp c

Dp t

Dt













6 Dynamic Games











a2

r2()

r1()

45deg

a1



















28

62 Markov Games













π i i ix y u x F x y ( ) = ( ) + ( )



































max x i i

i

x y V xπ δ( ) + ( )




















V y x y V xx1 1 1( ) = ( ) + ( )max ~π δ

~V x x R x V R x1 1 2 1 2( ) = ( )( ) + ( )( )[ ]π δ
















7 Bayesian Games









A Ai==prod i

n

1

T Ti==prod i

n

0















i(i) Existence of






























39






π θih

H


1











40




72 Global Games









0

1

h(θ)

θ θ



and 1 at

= See figure 7










~a ge h() ~a lt h()








1

2


















θττ

micro τ ττ

γθ ε

ε

θ

εθ

θ ε

ε

1

sˆ ˆ+ ( ) minus+

τ τ θτ micro τ

θ εθ θ εε

θ ετ τs

+









ττ

πτ τ

τθθ

ε

θ ε

ε





11 2

1minus+

prime ltρρ

τπθ g









τ τ θτ micro τ

τ τθ εθ θ ε

θ ε

s+ minus

++

= γ




722 Applications




45



dsd

rr

rlowast

=part part

minus primegt part

partmicromicro

microθ

θ

θ

1


















h mmθ

λθθ

( ) + minus minus

11











































REFERENCES
















































































































































a2

A+

A+

Aminus

Aminus

a

a

a1

r2()

r1()






















2 then the best





a2

a1acirc1 a1

r1 r2

a20


19









a2

r1()

r2()

a1



































52 Patent Races











π ii i

i j i j

h x V xh x h x r

=( ) minus

( ) + ( ) +ne












partpart part


+partpart

gt2 2

0π i

i ii i

i

i i

i

ip tp c

Dp t

Dt













6 Dynamic Games











a2

r2()

r1()

45deg

a1



















28

62 Markov Games













π i i ix y u x F x y ( ) = ( ) + ( )



































max x i i

i

x y V xπ δ( ) + ( )




















V y x y V xx1 1 1( ) = ( ) + ( )max ~π δ

~V x x R x V R x1 1 2 1 2( ) = ( )( ) + ( )( )[ ]π δ
















7 Bayesian Games









A Ai==prod i

n

1

T Ti==prod i

n

0















i(i) Existence of






























39






π θih

H


1











40




72 Global Games









0

1

h(θ)

θ θ



and 1 at

= See figure 7










~a ge h() ~a lt h()








1

2


















θττ

micro τ ττ

γθ ε

ε

θ

εθ

θ ε

ε

1

sˆ ˆ+ ( ) minus+

τ τ θτ micro τ

θ εθ θ εε

θ ετ τs

+









ττ

πτ τ

τθθ

ε

θ ε

ε





11 2

1minus+

prime ltρρ

τπθ g









τ τ θτ micro τ

τ τθ εθ θ ε

θ ε

s+ minus

++

= γ




722 Applications




45



dsd

rr

rlowast

=part part

minus primegt part

partmicromicro

microθ

θ

θ

1


















h mmθ

λθθ

( ) + minus minus

11











































REFERENCES
































































































































a2

A+

A+

Aminus

Aminus

a

a

a1

r2()

r1()






















2 then the best





a2

a1acirc1 a1

r1 r2

a20


19









a2

r1()

r2()

a1



































52 Patent Races











π ii i

i j i j

h x V xh x h x r

=( ) minus

( ) + ( ) +ne












partpart part


+partpart

gt2 2

0π i

i ii i

i

i i

i

ip tp c

Dp t

Dt













6 Dynamic Games











a2

r2()

r1()

45deg

a1



















28

62 Markov Games













π i i ix y u x F x y ( ) = ( ) + ( )



































max x i i

i

x y V xπ δ( ) + ( )




















V y x y V xx1 1 1( ) = ( ) + ( )max ~π δ

~V x x R x V R x1 1 2 1 2( ) = ( )( ) + ( )( )[ ]π δ
















7 Bayesian Games









A Ai==prod i

n

1

T Ti==prod i

n

0















i(i) Existence of






























39






π θih

H


1











40




72 Global Games









0

1

h(θ)

θ θ



and 1 at

= See figure 7










~a ge h() ~a lt h()








1

2


















θττ

micro τ ττ

γθ ε

ε

θ

εθ

θ ε

ε

1

sˆ ˆ+ ( ) minus+

τ τ θτ micro τ

θ εθ θ εε

θ ετ τs

+









ττ

πτ τ

τθθ

ε

θ ε

ε





11 2

1minus+

prime ltρρ

τπθ g









τ τ θτ micro τ

τ τθ εθ θ ε

θ ε

s+ minus

++

= γ




722 Applications




45



dsd

rr

rlowast

=part part

minus primegt part

partmicromicro

microθ

θ

θ

1


















h mmθ

λθθ

( ) + minus minus

11











































REFERENCES


















































































































a2

A+

A+

Aminus

Aminus

a

a

a1

r2()

r1()






















2 then the best





a2

a1acirc1 a1

r1 r2

a20


19









a2

r1()

r2()

a1



































52 Patent Races











π ii i

i j i j

h x V xh x h x r

=( ) minus

( ) + ( ) +ne












partpart part


+partpart

gt2 2

0π i

i ii i

i

i i

i

ip tp c

Dp t

Dt













6 Dynamic Games











a2

r2()

r1()

45deg

a1



















28

62 Markov Games













π i i ix y u x F x y ( ) = ( ) + ( )



































max x i i

i

x y V xπ δ( ) + ( )




















V y x y V xx1 1 1( ) = ( ) + ( )max ~π δ

~V x x R x V R x1 1 2 1 2( ) = ( )( ) + ( )( )[ ]π δ
















7 Bayesian Games









A Ai==prod i

n

1

T Ti==prod i

n

0















i(i) Existence of






























39






π θih

H


1











40




72 Global Games









0

1

h(θ)

θ θ



and 1 at

= See figure 7










~a ge h() ~a lt h()








1

2


















θττ

micro τ ττ

γθ ε

ε

θ

εθ

θ ε

ε

1

sˆ ˆ+ ( ) minus+

τ τ θτ micro τ

θ εθ θ εε

θ ετ τs

+









ττ

πτ τ

τθθ

ε

θ ε

ε





11 2

1minus+

prime ltρρ

τπθ g









τ τ θτ micro τ

τ τθ εθ θ ε

θ ε

s+ minus

++

= γ




722 Applications




45



dsd

rr

rlowast

=part part

minus primegt part

partmicromicro

microθ

θ

θ

1


















h mmθ

λθθ

( ) + minus minus

11











































REFERENCES




























































































































2 then the best





a2

a1acirc1 a1

r1 r2

a20


19









a2

r1()

r2()

a1



































52 Patent Races











π ii i

i j i j

h x V xh x h x r

=( ) minus

( ) + ( ) +ne












partpart part


+partpart

gt2 2

0π i

i ii i

i

i i

i

ip tp c

Dp t

Dt













6 Dynamic Games











a2

r2()

r1()

45deg

a1



















28

62 Markov Games













π i i ix y u x F x y ( ) = ( ) + ( )



































max x i i

i

x y V xπ δ( ) + ( )




















V y x y V xx1 1 1( ) = ( ) + ( )max ~π δ

~V x x R x V R x1 1 2 1 2( ) = ( )( ) + ( )( )[ ]π δ
















7 Bayesian Games









A Ai==prod i

n

1

T Ti==prod i

n

0















i(i) Existence of






























39






π θih

H


1











40




72 Global Games









0

1

h(θ)

θ θ



and 1 at

= See figure 7










~a ge h() ~a lt h()








1

2


















θττ

micro τ ττ

γθ ε

ε

θ

εθ

θ ε

ε

1

sˆ ˆ+ ( ) minus+

τ τ θτ micro τ

θ εθ θ εε

θ ετ τs

+









ττ

πτ τ

τθθ

ε

θ ε

ε





11 2

1minus+

prime ltρρ

τπθ g









τ τ θτ micro τ

τ τθ εθ θ ε

θ ε

s+ minus

++

= γ




722 Applications




45



dsd

rr

rlowast

=part part

minus primegt part

partmicromicro

microθ

θ

θ

1


















h mmθ

λθθ

( ) + minus minus

11











































REFERENCES



















































































































a2

a1acirc1 a1

r1 r2

a20


19









a2

r1()

r2()

a1



































52 Patent Races











π ii i

i j i j

h x V xh x h x r

=( ) minus

( ) + ( ) +ne












partpart part


+partpart

gt2 2

0π i

i ii i

i

i i

i

ip tp c

Dp t

Dt













6 Dynamic Games











a2

r2()

r1()

45deg

a1



















28

62 Markov Games













π i i ix y u x F x y ( ) = ( ) + ( )



































max x i i

i

x y V xπ δ( ) + ( )




















V y x y V xx1 1 1( ) = ( ) + ( )max ~π δ

~V x x R x V R x1 1 2 1 2( ) = ( )( ) + ( )( )[ ]π δ
















7 Bayesian Games









A Ai==prod i

n

1

T Ti==prod i

n

0















i(i) Existence of






























39






π θih

H


1











40




72 Global Games









0

1

h(θ)

θ θ



and 1 at

= See figure 7










~a ge h() ~a lt h()








1

2


















θττ

micro τ ττ

γθ ε

ε

θ

εθ

θ ε

ε

1

sˆ ˆ+ ( ) minus+

τ τ θτ micro τ

θ εθ θ εε

θ ετ τs

+









ττ

πτ τ

τθθ

ε

θ ε

ε





11 2

1minus+

prime ltρρ

τπθ g









τ τ θτ micro τ

τ τθ εθ θ ε

θ ε

s+ minus

++

= γ




722 Applications




45



dsd

rr

rlowast

=part part

minus primegt part

partmicromicro

microθ

θ

θ

1


















h mmθ

λθθ

( ) + minus minus

11











































REFERENCES


















































































































a2

r1()

r2()

a1



































52 Patent Races











π ii i

i j i j

h x V xh x h x r

=( ) minus

( ) + ( ) +ne












partpart part


+partpart

gt2 2

0π i

i ii i

i

i i

i

ip tp c

Dp t

Dt













6 Dynamic Games











a2

r2()

r1()

45deg

a1



















28

62 Markov Games













π i i ix y u x F x y ( ) = ( ) + ( )



































max x i i

i

x y V xπ δ( ) + ( )




















V y x y V xx1 1 1( ) = ( ) + ( )max ~π δ

~V x x R x V R x1 1 2 1 2( ) = ( )( ) + ( )( )[ ]π δ
















7 Bayesian Games









A Ai==prod i

n

1

T Ti==prod i

n

0















i(i) Existence of






























39






π θih

H


1











40




72 Global Games









0

1

h(θ)

θ θ



and 1 at

= See figure 7










~a ge h() ~a lt h()








1

2


















θττ

micro τ ττ

γθ ε

ε

θ

εθ

θ ε

ε

1

sˆ ˆ+ ( ) minus+

τ τ θτ micro τ

θ εθ θ εε

θ ετ τs

+









ττ

πτ τ

τθθ

ε

θ ε

ε





11 2

1minus+

prime ltρρ

τπθ g









τ τ θτ micro τ

τ τθ εθ θ ε

θ ε

s+ minus

++

= γ




722 Applications




45



dsd

rr

rlowast

=part part

minus primegt part

partmicromicro

microθ

θ

θ

1


















h mmθ

λθθ

( ) + minus minus

11











































REFERENCES










































































































































52 Patent Races











π ii i

i j i j

h x V xh x h x r

=( ) minus

( ) + ( ) +ne












partpart part


+partpart

gt2 2

0π i

i ii i

i

i i

i

ip tp c

Dp t

Dt













6 Dynamic Games











a2

r2()

r1()

45deg

a1



















28

62 Markov Games













π i i ix y u x F x y ( ) = ( ) + ( )



































max x i i

i

x y V xπ δ( ) + ( )




















V y x y V xx1 1 1( ) = ( ) + ( )max ~π δ

~V x x R x V R x1 1 2 1 2( ) = ( )( ) + ( )( )[ ]π δ
















7 Bayesian Games









A Ai==prod i

n

1

T Ti==prod i

n

0















i(i) Existence of






























39






π θih

H


1











40




72 Global Games









0

1

h(θ)

θ θ



and 1 at

= See figure 7










~a ge h() ~a lt h()








1

2


















θττ

micro τ ττ

γθ ε

ε

θ

εθ

θ ε

ε

1

sˆ ˆ+ ( ) minus+

τ τ θτ micro τ

θ εθ θ εε

θ ετ τs

+









ττ

πτ τ

τθθ

ε

θ ε

ε





11 2

1minus+

prime ltρρ

τπθ g









τ τ θτ micro τ

τ τθ εθ θ ε

θ ε

s+ minus

++

= γ




722 Applications




45



dsd

rr

rlowast

=part part

minus primegt part

partmicromicro

microθ

θ

θ

1


















h mmθ

λθθ

( ) + minus minus

11











































REFERENCES































































































































52 Patent Races











π ii i

i j i j

h x V xh x h x r

=( ) minus

( ) + ( ) +ne












partpart part


+partpart

gt2 2

0π i

i ii i

i

i i

i

ip tp c

Dp t

Dt













6 Dynamic Games











a2

r2()

r1()

45deg

a1



















28

62 Markov Games













π i i ix y u x F x y ( ) = ( ) + ( )



































max x i i

i

x y V xπ δ( ) + ( )




















V y x y V xx1 1 1( ) = ( ) + ( )max ~π δ

~V x x R x V R x1 1 2 1 2( ) = ( )( ) + ( )( )[ ]π δ
















7 Bayesian Games









A Ai==prod i

n

1

T Ti==prod i

n

0















i(i) Existence of






























39






π θih

H


1











40




72 Global Games









0

1

h(θ)

θ θ



and 1 at

= See figure 7










~a ge h() ~a lt h()








1

2


















θττ

micro τ ττ

γθ ε

ε

θ

εθ

θ ε

ε

1

sˆ ˆ+ ( ) minus+

τ τ θτ micro τ

θ εθ θ εε

θ ετ τs

+









ττ

πτ τ

τθθ

ε

θ ε

ε





11 2

1minus+

prime ltρρ

τπθ g









τ τ θτ micro τ

τ τθ εθ θ ε

θ ε

s+ minus

++

= γ




722 Applications




45



dsd

rr

rlowast

=part part

minus primegt part

partmicromicro

microθ

θ

θ

1


















h mmθ

λθθ

( ) + minus minus

11











































REFERENCES
























































































































π ii i

i j i j

h x V xh x h x r

=( ) minus

( ) + ( ) +ne












partpart part


+partpart

gt2 2

0π i

i ii i

i

i i

i

ip tp c

Dp t

Dt













6 Dynamic Games











a2

r2()

r1()

45deg

a1



















28

62 Markov Games













π i i ix y u x F x y ( ) = ( ) + ( )



































max x i i

i

x y V xπ δ( ) + ( )




















V y x y V xx1 1 1( ) = ( ) + ( )max ~π δ

~V x x R x V R x1 1 2 1 2( ) = ( )( ) + ( )( )[ ]π δ
















7 Bayesian Games









A Ai==prod i

n

1

T Ti==prod i

n

0















i(i) Existence of






























39






π θih

H


1











40




72 Global Games









0

1

h(θ)

θ θ



and 1 at

= See figure 7










~a ge h() ~a lt h()








1

2


















θττ

micro τ ττ

γθ ε

ε

θ

εθ

θ ε

ε

1

sˆ ˆ+ ( ) minus+

τ τ θτ micro τ

θ εθ θ εε

θ ετ τs

+









ττ

πτ τ

τθθ

ε

θ ε

ε





11 2

1minus+

prime ltρρ

τπθ g









τ τ θτ micro τ

τ τθ εθ θ ε

θ ε

s+ minus

++

= γ




722 Applications




45



dsd

rr

rlowast

=part part

minus primegt part

partmicromicro

microθ

θ

θ

1


















h mmθ

λθθ

( ) + minus minus

11











































REFERENCES



























































































































6 Dynamic Games











a2

r2()

r1()

45deg

a1



















28

62 Markov Games













π i i ix y u x F x y ( ) = ( ) + ( )



































max x i i

i

x y V xπ δ( ) + ( )




















V y x y V xx1 1 1( ) = ( ) + ( )max ~π δ

~V x x R x V R x1 1 2 1 2( ) = ( )( ) + ( )( )[ ]π δ
















7 Bayesian Games









A Ai==prod i

n

1

T Ti==prod i

n

0















i(i) Existence of






























39






π θih

H


1











40




72 Global Games









0

1

h(θ)

θ θ



and 1 at

= See figure 7










~a ge h() ~a lt h()








1

2


















θττ

micro τ ττ

γθ ε

ε

θ

εθ

θ ε

ε

1

sˆ ˆ+ ( ) minus+

τ τ θτ micro τ

θ εθ θ εε

θ ετ τs

+









ττ

πτ τ

τθθ

ε

θ ε

ε





11 2

1minus+

prime ltρρ

τπθ g









τ τ θτ micro τ

τ τθ εθ θ ε

θ ε

s+ minus

++

= γ




722 Applications




45



dsd

rr

rlowast

=part part

minus primegt part

partmicromicro

microθ

θ

θ

1


















h mmθ

λθθ

( ) + minus minus

11











































REFERENCES



















































































































6 Dynamic Games











a2

r2()

r1()

45deg

a1



















28

62 Markov Games













π i i ix y u x F x y ( ) = ( ) + ( )



































max x i i

i

x y V xπ δ( ) + ( )




















V y x y V xx1 1 1( ) = ( ) + ( )max ~π δ

~V x x R x V R x1 1 2 1 2( ) = ( )( ) + ( )( )[ ]π δ
















7 Bayesian Games









A Ai==prod i

n

1

T Ti==prod i

n

0















i(i) Existence of






























39






π θih

H


1











40




72 Global Games









0

1

h(θ)

θ θ



and 1 at

= See figure 7










~a ge h() ~a lt h()








1

2


















θττ

micro τ ττ

γθ ε

ε

θ

εθ

θ ε

ε

1

sˆ ˆ+ ( ) minus+

τ τ θτ micro τ

θ εθ θ εε

θ ετ τs

+









ττ

πτ τ

τθθ

ε

θ ε

ε





11 2

1minus+

prime ltρρ

τπθ g









τ τ θτ micro τ

τ τθ εθ θ ε

θ ε

s+ minus

++

= γ




722 Applications




45



dsd

rr

rlowast

=part part

minus primegt part

partmicromicro

microθ

θ

θ

1


















h mmθ

λθθ

( ) + minus minus

11











































REFERENCES


















































































































a2

r2()

r1()

45deg

a1



















28

62 Markov Games













π i i ix y u x F x y ( ) = ( ) + ( )



































max x i i

i

x y V xπ δ( ) + ( )




















V y x y V xx1 1 1( ) = ( ) + ( )max ~π δ

~V x x R x V R x1 1 2 1 2( ) = ( )( ) + ( )( )[ ]π δ
















7 Bayesian Games









A Ai==prod i

n

1

T Ti==prod i

n

0















i(i) Existence of






























39






π θih

H


1











40




72 Global Games









0

1

h(θ)

θ θ



and 1 at

= See figure 7










~a ge h() ~a lt h()








1

2


















θττ

micro τ ττ

γθ ε

ε

θ

εθ

θ ε

ε

1

sˆ ˆ+ ( ) minus+

τ τ θτ micro τ

θ εθ θ εε

θ ετ τs

+









ττ

πτ τ

τθθ

ε

θ ε

ε





11 2

1minus+

prime ltρρ

τπθ g









τ τ θτ micro τ

τ τθ εθ θ ε

θ ε

s+ minus

++

= γ




722 Applications




45



dsd

rr

rlowast

=part part

minus primegt part

partmicromicro

microθ

θ

θ

1


















h mmθ

λθθ

( ) + minus minus

11











































REFERENCES

























































































































28

62 Markov Games













π i i ix y u x F x y ( ) = ( ) + ( )



































max x i i

i

x y V xπ δ( ) + ( )




















V y x y V xx1 1 1( ) = ( ) + ( )max ~π δ

~V x x R x V R x1 1 2 1 2( ) = ( )( ) + ( )( )[ ]π δ
















7 Bayesian Games









A Ai==prod i

n

1

T Ti==prod i

n

0















i(i) Existence of






























39






π θih

H


1











40




72 Global Games









0

1

h(θ)

θ θ



and 1 at

= See figure 7










~a ge h() ~a lt h()








1

2


















θττ

micro τ ττ

γθ ε

ε

θ

εθ

θ ε

ε

1

sˆ ˆ+ ( ) minus+

τ τ θτ micro τ

θ εθ θ εε

θ ετ τs

+









ττ

πτ τ

τθθ

ε

θ ε

ε





11 2

1minus+

prime ltρρ

τπθ g









τ τ θτ micro τ

τ τθ εθ θ ε

θ ε

s+ minus

++

= γ




722 Applications




45



dsd

rr

rlowast

=part part

minus primegt part

partmicromicro

microθ

θ

θ

1


















h mmθ

λθθ

( ) + minus minus

11











































REFERENCES
























































































































π i i ix y u x F x y ( ) = ( ) + ( )



































max x i i

i

x y V xπ δ( ) + ( )




















V y x y V xx1 1 1( ) = ( ) + ( )max ~π δ

~V x x R x V R x1 1 2 1 2( ) = ( )( ) + ( )( )[ ]π δ
















7 Bayesian Games









A Ai==prod i

n

1

T Ti==prod i

n

0















i(i) Existence of






























39






π θih

H


1











40




72 Global Games









0

1

h(θ)

θ θ



and 1 at

= See figure 7










~a ge h() ~a lt h()








1

2


















θττ

micro τ ττ

γθ ε

ε

θ

εθ

θ ε

ε

1

sˆ ˆ+ ( ) minus+

τ τ θτ micro τ

θ εθ θ εε

θ ετ τs

+









ττ

πτ τ

τθθ

ε

θ ε

ε





11 2

1minus+

prime ltρρ

τπθ g









τ τ θτ micro τ

τ τθ εθ θ ε

θ ε

s+ minus

++

= γ




722 Applications




45



dsd

rr

rlowast

=part part

minus primegt part

partmicromicro

microθ

θ

θ

1


















h mmθ

λθθ

( ) + minus minus

11











































REFERENCES






































































































































max x i i

i

x y V xπ δ( ) + ( )




















V y x y V xx1 1 1( ) = ( ) + ( )max ~π δ

~V x x R x V R x1 1 2 1 2( ) = ( )( ) + ( )( )[ ]π δ
















7 Bayesian Games









A Ai==prod i

n

1

T Ti==prod i

n

0















i(i) Existence of






























39






π θih

H


1











40




72 Global Games









0

1

h(θ)

θ θ



and 1 at

= See figure 7










~a ge h() ~a lt h()








1

2


















θττ

micro τ ττ

γθ ε

ε

θ

εθ

θ ε

ε

1

sˆ ˆ+ ( ) minus+

τ τ θτ micro τ

θ εθ θ εε

θ ετ τs

+









ττ

πτ τ

τθθ

ε

θ ε

ε





11 2

1minus+

prime ltρρ

τπθ g









τ τ θτ micro τ

τ τθ εθ θ ε

θ ε

s+ minus

++

= γ




722 Applications




45



dsd

rr

rlowast

=part part

minus primegt part

partmicromicro

microθ

θ

θ

1


















h mmθ

λθθ

( ) + minus minus

11











































REFERENCES

































































































































V y x y V xx1 1 1( ) = ( ) + ( )max ~π δ

~V x x R x V R x1 1 2 1 2( ) = ( )( ) + ( )( )[ ]π δ
















7 Bayesian Games









A Ai==prod i

n

1

T Ti==prod i

n

0















i(i) Existence of






























39






π θih

H


1











40




72 Global Games









0

1

h(θ)

θ θ



and 1 at

= See figure 7










~a ge h() ~a lt h()








1

2


















θττ

micro τ ττ

γθ ε

ε

θ

εθ

θ ε

ε

1

sˆ ˆ+ ( ) minus+

τ τ θτ micro τ

θ εθ θ εε

θ ετ τs

+









ττ

πτ τ

τθθ

ε

θ ε

ε





11 2

1minus+

prime ltρρ

τπθ g









τ τ θτ micro τ

τ τθ εθ θ ε

θ ε

s+ minus

++

= γ




722 Applications




45



dsd

rr

rlowast

=part part

minus primegt part

partmicromicro

microθ

θ

θ

1


















h mmθ

λθθ

( ) + minus minus

11











































REFERENCES

























































































































V y x y V xx1 1 1( ) = ( ) + ( )max ~π δ

~V x x R x V R x1 1 2 1 2( ) = ( )( ) + ( )( )[ ]π δ
















7 Bayesian Games









A Ai==prod i

n

1

T Ti==prod i

n

0















i(i) Existence of






























39






π θih

H


1











40




72 Global Games









0

1

h(θ)

θ θ



and 1 at

= See figure 7










~a ge h() ~a lt h()








1

2


















θττ

micro τ ττ

γθ ε

ε

θ

εθ

θ ε

ε

1

sˆ ˆ+ ( ) minus+

τ τ θτ micro τ

θ εθ θ εε

θ ετ τs

+









ττ

πτ τ

τθθ

ε

θ ε

ε





11 2

1minus+

prime ltρρ

τπθ g









τ τ θτ micro τ

τ τθ εθ θ ε

θ ε

s+ minus

++

= γ




722 Applications




45



dsd

rr

rlowast

=part part

minus primegt part

partmicromicro

microθ

θ

θ

1


















h mmθ

λθθ

( ) + minus minus

11











































REFERENCES
























































































































7 Bayesian Games









A Ai==prod i

n

1

T Ti==prod i

n

0















i(i) Existence of






























39






π θih

H


1











40




72 Global Games









0

1

h(θ)

θ θ



and 1 at

= See figure 7










~a ge h() ~a lt h()








1

2


















θττ

micro τ ττ

γθ ε

ε

θ

εθ

θ ε

ε

1

sˆ ˆ+ ( ) minus+

τ τ θτ micro τ

θ εθ θ εε

θ ετ τs

+









ττ

πτ τ

τθθ

ε

θ ε

ε





11 2

1minus+

prime ltρρ

τπθ g









τ τ θτ micro τ

τ τθ εθ θ ε

θ ε

s+ minus

++

= γ




722 Applications




45



dsd

rr

rlowast

=part part

minus primegt part

partmicromicro

microθ

θ

θ

1


















h mmθ

λθθ

( ) + minus minus

11











































REFERENCES






















































































































A Ai==prod i

n

1

T Ti==prod i

n

0















i(i) Existence of






























39






π θih

H


1











40




72 Global Games









0

1

h(θ)

θ θ



and 1 at

= See figure 7










~a ge h() ~a lt h()








1

2


















θττ

micro τ ττ

γθ ε

ε

θ

εθ

θ ε

ε

1

sˆ ˆ+ ( ) minus+

τ τ θτ micro τ

θ εθ θ εε

θ ετ τs

+









ττ

πτ τ

τθθ

ε

θ ε

ε





11 2

1minus+

prime ltρρ

τπθ g









τ τ θτ micro τ

τ τθ εθ θ ε

θ ε

s+ minus

++

= γ




722 Applications




45



dsd

rr

rlowast

=part part

minus primegt part

partmicromicro

microθ

θ

θ

1


















h mmθ

λθθ

( ) + minus minus

11











































REFERENCES
























































































































i(i) Existence of






























39






π θih

H


1











40




72 Global Games









0

1

h(θ)

θ θ



and 1 at

= See figure 7










~a ge h() ~a lt h()








1

2


















θττ

micro τ ττ

γθ ε

ε

θ

εθ

θ ε

ε

1

sˆ ˆ+ ( ) minus+

τ τ θτ micro τ

θ εθ θ εε

θ ετ τs

+









ττ

πτ τ

τθθ

ε

θ ε

ε





11 2

1minus+

prime ltρρ

τπθ g









τ τ θτ micro τ

τ τθ εθ θ ε

θ ε

s+ minus

++

= γ




722 Applications




45



dsd

rr

rlowast

=part part

minus primegt part

partmicromicro

microθ

θ

θ

1


















h mmθ

λθθ

( ) + minus minus

11











































REFERENCES






































































































































39






π θih

H


1











40




72 Global Games









0

1

h(θ)

θ θ



and 1 at

= See figure 7










~a ge h() ~a lt h()








1

2


















θττ

micro τ ττ

γθ ε

ε

θ

εθ

θ ε

ε

1

sˆ ˆ+ ( ) minus+

τ τ θτ micro τ

θ εθ θ εε

θ ετ τs

+









ττ

πτ τ

τθθ

ε

θ ε

ε





11 2

1minus+

prime ltρρ

τπθ g









τ τ θτ micro τ

τ τθ εθ θ ε

θ ε

s+ minus

++

= γ




722 Applications




45



dsd

rr

rlowast

=part part

minus primegt part

partmicromicro

microθ

θ

θ

1


















h mmθ

λθθ

( ) + minus minus

11











































REFERENCES


















































































































39






π θih

H


1











40




72 Global Games









0

1

h(θ)

θ θ



and 1 at

= See figure 7










~a ge h() ~a lt h()








1

2


















θττ

micro τ ττ

γθ ε

ε

θ

εθ

θ ε

ε

1

sˆ ˆ+ ( ) minus+

τ τ θτ micro τ

θ εθ θ εε

θ ετ τs

+









ττ

πτ τ

τθθ

ε

θ ε

ε





11 2

1minus+

prime ltρρ

τπθ g









τ τ θτ micro τ

τ τθ εθ θ ε

θ ε

s+ minus

++

= γ




722 Applications




45



dsd

rr

rlowast

=part part

minus primegt part

partmicromicro

microθ

θ

θ

1


















h mmθ

λθθ

( ) + minus minus

11











































REFERENCES



















































































































40




72 Global Games









0

1

h(θ)

θ θ



and 1 at

= See figure 7










~a ge h() ~a lt h()








1

2


















θττ

micro τ ττ

γθ ε

ε

θ

εθ

θ ε

ε

1

sˆ ˆ+ ( ) minus+

τ τ θτ micro τ

θ εθ θ εε

θ ετ τs

+









ττ

πτ τ

τθθ

ε

θ ε

ε





11 2

1minus+

prime ltρρ

τπθ g









τ τ θτ micro τ

τ τθ εθ θ ε

θ ε

s+ minus

++

= γ




722 Applications




45



dsd

rr

rlowast

=part part

minus primegt part

partmicromicro

microθ

θ

θ

1


















h mmθ

λθθ

( ) + minus minus

11











































REFERENCES


















































































































0

1

h(θ)

θ θ



and 1 at

= See figure 7










~a ge h() ~a lt h()








1

2


















θττ

micro τ ττ

γθ ε

ε

θ

εθ

θ ε

ε

1

sˆ ˆ+ ( ) minus+

τ τ θτ micro τ

θ εθ θ εε

θ ετ τs

+









ττ

πτ τ

τθθ

ε

θ ε

ε





11 2

1minus+

prime ltρρ

τπθ g









τ τ θτ micro τ

τ τθ εθ θ ε

θ ε

s+ minus

++

= γ




722 Applications




45



dsd

rr

rlowast

=part part

minus primegt part

partmicromicro

microθ

θ

θ

1


















h mmθ

λθθ

( ) + minus minus

11











































REFERENCES





















































































































1

2


















θττ

micro τ ττ

γθ ε

ε

θ

εθ

θ ε

ε

1

sˆ ˆ+ ( ) minus+

τ τ θτ micro τ

θ εθ θ εε

θ ετ τs

+









ττ

πτ τ

τθθ

ε

θ ε

ε





11 2

1minus+

prime ltρρ

τπθ g









τ τ θτ micro τ

τ τθ εθ θ ε

θ ε

s+ minus

++

= γ




722 Applications




45



dsd

rr

rlowast

=part part

minus primegt part

partmicromicro

microθ

θ

θ

1


















h mmθ

λθθ

( ) + minus minus

11











































REFERENCES






















































































































ττ

πτ τ

τθθ

ε

θ ε

ε





11 2

1minus+

prime ltρρ

τπθ g









τ τ θτ micro τ

τ τθ εθ θ ε

θ ε

s+ minus

++

= γ




722 Applications




45



dsd

rr

rlowast

=part part

minus primegt part

partmicromicro

microθ

θ

θ

1


















h mmθ

λθθ

( ) + minus minus

11











































REFERENCES





















































































































τ τ θτ micro τ

τ τθ εθ θ ε

θ ε

s+ minus

++

= γ




722 Applications




45



dsd

rr

rlowast

=part part

minus primegt part

partmicromicro

microθ

θ

θ

1


















h mmθ

λθθ

( ) + minus minus

11











































REFERENCES




























































































































h mmθ

λθθ

( ) + minus minus

11











































REFERENCES
























































































































































REFERENCES








































































































































REFERENCES


















































































































REFERENCES






















































































































































































































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Complementarities and Games: New Developmentsblog.iese.edu/xvives/files/2011/09/111.pdfComplementarities and Games: New Developments X AVIER V IVES 437 ... gic situations is the theory