EGT and economics

1.EGT and economics: I.Optimality principles and models of behaviour dynamics Alexander A. Vasin

Introduction

Classical game theory

Game theory is widely used for description and analysis of economic players' behavior in microeconomics, public sector economics, political economics and otherfieldsof economic theory.

General ideas of classical game-theoretic analysis:

Game in normal form(as a model of players interaction)

- Severalparticipants (players)

-Several possible strategies for each player

-Payoff function s

Principle of Nash equilibrium(as a method to define agents strategies during their interaction)

-Nash equilibrium isa basicconcept of game theory . T he optimal outcome of a game isone whereno player has an incentive to deviate fromhis or herchosen strategy afterconsidering thechoice s of other players .

Principle of elimination of dominated strategies

-Strategy is called dominated if there existsanalternative strategy which provides agreater gain nomatter what strategies are chosen by other players

-Domination principle means that rational players will not use dominated strategies

-Dominance elimination can be made iterative

3. Evolution arygame theory: considered problems

1. Correspondenceof real behavior of economic agentstoNash equilibrium and domina nceelimination principles

In typicalconflicts thesearch of Nash equilibrium and sets of nondominated strategies deliveressophisticatedmath tasks. It is necessary to know all sets of strategies and payoff functions for their solution (seemodels of Cournout and Bertrand economical competition) .

A u sual participant of such interaction has precise information only about his own strategies and payoff function and often doesn't know about mentioned decision-making principles

Whyshould weexpect that his behavior will be relevantto the principles ?

Justification by meansof models of adaptive and imitative behavior (MAIB)

These models show that convergence to Nash equilibrium and domina nceeliminationproceedfrom generalpropertiesof evolution ary , adaptative and imitative mechanisms of behavior formation

In this casecompleteinformation awarness and rationality in choosing strategies are not required. Itsufficestocompare thepayoff sforcurrentbehavior strategy and chosen alternative.

Definition of evolutionary stable strategy and itsrelation toNash equilibrium .

2. How to de termineplayers utility functions forparticularinteraction s ?Standard approach: use concepts of particular sciences.

In economics : 'homo economicus concept(P.Samuelson)

In the role ofaproduce r:profit maximisation

In the role ofaconsumer :maximisation ofthedemandutility

Discrepancy to real behavior: russian labour market .

Alternative: consider evolution of preferences (L. Samuelson)

Model of evolution arymechanisms natural selection

In this modela societyof interacting populations with differentevolutionarymechanisms is considered

Analysis of this model shows that if replication is intheset of competingmechanismsthen behavior dynamics is coordinated withmaximization ofindividualfitness (Ch. Darwin). What about human populations?

3. Do individualsadjust to other interestsor extensively influencepayoff function sof otherplayer s? (agressive advertising, drug distribution)

Population game

Population gameis a static model of interaction ina largehomogenuous group of individuals.

Thisconceptis an analog of a game in normal form in classic game theory, so general noncooperative optimality principles are summarized:

Nash equilibrium

D ominance s olution

Idea ofevolutionary stable strategyis added

Formally apopulation gameG isa setof parametres

G = < >, where

Jis theset of players strategies

- standart simplex

i s the payoff function for players that use strategyjunder strategydistribution and other parametres of the model(e.g.t otal population size and environmental conditions). For social populations, the payoff function usually corresponds to consumer utility, income or profit. In this paragraphthefunction is exogenous.

Example of population game (M.Smith)

C onsider pair competitivecontes ts for one resource. Individuals ofapopulation try to find the desired object (food, lodging or a female). Some of them get it without collision, others conflict in pairs, where one of them is theowner and another one isinvader

-theset sof strategies in each role

- profits of individuals if choses and-

-p robability of collision.Itdoesn't depend on startegies andisde termin ed bythe sizeof population N

- profit of individuals avoided collision

Strategy of individual is a pair, whereand. Itis a rule of behavior choice according to the role .

Functionshows the averagepayoffof individuals with strategyj

7. Letanddenote distribution s over alternatives for the bothrolesandwhich correspond to strategy distribution. Then ,And for strategy C onsideralsothe situation whenplayers don'tdistinguish their roles . Then the set of strategiesisthe set of alternatives:In this case game G is equivalent to the game T he set of behavioralternativesand individualpayoffs maynot depend ontheroleintheprevious model too. However ,behavior models for those similar situations are completely different. 8. Main staticoptimality principlesNash equilibrium for the populational game Gis a distribution *such that any startegy which is used with positive frequence is an optimalreply to thegiven distributionunderany parameter , i.e. (2.1) Let payoff functions in gameGbe expansible, i.e.as inthemodelof pair contests .Notethat a part ofthepayoff function that depends onthestrategy chosen by a player is independent of the parameter . Then (2.1) is equivalent to the following condition, which doesn't include the parameter : The concept of Nash equilibrium is the best-known optimality criterion used in strategic behavior modeling. However, it is known from analysis of dynamic models, that among Nash equilibrium there can exist unstable states, thatare notrealized in practice. For this reason we alsoconsiderstrong eroptimality criteria. 9. Evolutionary stable strategy (ESS)for the populational game Gis a distribution * such thatistheaveragepayofffor mixed strategy or distribution when individuals inthepopulation are distributed in pure strategies according to . The concept of ESS can be interpreted in the following way. Let a small group of 'mutants' with strategy distribution be implemented in a stable population * . If distribution *is evolutionary stable then the implemented group can't fix inthepopulation, because its average fitness is less thenthefitness of initial strategy *. 10. Any ES Sis Nash equilibri um . Ifis notanequilibrium then mutants with pure strategy of bestreply to havegrater payoffthantheaveragepayoffinthemain population. Due to persistance offo ver it is correct for any group of such mutants. Thisstatement is correct if theshareofoneindividual inthepopulation is negligible , i.e. change of his strategy doesn't influnce payoff s order .Otherwise it is necessary torevisetheESS concept(Schaffer,1989). Forasymmetric game in normal form withnplayers, set of strategiesSand payoff function, ESS is defined as a symmetric situation suchthatany strategy changebyany player do es n't make his profitmorethantheprofit of other players withtheformer strategy.I.e.amutant doesn't have any profit benefits.S uchESS can be not a Nash equilibrium . In particular, in the game which correspond to symmetric Cournout oligopoly, players use 'market power' in Nash equilibrium andrealiselowerproduction volumesin comparison withthecompetitive equilibrium, whiletheESS corresponds tothecompetitive equilibrium. 11. Strict equilibriumof population gameGis distribution * such thatall players use one str a tegy, which is the bestreply to itself : Note, thatany strict equilibrium is ESS , even for groups withsufficiently large finite size. Selten(1988) showed that there are no ESS exceptforstrict equilibri a forrandom contest s with rolea symmetry of players ('owner' 'invader'). For general payoff functionsNash equilibriummay not exist . In other classesof gamesthere are a lot of equilibri a , some of them are unstable. In this case another optimality principle is appealing dominationthat isalsointernally bound to the Darwinian principle of natural selection. 12. Strategyjdominates strategyi on the set of distributionsif for any distributionover strategiesthe strategyjprovides a greater gainthan strategyiThe set is called adominating setif it can be obtained by iterativeexclusionof dominated strategies , i.e. there exists integerT >1such as The described procedure for iterative elimination of dominated strategies can be considered as a quasi-dynamic model of behavior microevolution within a population.Indeed, this procedure describes a sequential reduction of the set of strategies used by players: at each stage, more efficient (better fitted) strategies are substituted for less efficient ones. If >0thenstrategyjstrictly dominatesstrategyi ( ),J'is a strictly dominating set . Concepts of domination by mixed strategy and a set domination in mixed strategies are analogue 13. Search of Nash equilibrium and dominating sets ofapopulation game is generally asophisticated optimization problem .For random pair contests i t is possible toreducethem to the known computational tasksforappropriatebimatrix games. Proposition 2.1Distribution *is a Nash equilibrium of gamesuch thatand competitors don't differ conditions if and only if( * , *) isNash equilibrium in mixed strategies of symmetrical bimatrix game, i.e Proposition 2.2Distribution *such that is strict equilibrium of gameif and only if for any ,i.e.(s,s)is strict symmetric Nash equilibrium of gameinpurestrategies. Proposition 2.3Str a tegysdominates strategyr()in gameif and only ifin game , i.efor anyProposition 2.4Distributionis Nas hequilibrium of game G for assymetric paircontestsif and only ifis Nash equilibrium in mixed strategies of game = = 14.

So,for any random pair collisions Nash equilibrium of population game correspond to Na shequilibrium of bimatrix gamethat describes pair interaction

Analoguerela tion exists for random collisions withlarger numberof participants whenaseparate local interaction is characterizedbythe game of n participants.Theresults are easilygeneral ized in case of interpopulation collisions when individuals fromdifferentpopulations or social groups ('predator-prey', 'employer-employees' etc) play different roles .

The main condition of correspondence isindependenceof distributionin interacting groups from players' strategies

15. Model of adaptive- imit ative behavior (MA I B)

In which cases adeptive- imit ative mechanisms form population behavior that corresponds to Nash principle and principle ofdominatedstrategy elimination?

Let population gamedescribe interaction of population individuals that happens continiously, at every moment of time.

The numberof individuals and external factors arefixed ( do not depend on )

With intensity which depends on current player distributionoverst ra tegies and current payoff vectora player with strategyjturns into 'adaptive' status whererec onsiders his behavior

In adaptive status player with startegyjchoosesias alternative with probabilit y

Current and alternative strategies are compared. If chosen for comparison strategyiis better then initial strategyj(i.e. gives individual more profitunderthisdistribution over strategies)then the player changes his strategy toiwith probability

16. Thenis an averageshareof players who change their strategyjto strategy from setin a moment of time,-the shareof players who change their strategy from settoj . So, equations of distribution dynamics look like(2.2) Functionsare such asMentioned conditions guarantee that pathdoesn't come out set at any moment of timetand with any initial distribution 17. MA I B. Examples. Example 1.Let the intensity of status change to adaptive be constant. Alternative strategy is chosen bymeansof random imitation. And probability of changing current strategy to alternative isproportional to the difference between thecorresponding payoff functions. So,And (2.2) looks like This system is an analogue of autonomus continuous model of replicator dynamics (see part 3) Example 2.Alternative strategy is chosen with equal probabilities fromtheset of possible strategies, i.e.This example illustrates the mechanism of individual adaptation when each player knowsthe wholeset of possible strategies, and adaptation happens according to current payoffvalues . Adaptation doesn't depend on behavior of other population individuals. It is evident that there are many different MA I Bs.The f ollowing theoremsrelateMA I B stablestate s and solution softhecorresponding population game . Note thatany Nash equilibrium of population game G isa steady stateof MA I B. 18.

Relation of MA I B stablestatesand population game solutions

Denoteas a set ofbest replies todistribution.

Theorem 2.1:Let MA I Bmeet the followingconditions 1), 2) and 3) or 3') :

1)For anyand any

(intensity of changing status to adaptive is positive for all strategies)

2) For anyfunctionslook likewhere for any

(probability of strategy choice as alternative is function of corresponding payoffdifferenceandispositivefor apositive argument )

3)For any

(probability of strategy choice as alternative is positive for any strategy that gives maximum payoffunder thecurrent distributionoverstrategies)

3')For any andq>0

(for any pure strategy with maximum payoff, probability of this strategy choice as alternative is not less thanits shareinthepopulation multipliedbysome constant)

any stable(Lyapunov)point *of system (2.2) is Nash equilibrium of population game

b) if initial distributionand for paththere existsthen *is Nas hequilibriu m ,f or gameG

c) if *is a point of strict equilibrium for population gameGthen *is an assymptoticaly stablepointof system(2.2)

Theorem 2.2: Let MA I B (2.2)meetconditions 1 and 2 of theorem2.1and moreover

for any

(alternative strategy ischosen byrandom imitation)

2. ifthen

(intensity of changing status to adaptivedecreaseswith the raise of pa y off function)

3. increase smonotonicallyin x

(probability of strategy choice as alternative rises monotonically forpayoff remainder)

Ifisastrictly dominating set of strategies in populational gamethen ,for anyand initialdistribution

Other variants of such consistency conditions of MA I B dynamics with Nash and domination decisions (see Samuelson L., Zhang J.(1992) and Weibull (1995))relate to theconcept of monotonous dynamicss.t.

At the same time thereexistadaptation models that don'tmeettheorem s2.1 and 2.2. Models of natural selection of evolutional mechanisms considered in the next part explain whywe nevertheless should expectcoherence of real behavior dynamics withthenoted optimality principles. Moreover,payofffunctions of players are endogeneously defined in the frames of these models.

Replicator Dynamics Model (RDM)

Population is characterized with a setSof possible strategies

Strategy distribution of individuals at the current moment is set by vector

Individuals differ only inbehavior strateg ies , they don't change it during their life, strategy is inherited

In case of populations with both males an females, each of them should be considered asaseparate population

Genetical mechanism of inheritance : thestrategy isdeterminedby genes, connected withsex . Mechanism of imitation : thestrategy is defined by imitating behavior oftheparent ofthe samesex.

The r esult oftheinteraction in population during a period of time is characterized for players with strategys byfertility functionthatdetermines the average number ofoffsprings. It is also characterizedbysurvival functionthatdetermines the probability tosurviv e underdistributionand populationsize N .

21. Denote- the numberof those who use strategys. Then popultion dynamics, meetsthe following system: (3.1) Where is astrategy fitness function . It formalizes the Darwinian concept of individual fitness. At first sight a concept of payoff function is inapplicable to this model. Players strategies are fixed, the are not approaching to smt and don't choose anything. However,thepicture changesif welook at strategy distribution dynamics. The following theorem shows thatbehavior assymptotics in such population correspond sto fitness asapayoff functionfor this population .Particularly, if forstrategy distribution approaches to stationary, then there are only those strategies in population that maximize fitness (corresponds to Darwinian principle of natural selection: only most fitted survive). If in any distribution one strategy fits better than another, then a part oftheworst strategy in distribution (t)approaches to 0 while. In this case fitness is endogenuous objective function of this model. 22. Relation of Nash equilibr ia withstable MRD points Asymptotic stability of ES S Relation of dominating sets of strateg ieswith behavior dynamics Theorem 3.1 (on relation of Nash equilibriumwithstable points of RDM):Assume that the fitness functionis reperesentable in additive formThen1) any stable (Lyapunov) distribution * of system (3.1) is a Nash equilibrium in population game2) if for a certain path, the initial di s tributionandthen *is a Nash equilibrium of the specified population game Note that system (3.1) isn't closed, because its right part also depends on N(t). Conception of stable distribution for such system is formally defined in Bogdanov, Vasin (2002) Theorem 3.2 (on asymptotic stability of ESS):Assume that in theorem 3.1 *is evolutionary stable strategyfor population game. Then *is an asymptotically stable distribution of system (3.1) Theorem 3.3 (on the relation between dominating sets of strategies and behaviordynamics ):Assume thatSis a strictly dominating set of strategies in the game.Then, for anyand anyon the corresponding path of system 3.1 23. Random imitation Replicator Dynamics Modeldescribesactivity of evolution arymechanism that provides direct inheritance of strategies by children. In what degree are thestatedresults depend onconcrete evolution arymechanism ?Turns out thatit plays a very important role . Let's considermechanism of random imitationasanalternative exampleThis model differs from replicator dynamics only in one point : new individuals do not inherit strategy, they follow a strategy of randomly chosen adult.Then the population dynamics are describedSuch system dynamic corresponds topayrollfunctionin the sense of theorems 3.1-3.3. Thus, viability turns out to be an endogenous payoff function of individuals in the corresponding dynamical process. Proceeding from the previous example it seems that we have exchanged arbitrariness in the choice of payoff functions for arbitrariness in the choice of evolutionary mechanisms. However, actual evolutionary mechanisms are subject to natural selection. Only the most efficient mechanisms survive in the process of competition. 24.

Model of evolutionary mechanism scompetition

Consider the correspondingmodel of a society that includes several populationsthat differ only in their evolutionary mechanisms.

Individuals of all populations interact and do not distinguish population characters in this process. Thus, the evolutionary mechanism of an individual is an unobservable characteristic.

Fertility and viability functions describe the outcome of the interaction for each strategy and depend on the total distribution over strategies and the size of the society.

The set of strategiesSand the functions are the same for all populations.

Denote

L the set of populations

- the size of populationl

N the total population size

- distribution over strategies in populationl.

Then the total distribution over strategies is

Assume that operatorcorresponds to the evolutionary mechanism of populationland determines the dynamics of distribution.

(For example, in one population it is replication dynamics, in an other - random imitation, and so on. Particularly, dynamics may relate to maximization of some payoff function)

25. Then the dynamics of the society are governed by equations (3.2) Theorem 3.4.Let there exists a population of replicators in the society. Then the total distribution(t)over strategies meets the following analogs of theorems 3.1 and 3.2: 1) any stable distributionof system (3.2)is a Nash equilibrium of the population game2) if for pathinitial distributionandthen *is a Nash equilibrium of the population game G for path3) ifis a strict equilibrium of the gameGthenis an asymptotically stable distribution for system (3.2). Thus, the evolutionary mechanisms selection model confirms that individual fitness is an endogenous utility function for self-reproducing populations. 26.

Idea of proof of 1 stand 2d statements is quite easy: if stationary distribution over strategies is not Nash equilibrium of fitness function then nothing can prevent expansion of replicators that usethe best replystrategy. That is a contradiction to its stability

G eneralization of theorem 3 for elimination of dominated strategies is possible under more strict assumptions on the variety of evolutionary mechanisms. For any evolutionary mechanismand a pair of strategiess,r , let us call ans,r -substitute of mechanisma mechanism such that for strategies other thans and rthe shares of individuals who apply these strategies change as under mechanism, except that instead of strategysthey always playr . According to (Vasin, 1995), if for anys,r,lthe set of mechanisms includes all possible substitutes of, thena s for any strictly dominated strategys.

The r esult is stated for homogenuous population, without taking sex or age in consideration. It is easily generalized for populations with such structures. Fitness analogue in this case isa rateof balanced growth ofthepopulation, it isdetermined by theFrobenius number oftheLeslie matrix (see Semevskiy, Semenov, 1982)

Conclusion

Models and results of evolutionary game theory show that behavior evolution ina self-reproducingpopulation corresponds to well-known optimality principles Nash equilibrium and elimination of dominated strategies

Endogenously formed payoff function corresponds to Darwinian concept of individual fitness

Problems

However in biological and social populations cooperative and altruistic behavior are well-know n . It seems that they don't correspond to optimization of individual fitness

Problem of stability of mixed equilibrium, where more than one pure strategy is used with positive probability.This problem appears in case of interpopulation interaction where payoff for one population depends on distribution over strategies in another population. It also appears in case of randomcontestswith role asymmetry between participants. For such games mixed Nash equilibr iaare never evolutionary stable and strict equilibr ia may notexist. So, sufficient conditions of stability don't work

Relevanceof mentioned evolutiona rymodels to social populations. Superindividual is a self-rep roducingstructure that uses human population as a source forits reproduction . It can influence behavior dynamics in this population

28. II . Stability of equilibrium. Pecularities of social behavior evolution. 29. Stability problem of mixed strategies Let's consider game of two populations with sets of strategies and and payoff functionsand that show interaction result for all strategies Assume thatindividuals of the 1 stpopulation interact only with individuals of secon population and vice versa: individuals of the 2 ndpopulation interact only with individuals of the 1 st.At any moment of timeteach individual uses a chosen strategy Assume that arepopulation distribution sover strategies. PointisNash equilibrium of game if for anyi, jEquilibrium is calledmixedif for anyi, j 30.

It is easy to see that

For any game with indescrete payoff function there exist a Nash equilibrium

In nonsingular case amount of positive coordinatespandqis the same

p(t)andq(t)change according to the system

This system is called-if

1) Functionsandare such as

f or any distributionsand payoff vectors

That means that any Nash equilibrium is a stable point of system (4.1)

2) Functionscanddare measurable functions overtand continiously differentiable overp(0)andq(0).Derivatives are equibounded overt .

3) Setis invariant of system (4.1). Vector-functionsA, B, G and Hare continiously differentiable.

Note that MAIB,and a system with positive functions ..generally correspond to these assumptions

Note that system (4.1) can be converted to autonomous system

If for anyt, p(0), q(0)

This occasion takes place during interaction between populations of different sizes or between individuals of different roles in one population, for example, between 'owner' and 'invader' (Maynard Smith, 1982)

Let's consider game , system (4.1) and corresponding autonomous system (4.2)

Stable point of system (4.2) is calledsingularif there is a latent root of Jacobian matrix thatisequal to 0

Point is calledcentreif for any latent rootRe =0, Im 0

Point is calledsaddleif there is a latent root such thatRe >0

Theorem 4.1:Any mixed equilibrium(P*, Q*)is either

Singular point

Or centre point

Or saddle point of (4.2)

in this case(P*, Q*)is unstable point of a system (4.1) for any available functionsc, d.

This theorem doesn't solve the question of stability of'centre' points for which all latent roots of linearized matrix are purely imaginary. Let's use method that was deveped in work of Ritzberger and Vogelsberger in 1990 and based on Liouville theorem.

Consider system.According to this theorem, the field that is free of divergence ( ) saves any volume constant and can't be in asymptotically stable condition.

Let's describe MAIB class such that the shown method allows to make a conclusion about abscence of asymptotically stable equilibriums.

General MAIB equation:

Theorem 4.2:Assume that interpopulation MAIB (2.2) is such that

don't dependon(intensity of change to adaptive status and probability of strategy change don't depend on distribution over strategies in this population. However, it can depend on distribution over strategies in another population taking part in interacrion)

2)(alternative strategy is chosen with the way of random imtation of other individuals from population)

Any mixed equilibrium is asymptotically stable

If probability of strategy choice as of alternative for population individuals doen't depend on distribution over strategies in this population then in general assumption divergention of vector field of right parts of equation is negative an it is possible to consider convergence to mixed equilibrium

Problem of convergence to the mixed equilibrium has been considered in different literature also for iteration and indiscrete processes such as fictious play and, particularly, for Braun process for the game in normal form.

Denoteas mixed strategies on stept . Thendiscrete Braun process is described with such correlations :

whereis payoff function of playerain mixed strategies

is a set of singular mixed strategies of playerathat corresponds to a set of its pure strategies

Speaking about behavior dynamics in interaction population, process can be interpreted as adaptive in such way:after every period t>11/t part of each population changes its strategy to one of the best answers.

Braun offered and Robinson proved convergence of discrete process for antagonistiv bimatrix games

Danskin showed convergence of this process for antagonistic games with indescrete payoffs over multiplications of compact spaces

Fudenberg and Kreps showed convergence of adaptive play model for nonantagonistic bimatrix game 2x2 with one pure mixed Nash equilibrium

and Hirsch developed result for 2x2 games with several nash equilibrium (can be both mixed or not)

In Bogdanov's work that was done on the basis of Belenkiy and others results general rules for convergence of such processes for bimatrix games are shown. It is proved that convergence can be guaranteed for all bimatrix games that are converted to antagonistic with the following development:

a) adding constant to the column of payoff matrix of 1 stplayer

b)adding constant to the line of payoff matrix of 2 ndplayer

c)multiplication of payoff matrix and positive constant

It is known that those transformations set a game class with the same sets of Nash equilibrium

However thes results can't be widespread for nonantagonistic games of bif dimension

36. Process of fictious play doesn't convergence for Shepley example, where payoff matrix: If pure strategiesare taken as initial point then choice of players at other moments of time will follow a cycle with six strategies: (1,1) -> (1,3) -> (3,3) -> (3,2) -> (2,2) -> (2,1) -> (1,1) At that amount of periods, where process is in each of these conditions will rise exponentially. Obviously the process of fictious play doesn't convergence. For some nonantagonistic games, particularly for Shepley example, paths of fictious play process behave as average over time RDM. It can be shown that RDM is equivalent to MAIB particular case that corresponds to the theorem 4.2. At the same time not only paths of these MAIB but also their average over time don't convergence to equilibrium in Shepley example. 37. Best properties (in the meaning of convergence to mixed equilibrium) are shown by some more complex adaptive dynamics. Let's consider the following modification of indiscrete process of fictious play for the game of two players: (4.3) Whereis appraisal of mixed strategy of playeraat the moment of time. is the best answer of playeraover partners' strategyThe idea is thatapproximateswhenis quite big, i.e. The best answer is done for the future strategy. For Shepley example0,0413< /(1- )

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EGT and economics