HEC Paris. Majeure Economie, 2009-2010. Tristan Tomala.

Introduction to Game Theory.

This document summarizes the course: main definitions and results are givenas well as exercises. Most exercises illustrate important applications of thetheory. Classical textbooks are:

”Introduction to game theory”, by M.J. Osborne.”A course in game theory”, by M.J. Osborne and A. Rubinstein.”Game theory”, by D. Fudenberg and J. Tirole.

One can also consult the game theory sections of the following microeco-nomics textbooks:

”A course in microeconomic theory”, by D. Kreps. (French version, PUF).”Microeconomic theory”, by Mas-Colell, Whinston and Green.

1 Summary

1. What is game theory? What is a game? What is a player? Whatdoes ’strategic’ mean in game theory? A bit of history. Main fields ofapplication of game theory.

2. Preferences and Choices. Individual and collective choices. Pareto-optimality, aggregation.

3. Simultaneous move games. Why simultaneous move games? Classicalexamples of games. Dominant and dominated strategies. Nash equi-libria. Mixed strategies. Applications in oligopoly theory.

4. Sequential move games. Games of perfect information. Why do we playchess? Games of imperfect information. Back to simultaneous games.Repetition. Applications: negociation, Stackelberg competition.

5. Bayesian games. Games with incomplete information. What is the dif-ference between incomplete and imperfect information? Applications:auction theory, reputation.

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6. Mechanism design. The importance of the rules of the game in eco-nomic or social interactions, the revelation principle. Applications:voting, auctions, partnership, public goods.

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2 Introduction

Some dates.Pre-history:

• 1838, Cournot.

• 1871, Darwin.

• 1912, Zermelo: a theorem on Chess.

• 1921, Borel.

• 1928, Von Neumann.

History:

• 1944, Games and Economic Behavior by John Von Neumann and OskarMorgenstern.

• 1950, John Nash: Nash equilibrium.

• 1953, H.W. Kuhn: Extensive form games.

• 1967, J.C. Harsanyi: Games with incomplete information.

• 1976, Aumann and Shapley: the Folk Theorem.

Nobel Prizes (economics):

• 1994: Nash-Harsanyi-Selten.

• 2005: Aumann-Shelling.

• 2007: Hurwicz-Maskin-Myerson.

Concepts:

• Rational agent.

• Multi-player model: strategic thinking, common knowledge of rational-ity.

• Normal form games: reduction to simultaneous games.

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• Extensive games: modelling of dynamic strategies.

Applications:

• Biology, Computer Science.

• Political Science, Voting.

• Economics: Oligopoly, auctions, negociation, industrial organization,economics of information, contracts.

3 Preferences and choices

3.1 Individual preferences

Let X be a set of outcomes.

Definition 3.1 • A preference relation on X is a relation which is com-plete and transitive.

• A preference relation is represented by the utility function u : X → Rif,

x preferred to y iff u(x) ≥ u(y).

Properties.

• A preference relation on a finite (countable) set is always representedby a utility function.

• The utility function is not unique (increasing transformation).

• Some preferences are not representable (eg. lexicographic preferenceon [0, 1]× [0, 1]).

Theorem 3.2 Let X = Rn. The preference is continuous if the sets {y :y � x} and {y : y ≺ x} are open. Then it is representable by a continuousutility function.

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Choices Let X be a set of outcomes. A choice function over X is a mappingc which associates to every non-empty subset A of X another subset c(A)such that ∅ 6= c(A) ⊆ A. c(A) are the choices of the agent when she is giventhe option to choose an element of A.

A preference relation induces a choice function: c(A) is the set of elementsin A which are maximal for the relation among A (these are the best optionsin A according to the preference). In terms of utility, these are the optionsthat maximize the utility over A.

Conversely, one can derive a preference relation from a choice function asfollows: x is preferred to y iff x ∈ c({x, y}).

Exercise 1

Prove that a preference relation is complete and transitive iff the associatedchoice function satisfies the following two conditions:

(α) x ∈ B ⊂ A, x ∈ c(A) =⇒ x ∈ c(B).(β) x, y ∈ c(B), B ⊂ A, y ∈ c(A) =⇒ x ∈ c(A).

3.2 Collective choices

The question here is how to define a good outcome in a multi-agent context.Let X be a set of outcomes and N be a set of players. Let ui : X → R be autility function for player i (or a preference relation).

• An outcome x is Pareto-dominated if there exists an outcome y suchthat for each player i, ui(y) ≥ ui(x) with a strict inequality for at leastone player.

• An outcome is Pareto-optimal if it is not Pareto-dominated.

Remarks. Pareto optimality says nothing on equality or equity. SeveralPareto-optima may exist. Pareto-optima may be graphically represented inthe utility space.

3.3 Aggregation

How to go from individual preferences and choices to collective ones?

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Arrow’s impossibility theorem Let X be a finite set of outcomes andn be a number of players. Let R be the set of preferences over X. Anaggregation rule is a mapping from Rn to R which associates a social pref-erence to each profile of individual prerefences. For a profile of preferencesR = (R1, . . . , Rn), let Rs be the associated social preference.

• The aggregation rule is Paretian if, whenever all players prefer x to y,so does society.

• The aggregation rule satisifies independance of irrelevant alternativesIIA if the social ranking of x and y depends only on the individualrankings of these two outcomes.

• The aggregation rule is dictatorial if there is a player such that the socialpreference coincides with this player’s preference (for each profile).

Theorem 3.3 The only aggregation rules which are Paretian and satisfy IIAare the dictatorial ones. (Arrow’s Theorem).

Gibbard-Satterthwaite theorem Instead of aggregating preferences, onemay want simply to implement an outcome. Let X be a finite set of outcomesand n be a number of players. Let R be the set of preferences over X. Asocial choice function is a mapping f : Rn → X.

• f is Paretian if it selects only Pareto-optimal outcomes.

• f is Strategy-proof if for every player i, every preferences Ri, R′i of player

i and preferences R−i for the other players, player i of preference Ri

prefers f(Ri, R−i) to f(R′i, R−i).

• f is Dictatorial if there exists a player i such that f selects the outcomethat i prefers.

Theorem 3.4 If X is finite with at least three elements, the only socialchoice functions which are Paretian, strategy-proof and onto are the dictato-rial ones. (GS Theorem).

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4 Simultaneous move games

Definition 4.1 A n-player game is given by:

• A set of players N = {1, . . . , n}.

• A set of strategies Si for each player i. S = ×iSi is the set of strategyprofiles.

• A utility or payoff function ui : S → R for each player i.

Each player selects a strategy in her strategy set. Choices are simultaneous.Payoffs depend on the profile of strategies chosen.

Players are assumed: - to be rational (to maximize their payoff), - toknow the game and that other players are rational (common knowledge ofrationality and of the game).

Classical games.

C DC 3, 3 0, 3D 4, 0 1, 1

Prisoner’s Dilemma

F TF 2, 1 0, 0T 0, 0 1, 2

Battle of the Sexes

H TH 1,−1 −1, 1T −1, 1 1,−1

Matching Pennies

S GS 2, 2 1, 3G 3, 1 0, 0

Chicken

The Cournot oligopoly model. Auctions (see exercises).

4.1 Dominant/dominated strategies

Definition 4.2

• A strategy si of player i is dominant if for every other strategy ti andevery strategy profile of the other players s−i,

ui(si, s−i) ≥ ui(ti, s−i).

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• A strategy is weakly dominant if it is dominant and there is at leastone strategy profile of the other players s−i such that

ui(si, s−i) > ui(ti, s−i).

• A strategy si of player i is strictly dominant if for every other strategyti and every strategy profile of the other players s−i,

ui(si, s−i) > ui(ti, s−i).

Properties. Strictly dominant =⇒ weakly dominant =⇒ dominant. Theremay not exist a dominant strategy. There cannot be more than one weaklydominant strategy.

Definition 4.3

• A strategy si of player i is dominated if there exists another strategy tisuch that for every strategy profile of the other players s−i,

ui(si, s−i) ≤ ui(ti, s−i).

• A strategy is weakly dominated if it is dominated and there is at leastone strategy profile of the other players s−i such that

ui(si, s−i) < ui(ti, s−i).

• A strategy si of player i is strictly dominated if there exists anotherstrategy ti such that for every strategy profile of the other players s−i,

ui(si, s−i) < ui(ti, s−i).

Strictly dominated implies weakly dominated implies dominated. Theremay not exist a dominated strategy.

Iterated elimination of strictly dominated strategies (IEDS).

• Start with the initial game. Each player deletes her strictly dominatedstrategies. Consider the game with the remaining strategies.

• Iterate the deletion as long as at least one player finds a strictly domi-nated strategy is her strategy set.

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• If only one strategy remains for each player, the game is said to besolvable by IEDS.

Property. The order of deletion of strictly dominated strategies does notchange the outcome.

WARNING: This is not the case for deletion of weakly dominated strate-gies.

4.2 Exercises

Exercise 2

In a second-price auction, n-player compete for buying an indivisible object.The worth of the object is vi for player i, so that her utility is vi − p if shepurchases the object at price p and 0 if she does not purchase it. Each playersubmits a sealed bid to the auctioneer. The winner is the player that submitsthe highest bids and she pays the highest price among the other players (tiesare broken by the throw of a dice). Show that the bid bi = vi is a weaklydominant strategy. Show that this game is not solvable by IEDS (consider 2players, v1 = 1, v2 = 2, and restrict the possible bids to 0, 1, 2, 3).

Exercise 3

The auction setting is a collective choice problem where an outcome is: whogets the object and who pays what. Does the GS theorem apply to thisproblem?

Let us identify a preference with a valuation. Can you find a Strat-egy proof and Non-dictatorial social choice function which selects a Pareto-optimal outcome?

Exercise 4

There are n players. Each player submits a number in [0, 100] in a sealedenveloppe. The goal is to be as close as possible to half of the average of thenumbers chosen. Solve this game by IEDS.

Exercise 5

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A Cournot game is a n-player oligopoly game where: each competitor decidesthe quantity qi she produces (all produce the same good). The market priceis a decreasing function P (Q) of the total quantity Q =

∑i qi. Player i

substract a production cost ci(qi) from her profit.A linear Cournot game is such that: P (Q) = (A−Q)+ and ci(q) = Ciq.Assume n = 2, C1 = C2 (=0 for simplicity) and solve the game by IEDS.

Explain why it is not possible with three players.

Exercise 6

Show that in a simple majority rule voting procedure between two candidates,voting for one’s favorite candidate is a weakly dominant strategy (assume noindifferences). Give an example with three candidates where this fails.

Assume now that voters have to choose between m policies which lie ona uni-dimensional axis (say from Left-wing to Right-wing): each policy isidentified with a precise spot on the axis. Each voter has a favorite policyand her utility is a decreasing function of the distance from her favorite policyto the one actually implemented.

The voting procedure is the following: each voter names a policy and themedian policy is implemented (the one such that half voters stand at theleft and half at the right: assume for simplicity that the number of voters isodd).

Prove that it is a weakly dominant strategy for each voter to name herfavorite policy. Does this property hold if the average policy is implemented?

Relate this exercise to the GS theorem.

4.3 Nash equilibria

Definition 4.4 A Nash equilibrium is a profile of strategies s such that foreach player i and strategy ti,

ui(si, s−i) ≥ ui(ti, s−i)

Equivalent formulation.

• si is a Best-Reply to s−i if ui(si, s−i) ≥ ui(ti, s−i), ∀ti.

• si is a Nash equilibrium if and only if each player plays a best-reply tothe strategies of her opponents.

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Or,

• There does not exist a player i and a profitable deviation ti for thisplayer, ui(ti, s−i) > ui(si, s−i).

Exercise 7

Consider the 2-player linear Cournot game with zero unit cost. Draw thebest-reply curves. Deduce the Nash equilibrium.

Prove also that if the players alternatively play a best-reply to the strategyof the opponent, the strategies eventually converge to the Nash equilibrium.

Exercise 8

Price competition. In a Bertrand game, the structure of the market (demand,costs) is the same as in the Cournot game. Each competitor announces theprice at which she’s selling the good. The firm quoting the lowest priceserves all the demand (in case of equality the market splits equally). Assumesymmetric and linear cost and prove that there is a unique equilibrium tothis game where each firm sells at the marginal cost. What if the marginalcosts are different?

4.4 Mixed strategies

Definition 4.5 • A mixed strategy is a probability distribution over theset of (pure) strategies.

• The game played in mixed strategies is as follows: each player choosesa mixed strategy and draws a pure strategy at random. Random drawsare independent across players. The payoff is the expected payoff.

Theorem 4.6 Every game with finite action sets admits at least one equi-librium in mixed strategies. (Nash Theorem).

Proposition 4.7 A profile of mixed strategies is a Nash equilibrium if andonly if every pure strategy of player i that is played with positive probabilityis a best-reply to the strategies of the opponent. (Indifference principle).

Remarks:

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• In particular, all strategies played with positive probability yield thesame expected payoff.

• A strictly dominated strategy is not played at equilibrium. IEDS doesnot affect the set of equilibria.

• One may iteratively delete strategies which are strictly dominated bysome mixed strategy. Equilibria are not affected.

4.5 exercises

Exercise 9

Draw the best-reply curves and find the equilibria for: the prisonner’s dilemma,the Battle of the sexes, Matching Pennies. Find the equilibria. Find themagain using the indifference principle.

Exercise 10

Let t ∈ R be a parameter. Consider the game where each player has twoactions a, b. The payoff of the two players are the same and equal to: 0 ifthey play different actions, t if they both play a, 1 − t if they both play b.Write the matrix of this game. Compute the equilibria for each value of t.Draw the graph of the correspondence between t and the equilibria of theassociated game.

Exercise 11

The minority game. There are three players. Each of them has to choosebetween two options A, B. A player gets 1 if no other player chose thesame option as her, and 0 otherwise. Find all mixed equilibria of this game.(Hint: remark that there exists no equilibrium such that exactly two playersrandomize.)

Exercise 12

Two competing firms race for a prize. The one who invests more gets theprize V > 0. Investment is irreversible and the invested money is lost, nomatter the outcome. In case of equal investment, the prize is equally shared.Show that there is no equilibrium in pure strategies (the strategy set is [0, V ]).Can you find an equilibrium in mixed strategies?

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Exercise 13

There is an infinite sequence of players. Each can either stay home (0) or goto the beach (1). The payoff of a player who stays home is 0. The payoff ofa player who goes to the beach is 1 if the beach is not too crowded (i.e. thenumber of persons on the beach is finite) and -1 if the beach is too crowded(i.e. the number of persons on the beach is infinite). Show that this gamehas no equilibrium. First show this for pure strategies. Extend to mixedstrategies using the following: If (Xi)i is a sequence of independent binaryrandom variables, the probability that

∑i Xi = +∞ is either 0 or 1 (Borel

Cantelli’s lemma).

Exercise 14

Two players have to share N euros. Each of them demands an amount(integer). If the demands are feasible, the sharing is implemented. Otherwise,the player naming the least amount is served, the other takes the rest. Incase of unfeasible equal demands, the money is equally shared. Draw thepayoff matrix of this game for some values of N . Proceed to IEDS. Find thevalues of N for which the game is solvable. Find the Nash equilibria for eachvalue of N .

Exercise 15

Consider the game of guessing the average. Now, players can name arbitrarilylarge numbers. Show that this is not solvable by IEDS. Prove that there isa unique equilibrium.

Exercise 16

A congestion game. There are 6 persons driving from A to B. One road goesthrough C, the other goes through D. There are thus four road segments: AC,CB, AD, DB (all are one-way!). The travel time on a segment depends onthe number n of drivers on that segment. One has TAC(n) = TDB(n) = 10n,TCB(n) = TAD(n) = 50+n. Formulate this problem as a 6-player game whereeach players seeks to minimize her travel time. Find the Nash equilibium (itis unique). Does it minimize the total travel time of the six drivers?

A new road segment (one way) is opened from C to D, there is thus a newroad from A to B: A-C-D-B. One has TCD(n) = 10+n. Find the equilibriumof this new game. Is the traffic better?

Hint: to find the equilibria, use the symmetries of the road network andfind a strategy profile such that the travel time is the same on each road.

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5 Games of perfect information

Definition 5.1 A game of perfect information is given by:

• A set of players N = {1, . . . , n}.

• A game tree: a tree and a mapping assigning each node to a player.

• For each player, a utility functions on terminal nodes.

The game unfolds as follows. We start at the root of the tree. The playerto whom the root is assigned, chooses a branch. At the node reached, theplayer to whom the node is assigned chooses a branch and so on. When anode with no out-going branch is reached (a terminal node), the game is overand payoffs are distributed.

Examples: Chess, Checkers, Go...The definition can be extended to infinite trees: payoffs depend on the se-quence of nodes visited (the play of the game).

Definition 5.2 The Normal Form.

• A strategy of player i in a game with perfect information is a mappingwhich associates an action to each node of player i (an action is anout-going branch). Let Si be the set of strategies of player i.

• A strategy profile s induces a unique play of the game, i.e. a sequenceof visited nodes. The payoff ui(s) is the payoff of player i associatedwith this play.

• The Normal form (or strategic form) of the game with perfect informa-tion is the game with simultaneous moves (Si, ui)i.

Solution concepts for simultaneous games apply to games with perfect infor-mation. The following is a refinement of Nash equilibria.

Definition 5.3 Subgame Perfect Equilibria (SPE).

• Given a game with perfect information and a node z of the tree, thegame tree below z defines a subgame.

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• A SPE is a profile of strategies that induces a Nash equilibrium in everysubgame.

Theorem 5.4 Every finite game (finite game tree) with perfect informationhas a SPE (in pure strategies).

Backward induction.

• Solve each subgame of depth one. (This is a one-player game!!)

• Replace each subgame of depth one γ by a terminal node with theequilibrium payoff of γ.

• Iterate until the root is reached.

This algorithm computes the SPE. This also proves the theorem, by induction.

5.1 Exercises

Exercise 17

Stackelberg. Consider a linear Cournot game. The Stackelberg game is playedas follows: player 1 (the Leader) chooses her quantity, player 2 (the Follower)knowing the choice of player 1, chooses her own quantity. A Stackelbergequilibium is a SPE of this extensive form.

Compute it for a 2-player symetric linear Cournot model. Study the n-player case where player 1 moves first, player 2 moves second, player 3 movesthird,...

Compare the Stackelberg equilibirum with the Cournot equilibrium: fromthe point of view of the leader and of the follower. What would happen inthe Bertrand model?

Exercise 18

Negociation 1: Ultimatum. Two players have to share a surplus normalizedto 1. Player 1 proposes a sharing (x, 1− x). If player 2 accepts, the sharingis implemented and the payoffs are (x, 1− x). If player 2 refuses, payoffs are(0, 0). Find the SPE(s) of this game. Discuss the differences between thecontinuous and discrete verions (ie. what if cents are not splittable?). Findalso all Nash equilibrium outcomes.

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Exercise 19

Negociation 2: Rubinstein’s model of alternating offers. The descriptionstarts like the ultimatum. Two players have to share a surplus normalizedto 1.

Stage 1. Player 1 proposes a sharing (x1, 1 − x1). If player 2 accepts, thesharing is implemented and the payoffs are (x1, 1 − x1). If player 2 refuses,go to stage 2.

Stage 2. The surplus is now of size δ < 1. Player 2 proposes a sharing(x2, 1− x2). If player 1 accepts, the sharing is implemented and the payoffsare (δx2, δ(1− x2)). If player 1 refuses, go to stage 3.

Players alternate proposals and counter-proposals until an acceptance isrecorded. The size of the surplus is multiplied by δ at each stage.

Prove that (1/(1 + δ), δ/(1 + δ)) is a SPE payoff. (Hint: consider thestrategy that consists in offering precisely this share and in refusing any offerbelow δ/(1 + δ)). Prove also that it is the unique SPE payoff.

Exercise 20

Negociation 3: negociation with deadline. Consider Rubinstein’s game withthe additional rule: if no acceptance is recorded before stage T , the processends and the payoff is 0 for each player. Let (uT , vT ) be the SPE payoff ofthis game. The aim of this exercise is to prove that this exists, that it isindeed unique and to compute it. Define the game G(x, y) with only twostages and such that if player 1 refuses player 2’s offer at stage 2, the gameends and the payoffs are (δ2x, δ2y), where x, y are non-negative parameterssuch that x + y ≤ 1.

1. Prove that G(x, y) has a unique SPE payoff denoted F (x, y) (computeit).

2. Prove that (uT , vT ) = F (uT−2, vT−2).

3. Conclude.

Exercise 21

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Stones are placed on the cells of a n ×m board. Player alternately choosestones. When a player chooses a stone, she discard all stones in the North-East corner above this stone. The player taking the last stone (the South-West-most) has lost.

Prove that player 1 (the first to move) has a winning strategy. Find thisstrategy if n = m or if m = 2. What if the board extends infinitely to theNorth and East?

6 Games in extensive form

Definition 6.1 A game in extensive form (or game with imperfect informa-tion) is give by:

• A set of players, a game tree and payoff functions.

• For each player i, a partition of her nodes into information sets.

Information sets model the information available to a player when she choosesan action. Two nodes x, x′ are in the same information set when the playerdoes not know whether she is at x or at x′. This has no impact on theunfolding of the game. It only has an impact on the strategies.

• A strategy of player i is a mapping that associates an action to eachinformation set. (In particular, when two nodes are in the same infor-mation set, the player chooses the same action at these nodes.) Let Si

be the set of these strategies.

• The normal form of the extensive form game is (Si, ui)i where ui isdefined as in the previous section.

Each game admits a Normal form and an Extensive form.

• A mixed strategy is a probability distribution over pure strategies.

• A behavior strategy is a mapping that associates to each informationset, a probability distribution over the set of actions available.

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A player has perfect recall if she never forgets an information she onceknew or an action she once took: (1) if the nodes x, y, w, z are assignedto player i, if x, y are in disjoint information sets, if w is in the subgamefollowing x and z is in the subgame following y, then w and z are in disjointinformations sets. (2) If a, a′ are possible choices of player i at node x, theinformation sets following x, a are disjoint from those following x, a′.

Theorem 6.2 In a game with perfect recall, mixed and behavior strategiesare equivalent. (Kuhn’s theorem).

Definition 6.3 • A proper subgame is a subgame that cuts no informa-tion set.

• A SPE is a strategy profile that induces a Nash equilibrium in everyproper subgame.

Theorem 6.4 Every finite extensive form game admits a SPE in mixedstrategy.

This is a mix of Backward induction and of Nash’s Theorem.

6.1 Exercises

Exercise 22

Consider a symetric linear Cournot game. Study the equilibria of the gamewith one leader and two followers. Then with two leaders and one follower.

Exercise 23

Repeated Games. Let G be a finite simultaneous move game. In the repeatedgame, G is played at each stage t and the actions profiles are publicly ob-served. In the T -stage repeated game, G is played T times and the overallpayoff is the average payoff. In the δ-discounted game, G is infinitely repeatedand the overall payoff is the discounted average

∑t≥1(1− δ)δt−1ui(at).

1. Write the tree of the game (for a 2× 2 game repeated 2 times).

2. A history is a finite sequence of action profiles. Check that historiesare one-to-one associated with proper sub-games.

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3. Prove that if the Prisoner’s Dilemma is repeated T times, then at equi-librium, each player Defects at every stage.

4. Construct a SPE of the δ-discounted Prisoner’s Dilemma where eachplayer Cooperates at each stage (for δ large enough).

5. Using the same reasoning, prove that if two players engage in a repeatedand discounted Ultimatum game, there is a SPE for which the sharingis (1

2, 1

2) at each stage.

6. Prove that every repeated game admits a (simple) SPE.

Exercise 24

Cheap Talk. In a game with cheap talk, players are allowed to exchangecostless and non-binding messages before playing. A simple model of cheaptalk is the following. Let G be a simultaneous game and M be a set ofmessages.

• At the first stage, each player chooses a messages. Choices are simul-taneous and the profile of messages is publicly observed.

• The game G is played.

Let Γ be this game and σ be an equilibrium.

1. Prove that if a profile of messages m has positive probability under σ,the strategies following m form an equilibrium of G.

2. Deduce that the equilibrium payoffs in Γ are convex combinations ofthe equilibrium payoffs of G.

3. Assume M = {0, 1}. Let u, v be two equilibrium payoff vectors of G.Construct an equilibrium of Γ with payoff 1

2u + 1

2v.

4. Assume M = [0, 1]. Let u(1), . . . , u(n) be various equilibrium payoffvectors of G and λ(1), . . . , λ(n) be probability weights. Construct anequilibrium of Γ with payoff

∑k λ(k)u(k).

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Hints. Take X, Y independent random variables in {0, 1} and let Z = 1 ifX 6= Y and Z = 0 otherwise. Show that if X is uniform then so is Z.

Take X, Y independent random variables in [0, 1] and let Z = X + Y ifX + Y < 1 and Z = X + Y − 1 otherwise. Show that if X is uniform thenso is Z.

In both cases, consider the distribution of Z conditional on Y = y andverify that it is uniform.

7 Bayesian Games

Definition 7.1 A Bayesian game is defined by:

• A set of players N = {1, . . . , n}, an action set Ai for each player i.

• A set of types Θi for each player i.

• A belief pi(θ−i | θi) of player i of type θi on other player’s types. (Givenhis type, player i assigns probabilities to the other player’s types.)

• A payoff function ui : Θ× A → R. (The payoff of player i depends onactions and types.)

The type of a player represents her information about the game that is played.It also gives her beliefs about other player’s information. Special cases areoften considered:

• Private values: the payoff of player i depends only on actions and onher own type. This is restrictive: the type may be a partial informationon the common value of an object (eg. a financial asset).

• Common prior: There is a probability P on Θ such that pi(θ−i | θi) =P (θ−i | θi). In this case, the Bayesian game is a game of imperfectinfomation in which: -an extra player (Nature) selects the type profileaccording to P ; -player i is informed of the θi component; -playerschooses actions.

Definition 7.2 • A strategy of player i is a mapping that associates toevery type a (mixed) action.

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• A strategy profile is a Bayesian equilibrium if each player i of type θi

plays a best-reply to the strategies of the opponents (expected payoffsare evaluated with respect to the beliefs).

In the common prior case, a Bayesian equilibrium is simply a Nash equi-librium of the game with imperfect information.

7.1 Applications

Jury;Signalling;Choice of standard;Auctions (first and second price);Correlated equilibria.

8 Social choice and Mechanism design

An environment is:

• A set of players N .

• A set of outcomes.

• A set of types Θi for each player and a prior probability on Θ.

• Utilities ui(θi, x) that depend on types and outcomes (private values).

The problem: A benevolent designer chooses an outcome. His aim is tochoose an efficient outcome. He faces two problems: (1) How to aggregateindividual preferences, (2) how to induce players to reveal their preferences?

8.1 Mechanism design

Definition 8.1 • A mechanism is a family of strategy sets (Si)i and amapping g : ×iSi → X.

• An environment and a mechanism induce a Bayesian game where: -players learn their types, -choose strategies, -and the outcome is chosenaccording to g.

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• A social choice function f : Θ → X is implementable in dominantstrategies if there exists a mechanism (M, g) and a profile of weaklydominant strategies (σ∗i )i in the induced Bayesian game, such that f =g ◦ σ∗.

• A social choice function f : Θ → X is implementable in Bayesian equi-librium if there exists a mechanism (M, g) and a Bayesian equilibrium(σ∗i )i of the induced Bayesian game, such that f = g ◦ σ∗.

A mechanism represents the rules of the game set by the designer to induceplayers to choose the outcome collectively. For instance, an auction is just away of allocating an indivisible good.

• A mechanism is direct if Si = Θi for each i. (Each player is asked toannounce her type, ie. her preferences to the designer.)

• A direct mechanism is truthful if the truthful strategies (reporting thetrue type) form a Bayesian equilibrium (or are weakly dominant).

Theorem 8.2 A social choice function is implementable if and only if it isimplementable by a direct and truthful mechanism. (Revelation Principle)

This result applies both for implementation in weakly dominant strategies orin Bayesian equilibria. This allows to write necessary and sufficient conditionson f to be implementable.

Incentive compatibility

• f is implementable in weakly dominant strategies if and only if foreach player i, each pair of types θi, θ

′i, each profile of types of the other

players θ−i,ui(θi, f(θi, θ−i)) ≥ ui(θi, f(θ′i, θ−i))

• f is implementable in Bayesian equilibrium if and only if for each playeri, each pair of types θi, θ

′i,

Eθ−i[ui(θi, f(θi, θ−i)) | θi] ≥ Eθ−i

[ui(θi, f(θ′i, θ−i)) | θi]

Applications:

• Optimal auctions: revenu equivalence theorem.

• Public goods: Vickrey-Clarke-Groves mechanism.

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T.Tomala. HEC Majeure Economie

Examen virtuel de theorie des jeux. 2h00.

Exercice 1. Soit le jeu a deux joueurs suivant dans lequel le joueur 1 choisitla ligne et le joueur 2 la colonne.

G =

(3, 3) (1, 4) (6, 2) (1, 2)(4, 1) (0, 0) (6, 0) (3, 0)(2, 9) (0, 9) (6, 8) (5, 6)(2, 11) (0, 3) (5, 7) (10, 10)

1. Ce jeu est-il resoluble par elimiation iteree de strategies strictement

dominees ?

2. Determiner tous les equilibres de Nash (purs et mixtes) de ce jeu.

3. Les equilibres de Nash sont-ils Pareto-optimaux dans ce jeu? Est-cetoujours le cas?

Exercice 2. Alice et Bob ont le meme ordinateur portable. Malheureuse-ment, les deux ordinateurs ont ete voles. L’assurance veut leur rembourser aujuste prix et propose la regle suivante. Alice et Bob doivent annoncer chacunla valeur estimee de leur ordinateur. Les choix sont faits simultanements.Soit x la valeur annoncee par Alice et y la valeur annoncee par Bob.

• Si x = y alors chacun recoit cette somme.

• Si x < y, alors Alice recoit x + 2 et Bob recoit x− 2.

• Si x > y, alors Alice recoit y − 2 et Bob recoit y + 2.

On suppose que les valeurs annoncees doivent etre choisies parmi les nom-bres entiers compris entre 2 et 6.

1. Ecrire la matrice de ce jeu.

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2. Montrer que (2, 2) est le seul equilibre de Nash (pur ou mixte). Indi-cations: Le montrer d’abord en pur. Montrer ensuite que la strategie6 est forcement jouee avec probabilite zero dans un equilibre mixte.Conclure en poursuivant ce raisonnement.

3. On suppose maintenant qu’Alice joue avant Bob : Alice choisit x,l’annonce a Bob, qui choisit alors y.

Resoudre ce jeu par backward induction et comparer avec l’equilibre deNash du jeu simultane.

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Exercice 3. Deux firmes i = 1, 2 se font concurrence sur un meme marche.Chaque firme i doit choisir sa quantite de production qi ∈ R+, les choix etantsimultanes. Si la quantite totale produite est Q = q1 + q2, le prix de marchedu bien est fixe a P (Q) = max(1−Q, 0).

• La firme 1 qui produit la quantite q1 doit payer un cout de productionC1(q1) = q2

1.

• La firme 2 qui produit la quantite q2 doit payer un cout de productionC2(q2) = cq2

2 avec c > 0.

• Le but de chaque firme est de maximiser son benefice net,

benefice = quantite × prix de marche − cout.

1. Ecrire le jeu sous forme strategique associe (ensembles de strategies,fonctions de paiements).

2. Montrer que pour toute strategie q2 du joueur 2, le joueur 1 a uneunique meilleure reponse b1(q2) qui vaut:

b1(q2) = max{1− q2

4, 0}.

3. Montrer que pour toute strategie q1 du joueur 1, le joueur 2 a uneunique meilleure reponse b2(q1) qui vaut:

b2(q1) = max{ 1− q1

2(c + 1), 0}.

4. Determiner les equilibres de Nash de ce jeu (en justifiant la reponse).Calculer le prix d’equilibre en fonction du parametre c.

5. Comparer le prix limite quand c tend vers l’infini avec la situation oule joueur 1 est en situation de monopole sur le marche.

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HEC Majeure Economie 2008. Examen de theorie des jeux.

2h00. Documents et calculatrices autorises.Ordinateurs protables interdits. Anglais autorise.

Exercice 1. Bataille des sexes avec option d’entree. Soit la versionsuivante de la bataille des sexes (BoS), le joueur 1 choisit la ligne et le joueur2 la colonne.

F TF 3, 1 0, 0T 0, 0 1, 3

BoS

On considere le jeu Γ ou,

• Dans une premiere etape, le joueur 1 peut decider de jouer le jeu BoSou de sortir (S), ce choix etant annonce au joueur 2. Si il sort, lespaiements sont (2, 2).

• Si il decide de jouer, la bataille des sexes BoS est jouee.

1. Le jeu Γ est-il a information parfaite ? Ecrire precisement l’arbre de cejeu.

2. Donner la matrice du jeu Γ. On pourra regrouper les strategies equivalentesdu joueur 1 en une seule.

3. Determiner tous les equilibres de Nash de Γ, en strategies pures, puisen strategies mixtes. Donner les paiements des joueurs en chaqueequilibre.

4. Determiner les equilibres de Nash du jeu BoS, en strategies pures et enstrategies mixtes. Peut-on eliminer des equilibres de Γ par ”backwardinduction” ?

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5. Quel equilibre de Γ survit a l’elimination des strategies faiblementdominees ? Expliquer intuitivement cet equilibre.

Exercice 2. Sur l’ajout de strategies. Soit le jeu a deux joueurs suivantdans lequel le joueur 1 choisit la ligne et le joueur 2 la colonne.

G DH 6,1 2,0M 2,1 6,0B 3,0 3,1

1. Ce jeu est-il resoluble par elimination de strategies dominees ?

2. Soit A la strategie mixte qui joue H et M avec probabilites (1/2, 1/2).Calculer le paiement espere du joueur 1 quand il joue A, et ce pourchaque strategie du joueur 2. Determiner la matrice du jeu ou le joueur1 peut jouer H, M , B ou A.

3. Peut-on resoudre ce nouveau jeu par elimination de strategies dominees?

4. Determiner tous les equilibres, purs et mixtes du jeu de depart.

Exercice 3. Concurrence en quantite avec cout d’entree. Deuxfirmes i = 1, 2 se font concurrence sur un meme marche. Chaque firme i doitchoisir sa quantite de production qi ∈ R+, les choix etant simultanes. Si laquantite totale produite est Q = q1 + q2, le prix de marche du bien est fixe aP (Q) = 10−Q. Les fonctions de cout sont: C1(q) = C2(q) = 4q.

• 1. Donner le jeu associe, preciser les ensembles de strategies et lesfonctions de paiements.

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• 2. Determiner les courbes de meilleure reponse, puis les equilibres deNash.

• 3. Calculer les benefices a l’equilibre. Calculer egalement le profitoptimal d’une firme en position de monopole.

On suppose maintenant qu’avant de se faire concurrence, les entreprisesdoivent payer un cout d’entree sur le marche. Le jeu se deroule en deuxetapes:

- Etape 1 : chaque entreprise decide d’entrer sur le marche (In) ou derester en dehors (Out). Ces choix sont simultanes. Une entreprise qui restedehors a un gain de 0. Chaque entreprise qui entre doit payer un cout d’entreeF .

-Etape 2 : les entreprises qui sont entrees se font concurrence en quantite.

• 4. Resoudre le jeu en Etape 2. Montrer que pour determiner lesequilibres sous-jeu-parfaits (SPE) du jeu global, on peut se ramenera un jeu 2 × 2 dont les actions sont In et Out. Preciser la matrice dece jeu.

• 5. Determiner les equilibres (purs et mixtes) de ce jeu lorque : F < 4,4 < F < 9, F > 9. Commenter rapidement chaque cas.

• 6. On considere maintenant un grand nombre n de firmes identiques.Pour chaque valeur du cout d’entree F , determiner le nombre maximalde firmes qui entrent sur le marche a l’equilibre.

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