Mason. Game Theory

Game Theory1

Robin Mason

March 2, 2006

1These notes are based on to a large extent on the notes of Dirk Bergemannand Juuso Valimaki for similar courses at Yale University and the Helsinki Schoolof Economics. I am grateful to them both for the permission to use and edit thosenotes.

Contents

1 Game Theory: An Introduction 41.1 Game Theory and Parlour Games—a Brief History . . . . . . 51.2 Basic Elements of Noncooperative Games . . . . . . . . . . . . 6

2 Normal Form Games 72.1 Pure Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Dominance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.3 Nash equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3.1 Finite Normal Form Game . . . . . . . . . . . . . . . . 132.3.2 Cournot Oligopoly . . . . . . . . . . . . . . . . . . . . 142.3.3 Auctions with perfect information . . . . . . . . . . . . 15

2.4 Mixed Strategies . . . . . . . . . . . . . . . . . . . . . . . . . 152.4.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.4.2 Construction of a mixed strategy Nash equilibrium . . 202.4.3 Interpretation of mixed strategy equilibria . . . . . . . 23

2.5 Mixed Strategies with a Continuum of Pure Strategies . . . . 232.5.1 All pay auction . . . . . . . . . . . . . . . . . . . . . . 232.5.2 War of attrition . . . . . . . . . . . . . . . . . . . . . . 292.5.3 Appendix: Basic probability theory . . . . . . . . . . . 30

2.6 Appendix: Dominance Solvability and Rationalizability . . . . 332.6.1 Rationalizable Strategies . . . . . . . . . . . . . . . . . 35

2.7 Appendix: Existence of Mixed Strategy Equilibria . . . . . . . 36

3 Extensive Form Games 383.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.2 Definitions for Extensive Form Games . . . . . . . . . . . . . 40

3.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 403.2.2 Graph and Tree . . . . . . . . . . . . . . . . . . . . . . 40

1

CONTENTS 2

3.2.3 Game Tree . . . . . . . . . . . . . . . . . . . . . . . . . 403.2.4 Notion of Strategy . . . . . . . . . . . . . . . . . . . . 43

3.3 Subgame Perfect Equilibrium . . . . . . . . . . . . . . . . . . 463.3.1 Finite Extensive Form Game: Bank-Run . . . . . . . . 47

3.4 The Limits of Backward Induction: Additional Material . . . . 483.5 Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4 Repeated Games 514.1 Finitely Repeated Games . . . . . . . . . . . . . . . . . . . . . 514.2 Infinitely Repeated Games . . . . . . . . . . . . . . . . . . . . 54

4.2.1 The prisoner’s dilemma . . . . . . . . . . . . . . . . . . 544.3 Collusion among duopolists . . . . . . . . . . . . . . . . . . . 55

4.3.1 Static Game . . . . . . . . . . . . . . . . . . . . . . . . 554.3.2 Best Response . . . . . . . . . . . . . . . . . . . . . . . 564.3.3 Collusion with trigger strategies . . . . . . . . . . . . . 574.3.4 Collusion with two phase strategies . . . . . . . . . . . 58

5 Sequential Bargaining 605.1 Bargaining with Complete Information . . . . . . . . . . . . . 61

5.1.1 Extending the Basic Model . . . . . . . . . . . . . . . . 655.2 Dynamic Programming . . . . . . . . . . . . . . . . . . . . . . 65

5.2.1 Additional Material: Capital Accumulation . . . . . . . 68

6 Games of Incomplete Information 706.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

6.1.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 706.2 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

6.2.1 Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . 716.2.2 Belief . . . . . . . . . . . . . . . . . . . . . . . . . . . . 726.2.3 Payoffs . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

6.3 Bayesian Game . . . . . . . . . . . . . . . . . . . . . . . . . . 726.3.1 Bayesian Nash Equilibrium . . . . . . . . . . . . . . . 73

6.4 A Game of Incomplete Information: First Price Auction . . . 746.5 Conditional Probability and Conditional Expectation . . . . . 75

6.5.1 Discrete probabilities - finite events . . . . . . . . . . . 766.5.2 Densities - continuum of events . . . . . . . . . . . . . 78

6.6 Double Auction . . . . . . . . . . . . . . . . . . . . . . . . . . 806.6.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

CONTENTS 3

6.6.2 Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . 816.6.3 Equilibrium Analysis . . . . . . . . . . . . . . . . . . . 81

7 Adverse selection (with two types) 857.1 Monopolistic Price Discrimination . . . . . . . . . . . . . . . . 85

7.1.1 First Best . . . . . . . . . . . . . . . . . . . . . . . . . 867.1.2 Second Best: Asymmetric information . . . . . . . . . . 87

8 Theoretical Complements 898.1 Mixed Strategy Bayes Nash Equilibrium . . . . . . . . . . . . 898.2 Sender-Receiver Games . . . . . . . . . . . . . . . . . . . . . . 90

8.2.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 908.3 Perfect Bayesian Equilibrium . . . . . . . . . . . . . . . . . . 90

8.3.1 Informal Notion . . . . . . . . . . . . . . . . . . . . . . 908.3.2 Formal Definition . . . . . . . . . . . . . . . . . . . . . 91

8.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

Chapter 1

Game Theory: An Introduction

Game theory is the study of multi-person decision problems. The focus ofgame theory is interdependence, situations in which an entire group of peo-ple is affected by the choices made by every individual within that group.As such they appear frequently in economics. Models and situations of trad-ing processes (auction, bargaining), labor and financial markets, oligopolisticmarkets and models of political economy all involve game theoretic reason-ing. There are multi-agent decision problems within an organization, manyperson may compete for a promotion, several divisions compete for invest-ment capital. In international economics countries choose tariffs and tradepolicies, in macroeconomics, the central bank attempts to control price level.

1. What will each individual guess about the others’ choices?

2. What action will each person take?

3. What is the outcome of these actions?

In addition we may ask

1. Does it make a difference if the group interacts more than once?

2. What if each individual is uncertain about the characteristics of theother players?

3. What do observed actions tell about unobservable characteristics of theplayer?

1

CHAPTER 1. GAME THEORY: AN INTRODUCTION 2

Three basic distinctions can be made at the outset

1. non-cooperative vs. cooperative games

2. strategic (or normal form) games and extensive (form) games

3. games with complete or incomplete information

In all game theoretic models, the basic entity is a player. In noncoop-erative games the individual player and her actions are the primitives ofthe model, whereas in cooperative games coalition of players and their jointactions are the primitives. In other words, cooperative game theory can beinterpreted as a theory that allows for binding contracts between individuals.These notes will consider only non-cooperative game theory.

1.1 Game Theory and Parlour Games—a Brief

History

• E. Zermelo (1913) chess, the game has a solution, solution concept:backwards induction.

• E. Borel (1913) mixed strategies, conjecture of non-existence.

• J. v. Neumann (1928) existence of solutions in zero-sum games.

• J. v. Neumann and O. Morgenstern (1944) Theory of Games and Eco-nomic Behavior: Axiomatic expected utility theory, Zero-sum games,cooperative game theory.

• J. Nash (1950) Nonzero sum games and the concept of Nash equilib-rium.

• R. Selten (1965,75) dynamic games, subgame perfect equilibrium.

• J. Harsanyi (1967/68) games of incomplete information, Bayesian equi-librium.

CHAPTER 1. GAME THEORY: AN INTRODUCTION 3

1.2 Basic Elements of Noncooperative Games

A game is a formal representation of a situation in which a number of in-dividuals interact in a setting with strategic interdependence. The welfareof an agent depends not only on his own action but on the actions of otheragents as well. The degree of strategic interdependence may often vary.

Example 1 Monopoly, Oligopoly, Perfect Competition

To describe a strategic situation we need to describe the players, the rules,the outcomes, and the payoffs or utilities.

Example 2 Matching Pennies, Tick-Tack-Toe (Zero Sum Games).

Example 3 Meeting in New York with unknown meeting place (Non ZeroSum Game, Coordination Game, Imperfect Information).

Chapter 2

Normal Form Games

We start with games that have finitely many strategies for all players andwhere players choose their strategies simultaneously at the start of the game.In these games, the solution of the game can simply be obtained by searching(in a clever way) among all possible solutions.

2.1 Pure Strategies

We begin with a coordination game played simultaneously between Aliceand Bruce. Both Alice and Bruce have a dichotomous choice to be made:either go to an art exhibition or to ballet. This strategic interaction can berepresented by a double matrix as follows: The first entry in each cell is the

Alice

BruceArt Ballet

Art 2, 2 0, 0Ballet 0, 0 1, 1

Figure 2.1: Coordination

payoff of the “row player”. The second entry is the payoff of the “columnplayer”. How should this game be played? What should each individualplayer do?

But first we need to give a definition of a game. In order to describe agame, the set of players must be specified:

I = {1, 2, ..., I} . (2.1)

4

CHAPTER 2. NORMAL FORM GAMES 5

In order to describe what might happen in the interaction, we need to specifythe set of possible choices, called strategies for each player

∀i, si ∈ Si, (2.2)

where each individual player i has a set of pure strategies Si available to himand a particular element in the set of pure strategies is denoted by si ∈ Si.Finally there are payoff functions for each player i:

ui : S1 × S2 × · · · × SI → R. (2.3)

A profile of pure strategies for the players is given by

s = (s1, ..., sI) ∈I×i=1

Si

or alternatively by separating the strategy of player i from all other players,denoted by −i:

s = (si, s−i) ∈ (Si, S−i) .

A strategy profile is said to induce an outcome in the game. Hence for anyprofile we can deduce the payoffs received by the players. This representationof the game is known as normal or strategic form of the game.

Definition 4 (Normal Form Representation) The normal form ΓN rep-resent a game as:

ΓN = {I, {Si}i , {ui (·)}i} .

2.2 Dominance

The most primitive solution concept to games does not require knowledgeof the actions taken by the other players. A dominant strategy is the bestchoice for a player in a game regardless of what the others are doing. Thenext solution concepts assumes that rationality of the players is commonknowledge. Common knowledge of rationality is essentially a recursive no-tion, so that every player is rational, every player knows that every playeris rational, every player knows that every player knows that every playeris rational..., possibly ad infinitum. Finally, the Nash equilibrium conceptrequires that each player’s choice be optimal given his belief about the otherplayers’ behaviour, and that this belief be correct in equilibrium.


Cooperate DefectCooperate 3, 3 0, 4

Defect 4, 0 1, 1

Figure 2.2: Prisoner’s Dilemma

Consider next a truly famous game which figures prominently in the de-bate on the possibility of cooperation in strategic situations, the prisoner’sdilemma game: Besides its somewhat puzzling conclusion this game has an-other interesting property which comes to light when we analyze the payoff ofeach individual player, for the row player and for the column player In both

C DC 3 0D 4 1

Figure 2.3: Row player’s payoffs in Prisoner’s Dilemma

C DC 3 4D 0 1

Figure 2.4: Column player’s payoffs in Prisoner’s Dilemma

cases, one strategy is better the other strategy independent of all choicesmade by the opponent.

Definition 5 (Dominant Strategy) A strategy si is a dominant strategyfor player i in ΓN if for all s−i ∈ S−i and for all s′i 6= si,

ui (si, s−i) > ui (s′i, s−i) .

Definition 6 (Weakly Dominant Strategy) A strategy si is a dominantstrategy for player i in ΓN if for all s−i ∈ S−i and for all s′i 6= si,

ui (si, s−i) ≥ ui (s′i, s−i) ,

and ui

(si, s

′−i

)> ui

(s′i, s

′−i

), for some s′−i ∈ S−i.


The most straightforward games to analyze are the ones that have dom-inant strategies for all players. In those games, there is little need to thinkabout the strategic choices of others as the dominant strategy can never bebeaten by another choice of strategy.

Definition 7 (Dominant Strategy Equilibrium) If each player i in agame ΓN has a dominant strategy si, then s = (s1, ..., sN) is said to be adominant strategy equilibrium of ΓN .

Example 8 Second price auction (with complete information) is an exampleof a game with a dominant strategy equilibrium.

Definition 9 (Purely Strictly Dominated) A strategy si ∈ Si is a purelystrictly dominated strategy for player i in ΓN if there exists a pure strategys′i 6= si, such that

ui (s′i, s−i) > ui (si, s−i) (2.4)

for all s−i ∈ S−i.

Definition 10 (Purely Weakly Dominated) A strategy si is a weaklydominated strategy for player i in ΓN if there exists a pure strategy s′i 6= si,such that

ui (si, s−i) ≤ ui (s′i, s−i) , for all s−i ∈ S−i

and ui

(si, s

′−i

)< ui

(s′i, s

′−i

), for some s′−i ∈ S−i

Now consider a game Γ′ obtained from the original game Γ after elimi-nating all strictly dominated actions. Once again if a player knows that allthe other players are rational, then he should not choose a strategy in Γ′

that is never-best response to in Γ′. Continuing to argue in this way leadsto the outcome in Γ should survive an unlimited number of rounds of suchelimination:

Definition 11 (Iterated Pure Strict Dominance) The process of iter-ated deletion of purely strictly dominated strategies proceeds as follows: SetS0

i = Si. Define Sni recursively by

Sni =

{si ∈ Sn−1

i

∣∣@s′i ∈ Sn−1i , s.th. ui (s

′i, s−i) > ui (si, s−i) , ∀s−i ∈ Sn−1

−i

},


Set

S∞i =

∞⋂n=0

Sni .

The set S∞i is then the set of pure strategies that survive iterated deletion of

purely strictly dominated strategies.

Observe that the definition of iterated strict dominance implies that ineach round all strictly dominated strategies have to be eliminated. It isconceivable to define a weaker notion of iterated elimination, where in eachstep only a subset of the strategies have to be eliminated.

Here is an example of iterated strict dominance:

α2 β(3)2 γ

(1)2

α1 2, 3 3, 2 0, 1

β(2)1 0, 0 1, 3 4, 2

Figure 2.5: Iterated strict dominance

which yields a unique prediction of the game, and the superscript indi-cates the order in which dominated strategies are removed. We might betempted to suggest a similar notion of iteration for weakly dominated strate-gies, however this runs into additional technical problems as order and speedin the removal of strategies matters for the shape of the residual game asthe following example shows: If we eliminate first β1, then α2 is weakly

α2 β2

α1 3, 2 2, 2β1 1, 1 0, 0γ1 1, 1 0, 0

Figure 2.6: Iterated weak dominance

dominated, similarly if we eliminate γ1, then β2 is weakly dominated, butif we eliminate both, then neither α2 nor β2 is strictly dominated anymore.A similar example is The discussion of domination demonstrated that lessrestrictive notion of play may often give good prediction.. However as thefollowing example shows, it clearly is not enough for analyzing all strategicinteractions. The next subsection develops a solution concept that singlesout (M, M) as the relevant solution of the above game.


L RT 1, 1 0, 0

M 1, 1 2, 1B 0, 0 2, 1

Figure 2.7: Iterated weak dominance 2

L M RT 3, 0 0, 2 0, 3

M 2, 0 1, 1 2, 0B 0, 3 0, 2 3, 0

Figure 2.8: No dominated strategies

2.3 Nash equilibrium

Let us now go back to the coordination game. How to solve this game. Asolution should be such that every player is happy playing his strategy andhas no desire to change his strategy in response to the other players’ strategicchoices. A start is to do well given what the strategy chosen by the otherplayer. So let us look at the game form the point of one player, and as thegame is symmetric it is at the same time, the game of his opponent:

Art BalletArt 2 0

Ballet 0 1

Figure 2.9: Individual Payoffs in Battle of Sexes

Definition 12 (Best Response) In a game ΓN , strategy si is a best re-sponse for player i to his rivals strategies s−i if

ui (si, s−i) ≥ ui (s′i, s−i)

for all s′i ∈ Si. Strategy si is never a best response if there is no s−i forwhich si is a best response.

We can also put it differently by saying that si is a best response if

si ∈ arg maxs′i∈Si

ui (s′i, s−i) ⇔ si ∈ BR (s−i) ⇔ si ∈ BR (s) . (2.5)


We could now proceed and ask that the best response property holds forevery individual engaged in this play.

Definition 13 (Pure Strategy Nash Equilibrium) A strategy profile s =(s1, ..., sn) constitutes a pure strategy Nash equilibrium of the game ΓN if forevery i ∈ I,

ui (si, s−i) ≥ ui (s′i, s−i) , for all s′i ∈ Si. (2.6)

In other words, a strategy profile is a Nash equilibrium if each player’sstrategy si is a best response to the strategy profile s−i of all the remainingplayers. In the language of a best response, it says that

s ∈ BR (s) ,

and hence a pure strategy Nash equilibrium s is a fixed-point under the bestresponse mapping. Let us now consider some more examples to test ourability with these new notions.

2.3.1 Finite Normal Form Game

Consider the Hawk-Dove game where two animals are fighting over someprey. Each can behave like a dove or like a hawk. How is this game played

Hawk DoveHawk 3, 3 1, 4Dove 4, 1 0, 0

Figure 2.10: Hawk-Dove

and what is logic behind it.There are two conflicting interpretations of solutions for strategic and

extensive form games:

• steady state (“evolutionary”).

• deductive (“eductive”).

The latter approach treats the game in isolation and attempts to infer therestrictions that rationality imposes on the outcome. The games we are con-sidering next are economic applications. They are distinct from our previous


examples insofar as the strategy space is now a continuum of strategies. Thefirst problem is indeed very continuous and regular optimization instrumentsapply directly. The second example has continuous but not differentiable,payoff functions and we shall proceed directly without using the regular op-timization calculus.

2.3.2 Cournot Oligopoly

Consider the following Cournot model in which the sellers offer quantitiesand the market price is determined as a function of the market quantities.

P (q1, q2) = a− q1 − q2

The profit functions of the firms are:

Πi = (a− q1 − q2)qi

and differentiated we obtain:

Π′i = a− 2q1 − q2

⇔q1 (q2) =

a− q2

2and q2 (q1) =

a− q1

2, (2.7)

where q1 (q2) and q2 (q1) are the reaction functions or best response functionsof the players.

The steeper function is q1 (q2) and the flatter function is q2 (q1). Whilewe can solve (2.7) directly, we realize the symmetry of the problem, whichreduces the system of equation to a single equation, and a symmetric equi-librium, so that

q =a− q

2

andq =

a

3

and hence the profits are

Πi

(q1 =

a

3, q2 =

a

3

)=

a2

9. (2.8)


2.3.3 Auctions with perfect information

Consider the following setting in which there are n bidders with valuationsfor a single good

0 < v1 < v2 < .... < vn < ∞.

We shall consider two different auction mechanisms: the first and the secondprice auction. The object is given to the bidder with the highest index amongall those who submit the highest bid. In the first price auction, the winnerpays his own bid, in the second price auction, the winner pays the secondhighest bid.

The payoff functions in the first price auction are

ui (vi, b1, ..., bn) =

vi − bi if bi > bj,∀j 6= i,vi − bi if bi ≥ bj, and if bi = bj ⇒ i > j.0, otherwise

(2.9)

and in the second price auction they are

ui (vi, b1, ..., bn) =

vi − bk if bi > bj,∀j 6= i,vi − bk if bi ≥ bj, and if bi = bj ⇒ i > j.0, otherwise

(2.10)

where bk satisfies the following inequality

bi ≥ bk ≥ bj, ∀j 6= i, k.

We can then show that the equilibrium in the first-price auction has the prop-erty that the bidder with the highest value always wins the object. Noticethat this may not be true in the second price auction. However we can showthat in the second price auction, the bid bi = vi weakly dominates all otherstrategies.

Reading: Chapter 0 in Binmore (1992) is a splendid introduction intothe subject. Chapter b of MWG and Chapter 1-1.2 in Gibbons (1992) coversthe material of the last two lectures as does Chapter 1-1.2 in FT.

2.4 Mixed Strategies

So far we were only concerned with pure strategies, i.e. situations whereeach player i picks a single strategy si. However some games don’t have


equilibria in pure strategies. In these cases it is necessary to introduces anelement of “surprise” or “bluff” through considering randomized decisionsby the players. Consider the game of “matching pennies”:

Heads TailsHeads 1,−1 −1, 1Tails −1, 1 1,−1

Figure 2.11: Matching Pennies

It is immediate to verify that no pure strategy equilibrium exists. Asimilar game is “Rock, Paper, Scissors”:

R P SR 0, 0 −1, 1 1,−1P 1,−1 0, 0 −1, 1S −1, 1 1,−1 0, 0

Figure 2.12: Rock, Paper, Scissors

2.4.1 Basics

Let us now introduce the notation for mixed strategies. Besides playing apure strategy si each player is also allowed to randomize over the set of purestrategies. Formally, we have

Definition 14 A mixed strategy for player i, σi : Si → [0, 1] assigns to eachpure strategy si ∈ Si, a probability σi (si) ≥ 0 that it will be played, where∑

si∈Si

σi (si) = 1

The mixed strategy can be represented as a simplex over the pure strate-gies:

∆ (Si) =

{(σi (s1) , ..., σi (sn)) ∈ Rn : σi (sik) ≥ 0, ∀j, and

n∑k=1

σi (sik) = 1

}.

This simplex is also called the mixed extension of Si. For |Si| = 3, it can beeasily represented as a two-dimensional triangle sitting in a three dimensional


space. σi (si) is formally a probability mass function and σi is a probabilityvector. Such a randomization is called a mixed strategy σi which is a elementof set of distributions over the set of pure strategies:

σi ∈ ∆ (Si) .

An alternative notation for the set of mixed strategies, which you may findfrequently is

σi ∈ Σi.

In order to respect the idea that the players are choosing their strate-gies simultaneously and independently of each other, we require that therandomized strategies chosen by the different players satisfy statistical inde-pendence.1 In other words, we require that the probability of a pure strategyprofile s′ = (s′1, ..., s

′N) is chosen is given by

N∏i=1

σi (s′i) .

The expected utility of any pure strategy si when some of the remainingplayers choose a mixed strategy profile σ−i is

ui (σ1, ..., σi−1, si, σi+1, ..., σI) =∑

s−i∈S−i

(∏j 6=i

σj (sj)

)ui (s1, ..., si−1, si, si+1, ..., sI) .

Similarly, the expected utility of any mixed strategy σi is

ui (σ1, ..., σi−1, σi, σi+1, ..., σI) =∑s∈S

(∏j

σj (sj)

)ui (s1, ..., si−1, si, si+1, ..., sI) .

As another example, consider the famous “battle of the sexes” game:This game has clearly two pure strategy equilibria. To see if it has a mixed

1Much of what follows can be generalized to the case where this independence con-dition is dropped. The resulting solution concept is then called correlated equilibrium.Since independence is a special case of such correlated random strategies, the analysisbelow is a special case of correlated equilibrium. The notion of sunspots, familiar frommacroeconomic models is an example of correlated strategies. There all players observe asignal prior to the play of the game, and condition their play on the signal even thoughthe signal may be completely unrelated to the underlying payoffs.


Opera FootballOpera 2, 1 0, 0

Football 0, 0 1, 2

Figure 2.13: Battle of Sexes

strategy equilibrium as well, calculate the payoff to Sheila if Bruce is choosingaccording to mixed strategy σB (·):

uS (O, σB) = 2σB (O) + 0σB (F ) ,

and similarlyuS (F, σB) = 0σB (O) + 1σB (F ) .

Again this allows for an illuminating graphical representation by the best-response function. Consider the best-response mapping of the two players,where we plot the probability of attending the football game for Sheila on thex- axis and for Bruce on the y- axis. Every point of intersection indicates aNash equilibrium. An equilibrium mixed strategy by Sheila would be writtenas

σS =

(σS (O) =

2

3, σS (F ) =

1

3

)(2.11)

and if the ordering of the pure strategies is given, then we can write (2.11)simply as

σS =

(2

3,1

3

).

Before we give a systematic account of the construction of a mixed strategyequilibrium, let us extend our definitions for pure strategies to those of mixedstrategies.

Definition 15 (Normal Form Representation) The normal form repre-sentation ΓN specifies for each player i a set of strategies Σi, with σi ∈ Σi anda payoff function ui (σ1, ..., σn) giving the von Neumann-Morgenstern utilitylevels associated with the (possible random) outcome arising from strategies(σ1, ..., σn) :

ΓN ={I, (Σi, ui)i∈I

}.


Definition 16 (Best Response) In game ΓN ={I, (Σi, ui)i∈I

}, strategy

σi is a best response for player i to his rivals strategies σ−i if

ui (σi, σ−i) ≥ ui (σ′i, σ−i)

for all σ′i ∈ Σi.

Definition 17 (Nash Equilibrium) A mixed strategy profile σ∗ = (σ∗1, ..., σ∗N)

constitutes a Nash equilibrium of game ΓN ={I, (Σi, ui)i∈I

}if for every

i = 1, ..., I;ui

(σ∗i , σ

∗−i

)≥ ui

(σi, σ

∗−i

)(2.12)

for all σi ∈ Σi.

Lemma 18 Condition (2.12) is equivalent to:

ui

(σ∗i , σ

∗−i

)≥ ui

(si, σ

∗−i

), (2.13)

for all si ∈ Si, since any mixed strategy is composed of pure strategies.

Since this is the first proof, we will be somewhat pedantic about it.Proof. Suppose first that (2.12) holds. Then, it is sufficient (why?) to

show that for every s′i ∈ Si, there exists σ′i ∈ Σi such that

ui

(σ′i, σ

∗−i

)≥ ui

(s′i, σ

∗−i

)But since the set of pure strategies is a strict subset of the set of mixedstrategies, or Si ⊂ Σi, we can simply set σ′i = s′i, and hence it follows that(2.13) has to hold.

Next suppose that (2.13) holds and we want to show that this impliesthat (2.12) holds. It is then sufficient (again, why) to show that for everyσ′i ∈ Σi, there exists s′i ∈ Si such that

ui

(s′i, σ

∗−i

)≥ ui

(σi, σ

∗−i

).

But since every mixed strategy σi is composed out of pure strategies, inparticular since

ui

(σi, σ

∗−i

)=∑si∈Si

σi (si) ui

(si, σ

∗−i

),


consider the pure strategy si among all those with σi (si) > 0 which achievesthe highest payoff ui

(si, σ

∗−i

). Then it must be the case that

ui

(si, σ

∗−i

)≥ ui

(σi, σ

∗−i

),

and hence the conclusion.The following proposition indicates how to construct mixed strategy equi-

libria. LetS+

i (σi) = {si ∈ Si |σi (si) > 0} ,

then S+i (σi) is the set of all those strategies which receive positive probability

under the mixed strategy σi. As S+i (σi) contains all points for which σi

assigns positive probability, it is mathematically referred to as the supportof Si.

Proposition 19 (Composition) Let S+i (σ∗i ) ⊂ Si denote the set of pure

strategies that player i plays with positive probability in a mixed strategyprofile σ∗ = (σ∗1, ..., σ

∗I ). Strategy profile σ∗ is a Nash equilibrium in game

ΓN ={I, (Σi, ui)i∈I

}if and only if for all i = 1, ..., I

(i) ui

(si, σ

∗−i

)= ui

(s′i, σ

∗−i

)for all si, s

′i ∈ S+

i (σ∗i ) ;

(ii) ui

(si, σ

∗−i

)≥ ui

(s′i, σ

∗−i

)for all si ∈ S+

i (σ∗i ) , s′i ∈ S.

2.4.2 Construction of a mixed strategy Nash equilib-rium

We return to the example of the battle of sexes.

Sheila

BruceOpera Football

Opera 2, 1 0, 0Football 0, 0 1, 2

Figure 2.14: Battle of Sexes

Then a strategy for Sheila has the following payoff if Bruce is playing amixed strategy σB (·)

uS (σB, O) = 2σB (O) + 0σB (F ) ,


and similarlyuS (σB, F ) = 0σB (O) + 1σB (F ) .

We want to create a mixed strategy for Sheila and Bruce. If Sheila is supposedto use a mixed strategy, it means that she picks either alternative with apositive probability. As the strategy is supposed to be a best-response toher, it has to be the case that the expected payoffs are identical across thetwo alternative, for otherwise Sheila would clearly be better off to use onlythe pure strategy which gives her the strictly higher payoff. In short, it hasto be that

uS (σB, O) = uS (σB, F )

or explicitly2σB (O) + 0σB (F ) = 0σB (O) + 1σB (F ) . (2.14)

Thus when we consider whether Sheila will randomize, we examine conditionson the mixing behaviour of Bruce under which Sheila is indifferent. Let

σB (F ) = σB

and consequentlyσB (O) = 1− σB.

The condition (6.13) can then be written as

2 (1− σB) = σB ⇔ σB =2

3. (2.15)

Thus if Sheila is to be prepared to randomize, it has be that Bruce is random-izing according to (6.14). Now Bruce is willing to randomize with σB = 1

3in

equilibrium if and only if he is indeed indifferent between the pure strategyalternatives. So, now we have to ask ourselves when is this the case. Well,we have to investigate in turn how the indifference condition of Bruce deter-mines the randomizing behaviour of Sheila. (The construction of the mixedstrategy equilibrium highlights the interactive decision problem, which is es-sentially solved as a fixed point problem.) A similar set of conditions asbefore gives us

uB (σS, O) = uB (σS, F )

or explicitlyσS (O) + 0σS (F ) = 0σS (O) + 2σS (F ) . (2.16)


Thus as we consider whether Bruce is willing to randomize, we examineconditions on the mixing behaviour of Sheila to make Bruce indifferent. Let

σS (F ) = σS

and consequentlyσS (O) = 1− σS.

The condition (6.19) can then be written as

(1− σS) = 2σS ⇔ σS =1

3. (2.17)

Thus we reached the conclusion that the following is a mixed strategy Nashequilibrium

σ∗B =

(σB (O) =

1

3, σB (F ) =

2

3

)σ∗S =

(σS (O) =

2

3, σS (F ) =

1

3

).

The payoff to the two players is then 23

in this mixed strategy equilibrium.This examples presents the general logic behind the construction of a mixedstrategy Nash equilibrium. The only issue which we still have to discuss isthe support of the mixed strategy equilibrium.

The following game is a helpful example for this question:

α β γα −1, 1 1,−1 x, xβ 1,−1 −1, 1 x, xγ x, x x, x x, x

Figure 2.15: Augmented Matching Pennies

For x < 0, γ cannot be part of any mixed strategy equilibrium, and forx > 0, the original matching pennies equilibrium cannot be a mixed strategyequilibrium anymore. However for x = 0, there can be many mixed strategyequilibria, and they differ not only in the probabilities, but in the support aswell.2

2Verify that there can’t be an equilibrium in which randomization occurs between αand γ only, or alternatively between β and γ.


2.4.3 Interpretation of mixed strategy equilibria

Finally, consider the competing interpretations of a mixed strategy Nashequilibrium:

1. mixed strategies as objects of choice

2. steady state, probability as a frequency of certain acts

3. mixed strategies as pure strategies in a game where the randomnessis introduced through some random private information (Harsanyi’spurification argument)

Remark 20 In the battle of sexes, the mixed strategy equilibrium used pri-vate randomization by each player.

Suppose instead they would have a common (or public) randomizationdevice, say a coin, then they could realize a payoff of (3

2, 3

2) by e.g. both going

to opera if heads and both going to football if tails. Observe that this payoffis far superior to the mixed strategy equilibrium payoff The notion of publicrandomization is used in the correlated equilibrium.

Finally we consider games where each player has a continuum of actionsavailable. The first one is an example of an “all pay auction”.

2.5 Mixed Strategies with a Continuum of

Pure Strategies

2.5.1 All pay auction

Two investors are involved in a competition with a prize of v. Each investorcan spend any amount in the interval [0, v]. The winner is the investor whospends the most. In the event of tie each investor receives v/2. Formulatethis situation as a strategic game and find its mixed strategy Nash equilibria.

The all pay auction is, despite its initially funny structure, a rather com-mon strategic situation: Raffle, R&D patent races, price wars of attrition,wars, winner take all contests with participation costs.


The payoff function of the very agent is therefore given by

ui (v, bi, bj) =

v − bi for bj < bi12v − bi, for bi = bj

−bi for bi < bj

The mixed (or pure) strategy by player i can be represented by the distrib-ution function of the bid of i:

Fi (bi) ,

where we recall that the distribution function (or cumulative distributionfunction) Fi (bi) is defining the event

Fi (bi) = Pr (b ≤ bi)

We recall the following notation:

F(b−)

= lima↓b

F (a) ,

andF(b+)

= lima↑b

F (a) .

Before we can analyze and derive the equilibrium, we have to ask what isthe expected payoff from submitting a bid b1 given that player 2 uses anarbitrary distribution function F2. From the payoff functions, we see that itcan be calculated as:

E [u1 (v, b1) |F2 ] = F2 (b2 < b1) (v − b1) +

(F2 (b2 ≤ b1)− F2 (b2 < b1))

(1

2v − b1

)+ (1− F (b2 ≤ b1)) (−b1) ,

which can be written more compactly as

E [u1 (v, b1) |F2 ] = F2

(b−1)v +

(F2 (b1)− F2

(b−1)) 1

2v − b1. (2.18)

This expected payoff function shows that in this auction the bidder has tofind the best trade-off between increasing the probability of winning and thecost associated with increasing the bid.


We suppose for the moment that the distribution function F2 (·) is continuous-(ly differentiable) everywhere. Then (2.18) can be rewritten as

E [u1 (v, b1) |F2 ] = F2 (b1) v − b1.

As agent i chooses his bids optimally in equilibrium, his bid must maximizehis payoff. In other words, his optimal bidding strategy is characterizedby the first order conditions which represent the marginal benefit equalsmarginal cost condition

f2 (b1) v − 1 = 0, (2.19)

which states that the increase in probability of winning which occurs at therate of f (b1) multiplied by the gain of v must equal the marginal cost ofincreasing the bid which is 1. As we attempt to construct a mixed strat-egy equilibrium, we know that for all bids with f1 (b1) > 0, the first ordercondition has to be satisfied. Rewriting (2.19) we get

f2 (b1) =1

v. (2.20)

As we look for a symmetric equilibrium, in which agent i and j behaveidentically, we can omit the subscript in (6.4) and get a uniform (constant)density

f ∗ (b) =1

v

with associated distribution function:

F ∗ (b) =b

v. (2.21)

As F (b = 0) = 0 and F (b = v) = 1, it follows that F ∗ (b) in fact constitutesa mixed equilibrium strategy.

We went rather fast through the derivation of the equilibrium. Let usnow complete the analysis. Observe first that every pure strategy can berepresented as a distribution function. We start by explaining why therecan’t be a pure strategy equilibrium in the all-pay auction, or in other wordswhy there can be no atoms with probability 1.

Lemma 21 (Non-existence of pure strategy equilibrium) @ a pure strat-egy equilibrium in the all pay auction.


Proof. We argue by contradiction. Suppose therefore that there is a purestrategy equilibrium. Then either (i) bi > bj or (ii) bi = bj. Suppose firstthat

bi > bj

then bi would lower his bid and still win the auction. Suppose then that

bi = bj.

The payoff of bidder i would then be

1

2v − bi,

but by increasing the bid to bi + ε for small ε, he would get the object forsure, or

v − bi − ε.

As ε → 0, and when taking the difference, we see that i would have a jumpin his net utility of 1

2v as

v − bi − ε−(

1

2v − bi

)=

1

2v − ε → 1

2v,

which concludes the proof.�As we have now shown that there is no pure strategy equilibrium, we

can concentrate on the derivation of a mixed strategy equilibrium. In thederivation above, we omitted some possibilities: (i) it could be that the distri-bution function of agent i is not continuous and hence has some mass pointsor atoms, (ii) we have not explicitly derived the support of the distribution.We will in turn solve both problems.

A similar argument which excludes a pure strategy (and hence atomswith mass one) also excludes the possibility of atoms in mixed strategy equi-librium.

Lemma 22 (Non-existence of mass points in the interior) @ a mixedstrategy equilibrium with a mass point b in the interior of the support of eitherplayer’s mixed strategy ( i.e. Fi (b) − Fi (b

−) > 0 for some i) in the all payauction.


Proof. Suppose to the contrary that the strategy of player j contains anatom so that for some b,

Fj (b)− Fj

(b−)

> 0,

If Fj (·) has an atom at b, then for any ε > 0, it must be that

Fi (b)− Fi (b− ε) > 0,

otherwise j could lower the bids at the atom without changing the probabilityof winning. But consider the winnings of i at b′ < b. They are at best

vFj

(b−)− b−

but by increasing the bid to b′′

> b, i would receive

vFj

(b′)− b′.

But letting b′ ↑ b and b

′′↓ b, we can calculate the expected payoff difference

between

limb′↑b, b

′′↓b

(vFj

(b′′)− b

′′)−(vFj

(b′)− b

′)

= v(Fj (b)− Fj

(b−))

> 0,

(2.22)and hence would offer a profitable deviation for player i. �

Notice the argument applies to any b < v, but not for b ≥ v, and hencethere could be atoms at the upper bound of the support of the distribution.3

It then remains to find the mixed strategy equilibrium which has an intervalas its support. Suppose Fi (bi) is the distribution functions which defines themixed strategy by player i. For her to mix over any interval

bi ∈[bi, bi

](2.23)

3Recall the support of a (density) function is the closure of the set

{x : f (x) 6= 0} ,

and thus the support issupp f = {x : f (x) 6= 0}


she has to be indifferent:vFj (b)− b = c, (2.24)

where c is the expected payoff from any mixed strategy equilibrium. Arguethat c = 0, then we solve the equation to

Fj (b) =b

v, (2.25)

which tells us how player j would have to randomize to make player i indif-ferent. The argument is actually already made by the derivation of the firstorder conditions and the observation that there can’t be any atom in theinterior of [0, v]. By symmetry, (2.25) then defines the equilibrium pricingstrategy by each player.

Suppose we were to increase the number of bidders, then (6.10) changesunder the assumption of symmetry to

v (F (b))n−1 − b = c, (2.26)

where c is the expected payoff from any mixed strategy equilibrium. Argueagain that c = 0, then we solve the equation to

F (b, n) =

(b

v

) 1n−1

(2.27)

and the expected payment of each individual bidder is∫ v

0

bdF (b) =

∫ v

0

bf (b) db

and since

F ′ (b) = f (b) =

(bv

) 1n−1

(n− 1) b

we obtain ∫ v

0

(bv

) 1n−1

(n− 1)db =

v

n. (2.28)


2.5.2 War of attrition

Suppose two players are competing for a prize of value v. They have to pay cper unit of time to stay in the competition. If either of the two player dropsout of the competition, then the other player gets the object immediately.This is a simple example of a timing game, where the strategy is defined bythe time τi at which the player drops out. There are pure strategy Nashequilibria, which have

(τi = 0, τj ≥ c

v

), however the only symmetric equilib-

rium is a mixed strategy equilibrium, where each player must be indifferentbetween continuing and dropping out of the game at any point in time, or

fi (t)

1− Fi (t)v = c ⇔ fi (t)

1− Fi (t)=

c

v,

where the indifference condition may be interpreted as a first order differentialequation

dFi (t)

(1− Fi (t))=

c

v.

It is very easy to solve this equation once it is observed that the left handside in the equation is

− d

dtln (1− Fi (t)) .

Hence integrating both sides and taking exponentials on both sides yields

1− Fi (t) = e−cvt, or Fi (t) = 1− e−

cvt.

Alternatively, we may appeal to our knowledge of statistics, recognize

fi (t)

1− Fi (t)

as the hazard rate4, and notice that only the exponential distribution function

4To understand why this is called the hazard rate, think of t as the time a machinefails. Then as time t passes, and moving from t to t + dt, one finds that the conditionalprobability that the machine fails in this interval, conditional on not having failed beforeis

f (t) dt

1− F (t).

Als recall that the conditional probability is

P (A |B ) =P (A ∩B)

P (B)


has a constant hazard ratio λ

F (t) = 1− e−λt

andf (t) = λe−λt.

The symmetric equilibrium is then given by:

F (t) = 1− e−cvt

and the expected time of dropping out of the game, conditional on the otherplayer staying in the game forever is∫ ∞

0

tdF (t) =

∫ ∞

0

tf (t) dt =v

c.

2.5.3 Appendix: Basic probability theory

The general framework of probability theory involves an experiment that hasvarious possible outcomes. Each distinct outcome is represented by a pointin a set, the sample space. Probabilities are assigned to certain outcomesaccording to certain axioms. More precisely X is called a random variable,indicating that the value of X is determined by the outcome of the experi-ment. The values X can take are denoted by x. We can then refer to eventssuch as {X = x} or {X ≤ x} and the sample space is X .

We distinguish two cases of probability distributions: discrete and con-tinuous.

Discrete Case

In the discrete case, the numbers of possible outcomes is either finite orcountable and so we can list them. Consider the following experiment inwhich the numbers x ∈ {1, 2, 3, 4} are drawn with probabilities p (x) as in-dicated. Any point x with p (x) > 0 is called a mass point or an atom. Thefunction p (·) is a probability mass function

p : X → [0, 1] ,

such that ∑x∈X

p (x) = 1.


This is an example of discrete probabilities. The cumulative distributionfunction is then given by

F (z) =∑x≤z

p (z) .

Notice that F (x) is a piecewise constant function, which is monotonicallyincreasing and at the points x ∈ {1, 2, 4} discontinuous. The function F (x)is zero before 1 and 1 at and after 4.

Continuous Case

Suppose next that we wish to choose any number between x ∈ [0, 4]. Theassignment of probabilities occurs via the density function f (x) ≥ 0, so that∫ 4

0

f (x) dx = 1.

The likelihood that we pick a number x ≤ x′ is then given by

F (x′) = F (x ≤ x′) =

∫ x′

0

f (x) dx

The likelihood that a number x ∈ [x′, x′′] is chosen in the experiment is givenby

F (x′′)− F (x′) =

∫ x′′

x′f (x) dx

In other words, any interval of numbers [x, x′] has some positive probabilityof being chosen. An example of such a distribution function is then given by:

F (x) =

0, for x < 0

1−(1− 1

4x)3

, for 0 ≤ x ≤ 4

1, for x > 4.

(2.29)

We may then ask what is the probability that a number is drawn out ofthe interval (x, x′), and it is given by

F (x′)− F (x) .


The density function then identifies the rate at which probability is added tothe interval by taking the limit as ∆ = x′ − x converges to zero:

f (x) = lim∆→0

F (x′)− F (x)

x′ − x=

F (x + ∆)− F (x)

∆.

Notice that the density is only the rate at which probability is added to theinterval, but is not the probability of a particular x, call it x̂ occurring sincethe probability of x̂ occurring is of course 0 under a continuous distributionfunction, since ∫ x̂

x̂

f (x) dx = 0.

Thus we have to distinguish between events which have probability zero, butare possible and impossible events. The density function associated with thedistribution function F (x) is given by

f (x) =

0, for x < 034

(1− 1

4x)2

, for 0 ≤ x ≤ 4

0 for x > 4.

Mixed Case

Finally, we can of course have distribution functions with continuous anddiscrete parts. We saw an example of this in the case of a mixed strategywith an atom.

Notice that the discrete probability mass introduces a discontinuity inthe distribution function so that

F(p−)

= limp′↑p

F (p′) 6= F (p) .

The point x = 2 is often called an atom as it carries strictly positive proba-bility to the distribution function.

Expectations

The density function f (x) is of course

F (x) =

∫ x

−∞f (z) dz


as we had in the discrete case the analogous expression in the sum

F (x) =∑z≤x

p (z) .

Finally we can take the expectation of the lottery as

E [x] =

∫ ∞

−∞zf (z) dz

or for discrete random variables

E [x] =∑

z

zp (z)

Reading: Recall: A brief review of basic probability concepts, includingsample space and event, independent events, conditional probabilities andBayes’ formula, discrete and continuous random variables, probability massfunction, density and distribution function, all introduced in Ross (1993),Chapter 1–2.4 is highly recommended. Osborne and Rubinstein (1994), chap-ter 3 contains a very nice discussion of the interpretation and conceptualaspects of the equilibrium notion, especially in its mixed version.

2.6 Appendix: Dominance Solvability and Ra-

tionalizability

In this section, we elaborate on the notion of iterated deletion of dominatedstrategies. We define dominance in a way that allows for the use of mixedstrategies as well and we also connect this concept to the notion of ratio-nalizable strategies where the question is phrased in terms of finding a setof mutually compatible strategies in the sense that all such strategies arebest responses to some element in the other players’ allowed set of strategies.Nash equilibrium is a special case of this concept where it is also requiredthat the sets of allowed strategies are singletons for all players.

Definition 23 (Strictly Dominated) A strategy si ∈ Si is a strictly dom-inated strategy for player i in game ΓN =

{I, (Σi, ui)i∈I

}if there exists a

mixed strategy σ′i 6= si, such that

ui (σ′i, s−i) > ui (si, s−i) (2.30)


for all s−i ∈ S−i.

Remark 24 This notion of dominance is based on domination by pure ormixed strategies. Clearly, more strategies can be dominated if we allow mixedstrategies to be in the set of dominating strategies.

Definition 25 (Never-Best response) A strategy si is never-best responseif it is not a best response to any belief by player i. In other words, for allσ−i ∈ Σ−i, there is a s′i such that

ui

(s′i, σ

′−i

)> ui

(si, σ

′−i

).

Lemma 26 In a two player game, a strategy si of a player is never-bestresponse if and only if it is strictly dominated.

The notion of strict dominance has hence a decision-theoretic basis thatis independent of the notion of mixed strategy.5

Now consider a game Γ′ obtained from the original game Γ after elimi-nating all strictly dominated actions. Once again if a player knows that allthe other players are rational, then he should not choose a strategy in Γ′

that is never-best response to in Γ′. Continuing to argue in this way leadsto the outcome in Γ should survive an unlimited number of rounds of suchelimination:

Definition 27 (Iterated Strict Dominance) The process of iterated dele-tion of strictly dominated strategies proceeds as follows: Set S0

i = Si, andΣi = Σ0

i . Define Sni recursively by

Sni =

{si ∈ Sn−1

i

∣∣@σi ∈ Σn−1i , s.th. ui (σi, s−i) > ui (si, s−i) , ∀s−i ∈ Sn−1

−i

},

and similarly

Σni = {σi ∈ Σi |σi (si) > 0 only if si ∈ Sn

i } .

Set

S∞i =

∞⋂i=1

Si,

5The above Lemma extends to more than two players if all probability distributions onS−i and not only the ones satisfying independence are allowed.


and

Σ∞i =

{σi ∈ Σi

∣∣@σ′i ∈ Σi s.th. ui (σ′i, s−i) > ui (σi, s−i) , ∀s−i ∈ S∞

−i

}The set S∞

i is then the set of pure strategies that survive iterated deletionof strictly dominated strategies. The set Σ∞

i is the set of player’s i mixedstrategies that survive iterated strict dominance.

2.6.1 Rationalizable Strategies

The notion of rationalizability begins by asking what are all the strategiesthat a rational player could play? Clearly, a rational player will play a bestresponse to his beliefs about the play of other players and hence she willnot use a strategy that is never a best response. Hence such strategies maybe pruned from the original game. If rationality of all players is commonknowledge, then all players know that their opponents will never use strate-gies that are never best responses. Hence a player should use only strategiesthat are best responses to some belief of other players’ play where the othersare not using never best response strategies. Iteration of this reasoning leadsto a definition of rationalizable strategies. Denote the convex hull of A byco (A) . In other words, co (A) is the smallest convex set containing A.

Definition 28 (Rationalizable Strategies) Set Σ0i = Σi and for each i

define recursively

Σni =

{σi ∈ Σn−1

i

∣∣∃σ−i ∈(×j 6=ico

(Σn−1

j

)), s.th. ui (σi, σ−i) ≥ ui (σ

′i, σ−i) ,∀σ′i ∈ Σn−1

i

}.

(2.31)The rationalizable strategies for player i are

Ri =∞⋂

n=0

Σni . (2.32)

{I, (Σi, ui)i∈I

}, the strategies in ∆ (Si) that survive the iterated removal of

strategies that are never a best response are known as player i′s rationalizablestrategies.

The reason for having the convex hull operator in the recursive defini-tion is the following. Each point in the convex hull can be obtained as arandomization over undominated strategies for j 6= i. This randomization


corresponds to i′s subjective uncertainty over j′s choice. It could fully wellbe that this random element is dominated as a mixed strategy for player j.Hence co

(Σn−1

j

)6= Σn−1

j in some cases.Rationalizability and iterated strict dominance coincide in two player

games. The reason for the coincidence is the equivalence of never-best re-sponse and strictly dominated in two player games which does not hold inthree player games as the following example suggests. In the game below,we have pictured the payoff to player 3, and player 3 chooses either matrixA, B, C or D.

α2 β2

α1 9 0β1 0 0

A

α2 β2

α1 0 9β1 9 0

B

α2 β2

α1 0 0β1 0 9

C

α2 β2

α1 6 0β1 0 6

D

Figure 2.16: Rationalizable vs. undominated strategies

where we can show that action D is not a best response to any mixedstrategy of players 1 and 2, but that D is not dominated. If each player i canhave a correlated belief about the others’ choices, the equivalence of ‘corre-lated rationalizability’ and iterated deletion of strictly dominated strategiescan be established.

2.7 Appendix: Existence of Mixed Strategy

Equilibria

Up to this point, we have not shown that mixed strategy equilibria exist forgeneral games. We have simply assumed their existence and characterizedthem for some games. It is important to know that for most games, mixedstrategy Nash equilibria do exist.

The first existence result was proved by Nash in his famous paper Nash(1950).

Theorem 29 (Existence in Finite Games) Finite games, i.e. games witha finite number of players and a finite number of strategies for each player,have a mixed strategy Nash equilibrium.


The proof of this result is a standard application of Kakutani’s fixed pointtheorem, and it is given in almost all graduate level textbooks, for instancethe appendix to Chapter 8 in MWG give the proof.

Many strategic situations are more easily modeled as games with a con-tinuum of strategies for each player (e.g. the Cournot quantity competi-tion mode above). For these games, slightly different existence theorems areneeded. The simplest case that was also proved by Nash is the following (andalso covered in the same appendix of MWG).

Theorem 30 (Continuous-Quasiconcave case) Let ΓN ={I, (Si, ui)i∈I

}be a normal form game where I = {1, ..., N} for some N < ∞ and Si =[ai, bi] ⊂ R for all i. Then if ui : S → R is continuous for all i and quasicon-cave in si for all i, ΓN has a pure strategy Nash equilibrium.

Observe that this theorem actually guarantees the existence of a purestrategy equilibrium. This results from the assumption of continuity andquasiconcavity. If either of these requirements is dropped, pure strategyequilibria do not necessarily exist.

Unfortunately, the payoffs in many games of interest are discontinuous(e.g. the all pay auction and the war of attrition examples above show this).For these games, powerful existence results have been proved, but the math-ematical techniques are a bit too advanced for this course. State of the artresults are available in the recent article Reny (1999). The basic conclusionof that paper is that it will be very hard to find economically relevant gameswhere existence of mixed strategy equilibria would be doubtful.

An artificial game where no equilibria (in either pure or mixed strategies)exists is obtained by considering a modification of the Cournot quantitygame. Recall that in that game, the payoffs to the two players are given by

Πi(qi, qj) = (a− q1 − q2) qi.

If the payoffs are modified to coincide with these at all (q1, q2) except(

a3, a

3

)and πi

(a3, a

3

)= 0 for both firms, then one can show that the game has no

Nash equilibria. The argument to show this is that all of the strategies exceptfor qi = a

3are deleted in finitely many rounds of iterated deletion of strictly

dominated strategies. Hence none of those strategies can be in the supportof any mixed strategy Nash equilibrium. But it is also clear that

(a3, a

3

)is

not a Nash equilibrium of the game.Reading: Chapter 8 in MWG, chapter 2 in FT and to a lesser extentChapter 1 in Gibbons (1992) cover the material of this chapter.

Chapter 3

Extensive Form Games

3.1 Introduction

This lecture covers dynamic games with complete information. As dynamicgames, we understand games which extend over many periods, either finitelymany or infinitely many. The new concept introduced here is the notionof subgame perfect Nash equilibrium (SPE), which is closely related to theprinciple of backward induction.

The key idea is the principle of sequential rationality : equilibrium strate-gies should specify optimal behaviour from any point in the game onward.The notion of subgame perfect Nash equilibrium is a strengthening of theNash equilibrium to rule out incredible strategies. We start with finite games,in particular games with a finite horizon.

We then consider two of the most celebrated applications of dynamicgames with complete information: the infinite horizon bargaining game ofRubinstein and the theory of repeated games.

Example 31 Predation. There is an entrant firm and an incumbent firm.Sharing the market (1, 1) is less profitable than monopoly (0, 2), but it isbetter than a price war (−3,−1). Depict the game in the normal form andthe extensive form. Characterize the equilibria of the game. Distinctionbetween subgame perfect and Nash equilibrium. Credible vs. incredible threat.Future play (plan) affects current action.

Example 32 Cournot Model. Stackelberg Equilibrium. Consider

35

CHAPTER 3. EXTENSIVE FORM GAMES 36

again the quantity setting duopoly with aggregate demand

P (q1, q2) = a− q1 − q2

and the individual profit function:

Πi (qi, qj) = (a− q1 − q2)qi

The best response function in the quantity setting game are, as we observedbefore,

qj = R (qi) =a− qi

2Suppose player i moves first, and after observing the quantity choice of playeri , player j chooses his quantity optimal. Thus i takes into account thereaction (function) of player j, and hence

Πi (qi, R (qi)) = (a− qi −a− qi

2)qi = (

a

2− qi

2)qi

and the associated first-order conditions are

Π′i (qi, R (qi)) =

a

2− qi = 0.

The Stackelberg equilibrium is then

qi =a

2, Πi

(a

2,a

4

)=

a2

8

qj =a

4, Πj

(a

2,a

4

)=

a2

16The sequential solution method employed here is called “backward induction”and the associated Nash equilibrium is called the “subgame perfect” Nashequilibrium. Comparing with the Cournot outcome we find

qSi + qS

j ≥ qCi + qC

i =a

3+

a

3

but

ΠSi > ΠC

i =a2

9, ΠS

j < ΠCj .

This example illustrates the value of commitment, which is in this game sim-ply achieved by moving earlier than the opponent.Notice that the Cournot equilibrium is still a Nash equilibrium. But is it rea-sonable? One has to think about the issue of “non credibility” and “emptythreats”.


3.2 Definitions for Extensive Form Games

3.2.1 Introduction

The extensive form of the game captures (i) the physical order of play, (ii)the choices each player can take, (iii) rules determining who is to move andwhen, (iv) what players know when they move, (v) what the outcome is asa function of the players’ actions and the players’ payoff from each possibleoutcome, (vi) the initial condition that begin the game (move by nature).

3.2.2 Graph and Tree

We introduce all the elements of the conceptual apparatus, called the gametree. The language of a tree is borrowed from graph theory, a branch ofdiscrete mathematics and combinatorics.

A graph G is given by (V, E), where V = {v1, ..., vn} is a finite set ofnodes or vertices and E = {vivj, vkvl, ..., vrvs} is a set of pairs of vertices (or2-subsets of V), called branches or edges which indicates which nodes areconnected to each other. In game theory, we are mostly interested in graphsthat are directed. Edges in a directed graph are ordered pairs of nodes (v1, vj)and we say that the edge starts at vi and ends at vj. A walk in a graph Gis a sequence of vertices,

v1, v2, ...., vn

such that vi and vi+1 are adjacent, i.e., form a 2-subset. If all its vertices aredistinct, a walk is called a path. A walk v1, v2, ...., vn such that v1 = vn iscalled a cycle.

A graph T is a tree if it has two properties

(T1) T is connected,

(T2) there are no cycles in T,

where connected means that there is a path from every vertex to every othervertex in the graph. A forest is a graph satisfying (T2) but not necessarily(T1).

3.2.3 Game Tree

The game form is a graph T endowed with a physical order represented by≺, formally (T,≺) The order ≺ represents precedence on nodes. We write


y ≺ x if y precedes x in the game form, i.e. if every path reaching x goesthrough y. The relation ≺ totally orders the predecessors of each member ofV . Thus ≺ is asymmetric and transitive.

The following notation based on the game form (T,≺) is helpful:

name notation definition

terminal nodes (outcomes) Z {t ∈ T : s (t) = ∅}decision nodes X {T\Z}initial nodes W {t ∈ T : p (t) = ∅}predecessors of t p (t) {x ∈ T : x ≺ t}immediate predecessor of t p1 (t) {x ∈ T : x ≺ t and y ≺ t ⇒ y ≺ x}

n-th predecessor pn (t) p1 (pn−1 (t)) with pn−1 (t) /∈ W, p0 (t) = tnumber of predecessors l (t) pl(t) (t) ∈ W

immediate successor s (x) {y ∈ T : x ≺ y and t ∈ y ⇒ y ≺ t}terminal successor z (x) {z ∈ Z : x ≺ z} for x ∈ X.

In order to define formally the extensive form game, we need the followingingredients.

(i) The set of nodes V consists of initial nodes W , decision nodes X, andterminal nodes Y .

T = {W, X, Y } .

(ii) A mappingp : T\W → T,

which is called the predecessor mapping (it is a point to set mapping). Theimmediate predecessor nodes are p1 (x) and the immediate successor nodesof x are

s (x) .where p1 (s (x)) = x.

(iii) An assignment function

α : T\W → A,

where we require that α is a one-to-one function onto the set s (x) of x. Theterm α (s (x)) represents the sets of all actions which can be taken at x.


(iv) A functionι : X → I

assigning each decision node to the player who moves at that node.

Definition 33 A partition H of a set X is a collection of subsets of X, with

H = {h1, ..., hk} ,

s.th.hk ∩ h′k = ∅,

and ⋃k

hk = X.

Definition 34 A partition H of X is called a collection of information setsif ∀x′, x′′

x′′ ∈ h (x′) ⇒ ι (x′) = ι (x′′) and α (s (x′)) = α (s (x′′)) . (3.1)

Information sets as histories. By the consistency requirement im-posed in (6.3) we define the set of actions which can be taken at an informa-tion set h as A (h), where

A (h) , α (s (h)) ⇔ A (h) , α (s (x′)) = α (s (x′′)) , for all x′′ ∈ h (x′) .

The partition H can be further partitioned into Hi such that Hi = ι−1 (i),where Hi is the set of all decision nodes at which i is called to make a move.A prominent way of identifying an information set is through the historyleading to it. Hence h ∈ H is often called a particular history of the game.1

(vi) A collection of payoff functions

u = {u1 (·) , ..., uI (·)} ,

assigning utilities to each terminal node that can be reached

ui : Z → R.

1For notational convenience, we assume that α is onto and that for each a ∈ A, A−1 (a)is a singleton on H, but not on X of course. That is, each action can be taken only in asingle information set.


Observe that since each terminal node can only be reached through a singlepath, defining payoffs on terminal nodes is equivalent to defining payoffs onpaths of actions.

(vii) An initial assessment ρ as a probability measure over the set ofinitial nodes

ρ : W → [0, 1] ,

assigning probabilities to actions where nature moves.

Definition 35 (Extensive Form Game) The collection {T,≺; A, α; I, ι; H; ui; ρ}is an extensive form game.

Definition 36 (Perfect Information) A game is one of perfect informa-tion if each information set contains a single decision node. Otherwise, it isa game of imperfect information.

Definition 37 (Almost Perfect Information) A game is of “almost” per-fect information if the player’s choose simultaneously in evert period t, know-ing all the actions chosen by everybody at dates 0 to t− 1.

3.2.4 Notion of Strategy

A strategy is a complete contingent plan how the player will act in everypossible distinguishable circumstance in which she might be called upon tomove. Consider first the following game of perfect information:

Definition 38 (Pure Strategy) A strategy for player i is a function

si : Hi → A, (3.2)

where A = ∪hA (h) , such that si (h) ∈ A (h) for all h ∈ Hi.

Definition 39 (Mixed Strategy) Given player i′s (finite) pure strategyset Si, a mixed strategy for player i,

σi : Si → [0, 1]

assigns to each pure strategy si ∈ Si a probability σi (si) ≥ 0 that it will beplayed where ∑

si∈Si

σi (si) = 1.


Definition 40 (Outcome) The outcome induced by a mixed strategy pro-file σ in an extensive form game is the distribution induced by the initialassessment ρ and the mixed strategies on the terminal nodes.

In general the number of pure strategies available to each player is givenby

|Si| =∏

hi∈Hi

|A (hi)|

where |S| represents in general the cardinality of a set S, i.e. the number ofmembers in the set. If the number of actions available at each informationset hi is constant and given by |A (hi)|, then |Si| simplifies to

|Si| = |A (hi)||Hi| .

In extensive form games, it is often customary to use a slightly differentnotion for a mixed strategy. In the definition above, the number of com-ponents needed to specify a mixed strategy is |Si| − 1. This is often a verylarge number. A simpler way to define a random strategy is the following.Rather than considering a grand randomization at the beginning of the gamefor each player, we could consider a sequence of independent randomizations,one at each information set for each of the players. This leads to the notionof a behaviour strategy.

Definition 41 (Behaviour Strategy) A behaviour strategy of player i isa mapping

bi : Hi → ∆ (A)

such that for all hi ∈ Hi, b (hi) ∈ ∆ (A (hi)) .

Observe that in order to specify a behaviour strategy for player i, only∑hi∈Hi

|A (hi)− 1|

components must be specified. It is immediately obvious that each behaviourstrategy generates a mixed strategy in the sense of the above definition. Muchless obvious is that most often it is enough to consider behaviour strategiesin the sense that each outcome induced by a mixed strategy profile is alsoinduced by some behaviour strategy profile. This is true as long as the gameis one of perfect recall.


Definition 42 (Perfect Recall) An extensive form game {T,≺; A, α; I, ι; H; ui; ρ}has perfect recall if

i) ∀i,∀hi ∈ Hi, x, x′ ∈ hi ⇒⇁ (x ≺ x′).ii) If x′′ ∈ h (x′) and x ∈ p (x′) , then ∃x̂ ∈ h (x) such that x̂ ∈ p (x′′) and

all the actions on the path from x to x′ coincide with the actions from x̂ tox′′.

The first requirement says intuitively that no player forgets his moves,and the second says that no player forgets past knowledge.

Theorem 43 (Kuhn’s Theorem) If the extensive form game {T,≺; A, α; I, ι; H; ui; ρ}has perfect recall and an outcome µ ∈ ∆ (Z) is induced by ρ and σ =(σ1, ..., σN) , then there is a behaviour strategy profile b = (b1, ..., bN) suchthat µ is induced by ρ and b.

In other words, in games of perfect recall, there is no loss of generality inrestricting attention to behaviour strategies.

The following example is due to Rubinstein and Piccione (see the specialissue of Games and Economic Behavior, July 1997, on games of imperfectrecall).

Example 44 (Absent Minded Driver) This is a single person game withtwo decision nodes for the player. The player must decide whether to keepon driving D on the highway or whether to exit E. If the driver chooses Dtwice, he ends in a bad neighborhood and gets payoff of −1. If he exits at thefirst node, he gets 0. M If he chooses D followed by E, then his payoff is 2.

The twist in the story is that the driver is absent minded. He cannotremember if he has already decided or not in the past. This absent minded-ness is represented by a game tree where both of the nodes are in the sameinformation set even though the two nodes are connected by an action by thesingle player. Hence the game is not one of perfect recall as defined above.

It is a useful exercise to draw the game tree for this problem and considerthe various notions of strategies. The first step in the analysis of any gamesis to decide what the possible strategies are. As usual, a strategy should bean assignment of a (possibly mixed) action to each information set. Withthis definition for a strategy, it is easy to show that the optimal strategy inthis game cannot be a pure strategy. Observe also that if the driver were ableto commit to a mixed strategy, then she would choose to exit with probabilityp = 2

3resulting in a payoff of 2

9.


If the driver were to adopt the strategy of exiting with probability 23, then

she would assign probabilities 3/4 and 1/4 respectively to the first and the sec-ond node in her information set. In order to calculate correctly the payoffsto the driver, behaviour at other nodes must be kept constant when calculat-ing the current payoff to exiting or driving on. Thus choosing D and thenfollowing the proposed strategy yields a payoff of

3

4(2

2

3− 1

3)− 1

4=

1

2.

Choosing E yields similarly a payoff of 12.

The key to understanding the workings of the game lies in the inter-pretation of a deviation from the proposed equilibrium strategy. Most gametheorists believe that in games of this type, a deviation should be modeled asa single deviation away from the proposed strategy. In particular, this wouldimply that a deviation can take place at a single node in the information setwithout necessarily happening at the other. Piccione and Rubinstein offer adifferent interpretation of the game, see the discussions in the issue of GEBas cited above.

3.3 Subgame Perfect Equilibrium

Before we can define the notion of a subgame perfect equilibrium we need toknow what a subgame is. The key feature of a subgame is that, contemplatedin isolation it forms its own game.

Definition 45 (Subform) A subform of an extensive form is a collection ofnodes T̂ ⊆ T , together with {≺; A, α; I, ι; H} all defined on the restriction toT̂ , satisfying closure under succession and preservation of information sets:

if x ∈ T̂ , then s (x) ⊆ T̂ , h (x) ⊆ T̂ ,

A proper subform is a subform T̂ consisting solely of some node x and itssuccessors.

Definition 46 (Subgame) Given a proper subform, the associated propersubgame starts at x, with the payoffs restricted to T̂ ∩ Z, and the initialassessment ρ̂ (x) = 1.


Definition 47 (Subgame Perfect Nash Equilibrium) A profile of strate-gies

σ = (σ1, ..., σn)

in an extensive form game ΓE is a SPE if it induces a Nash equilibrium inevery subgame of ΓE.

Clearly, every SPE is an NE, but the converse doesn’t hold. Considerfirst the extensive form games with perfect information. Chess is a finiteextensive form game as is centipede game introduced above. What is theunique subgame perfect equilibrium in this game?

It is a description of the equilibrium strategy for each player:

s∗1 = (s, s′, s′′) , s∗2 = (s, s′) ,

on and off the equilibrium path. The equilibrium path is the set of connectededges which lead from an initial node to a final node. The equilibrium pathis thus {s} and the equilibrium outcome is the payoff realization (1, 1).Thus a description of the equilibrium strategy asks for the behaviour of everyplayer on and off the equilibrium path.

Theorem 48 (Zermelo, Kuhn) Every finite game of perfect informationΓE has a pure strategy subgame perfect Nash equilibrium.

3.3.1 Finite Extensive Form Game: Bank-Run

Consider the following bank-run model with R > D > r > D/2. Theinterpretation is as follows. The bank acts as an intermediary between lenderand borrower. Each lender deposit initially D. The bank also attempts totransform maturity structures. The deposits are short-term, but the loan islong-term. Suppose the deposits are applied towards an investment projectwhich has a return of 2r if the project is liquidated after one period and 2Rif liquidated after two periods. The return is 2D if the project is continuedfor more than two terms. If either of the depositor asks for a return of itsdeposit, then the project is liquidated, and the withdrawing depositor eithergets D or his share in liquidation proceeds, whichever is larger. Denote thefirst stage actions by w for withdrawing and n for not withdrawing. Let thesecond period actions for the two players be W and N respectively. Thusthe stage payoffs are given by the two matrices There are two pure strategy


w nw r, r D, 2r −Dn 2r −D, D second period

Figure 3.1: First period

W NW R,R 2R−D, DN D, 2R−D D, D

Figure 3.2: Second period

subgame perfect equilibria of this game

s∗1 = {w, W} , s∗2 {w,W}

ands∗1 = {n,W} , s∗2 {n, W}

Consider the following two-period example with imperfect information.The first period game is a prisoner’s dilemma game and the second game iscoordination game: and then Specify all subgame perfect equilibria in this

α βα 2, 2 −1, 3β 3,−1 0, 0

Figure 3.3: First period

game. Does the class of subgame perfect equilibria coincide with the class ofNash equilibria? For x ≥ 2, cooperation is sustainable in a subgame perfectequilibria by switching from the pure to the mixed strategy equilibrium inthe second stage game.

3.4 The Limits of Backward Induction: Ad-

ditional Material

Consider the following example of game, referred to as “Centipede Game”.There are two players each start with 1 dollar. They alternate saying “stop”


γ δγ x, x 0, 0δ 0, 0 x, x

Figure 3.4: Second period

or “continue”, starting with player 1. If they say continue, 1 dollar is takenfrom them and 2 dollar is given to the other player. As soon as either playersays stop the game is terminated and each player receives her current pile ofmoney. Alternatively, the game ends if both players pile reach 100 dollars:

(1) −→ (2) −→ (1) −→ (2) −→ (1) .... (2) −→ (1) −→ (100, 100)↓ ↓ ↓ ↓ ↓ ↓

(1, 1) (0, 3) (2, 2) (1, 4) (98, 101) (100, 100)

The first entry in the payoff vector refers to player 1, the second to player2. Instead of looking at the ‘true’ centipede game, let us look at a smallerversion of the game:

(1) c (2) c (1) c′ (2) c′ (1) c′′ (2, 4)s s s′ s′ s′′

(1, 1) (0, 3) (2, 2) (1, 4) (3, 3)

What is a strategy for player 1 in this small centipede game? A strategy isa complete description of actions at every information set (or history). Player1 has two histories at which he could conceivable make a choice H1 = {∅, cc}and likewise H2 = {c, ccc′}. Thus a possible strategy for player 1 is

h1 = ∅ h1 = cc h1 = ccc′c′

↓ ↓ ↓s1 = { c , c′ s′′ }.

Another possible strategy is however:

h1 = ∅ h1 = cc h1 = ccc′c′

↓ ↓ ↓s1 = { s , c′ s′′ }.

The notion of a strategy in the extensive form game is stretched beyondwhat we would call a plan. We saw in the centipede game that it requires


a player to specify his action after histories that are impossible if he carriesout his plan. A different interpretation of the extensive form game may helpour understanding. We may think of each player having as many agents (ormultiple selves) as information sets. The strategy of each player is to giveinstructions to each agents (which he may not be able to perfectly control)as to what he should do if he is called to make a decision. All agents of thesame player have the same payoff function. A plan is then a complete set ofinstructions to every agent who acts on behalf of the player.

Notice that we did not discuss notions similar to rationalizability in thecontext of extensive form games. The reason for this is that making senseof what common knowledge of rationality means in extensive form games ismuch more problematic than in normal form games. A simple example willillustrate.

Robert Aumann has maintained that common knowledge of rationalityin games of perfect information implies backwards induction. Hence e.g., thecentipede game should have a unique solution where all players stop the gameat all nodes. Suppose that this is the case. Then after observing a move wherethe first player continues the game, the second player ought to conclude thatthe first player might not be rational. But then it might well be in the secondplayer’s best interest to continue in the game. But if this is the case, then itis not irrational for the first player to continue at the first node. It shouldalso be pointed out that knowledge and belief with probability 1 are more orless the same in normal form games, but in extensive form games these twonotions yield very different conclusions. Appropriate models of knowledgefor extensive form games form a very difficult part of interactive logic (orepistemic logic) and Chapter 14 in FT is a slightly outdated introductioninto this area.

3.5 Reading

The notion of extensive form game is first (extensively) developed in Kuhn(1953) and nicely developed in Kreps (1990) and Kreps and Wilson (1982).The relation between mixed and behaviour strategies is further discussed inFT, Ch.3.4.3.

Chapter 4

Repeated Games

4.1 Finitely Repeated Games

We first briefly consider games which are repeated finitely many times. Thenwe move on the theory of infinitely repeated games. We first introduce somenotation. Consider the prisoner’s dilemma game: and repeat the stage game

C DC 4, 4 0, 5D 5, 0 1, 1

Figure 4.1: Prisoner’s Dilemma

T times. A pure strategy si =(s1

i , ..., sTi

)for a player i in the repeated game

is then a mapping from the history of the game H t into the strategies in thestage game Si:

sti : H t−1 → Si.

A complete specification of a repeated game strategy has to suggest a strategychoice after every possible history of the game, (and not only the historieswhich will emerge in equilibrium). A complete specification can be thoughtof as a complete set of instructions written before playing the game, so thatno matter what happens in the game, once it started a prescription of playis found the set of instructions. The history of the game is formed by thepast observed strategy choices of the players:

H t−1 =(s11, ..., s

1I ; ...; s

t−11 , ..., st−1

I

).

48

CHAPTER 4. REPEATED GAMES 49

Definition 49 Given a stage game Γ ={I, (Si)i∈I , (ui)i∈I

}, we denote by

Γ (T ) the finitely repeated game in which Γ is played T times.

The outcomes of all preceding plays are observed before the next playbegins. The payoffs for Γ (T ) are simply the sums of the payoffs from the Tstage games.

Definition 50 In a repeated game Γ (T ) a player’s strategy specifies theaction the player will take in each stage, for each possible history of playthrough the previous stage:

si ={s0

i , ...., sTi

}where:

sti : H t−1 → Si.

Theorem 51 If the stage game Γ has a unique Nash equilibrium then, forany finite T , the repeated game Γ (T ) has a unique subgame-perfect outcome:the Nash equilibrium of Γ is played in every stage.

Corollary 52 Every combination of stage game Nash equilibria forms a sub-game perfect equilibrium.

Consider now the following game with multiple equilibria in the stagegame Can coordination among the players arise in a stage game?

C D EC 4, 4 0, 5 −3,−3D 5, 0 1, 1 −3,−3E −3,−3 −3,−3 -2,-2

Figure 4.2: Modified Prisoner’s Dilemma

Example 53 To understand the complexity of repeated games, even if theyare repeated only very few times, let us consider the prisoner’s dilemma playedthree times. What are all possible histories of a game? How many strategychoice have to be made in order for the repeated game strategy to be a completespecification?


The history H−1 is naturally the empty set:

H−1 = {∅} .

The history H0 is the set of all possible outcomes after the game has beenplayed once. Each outcome is uniquely defined by the (pure) strategy choicesleading to this outcome, thus

H0 = {CC, CD,DC, DD} .

The cardinality of possible outcomes in period 0 is of course∣∣H0∣∣ = |S1| × |S2| = 4.

We then observe that the number of possible histories grows rather rapidly.In period 1, it is already∣∣H1

∣∣ = |S1| × |S2| × |S1| × |S2| = 16,

and in general it is ∣∣H t∣∣ = (|S1| × |S2|)t+1 .

However the complexity of strategies grows even faster. A complete strategyfor period t specifies a particular action for every possible history leading toperiod t. Thus the number of contingent choices to be made by each playerin period t is given by∣∣St

i

∣∣ = |Si||Ht−1| = |Si||S1|×|S2|t+1

.

Notice that a strategy even in this two period game is not merely an instruc-tion to act in a specific manner in the first and second period, but needs tospecify the action after each contingency.

In order to sustain cooperation, two conditions need to be satisfies: (i)there are rewards from cooperation, and (ii) there are credible strategies topunish deviators. This theme of “carrots and sticks” is pervasive in the entiretheory of repeated games. As the punishment occurs in the future, agentsmust value the future enough so that lower payoffs in the future would in-deed harm the agent’s welfare sufficiently. If the discount factor representsthe agent’s attitude towards the future, then higher discount factors makecooperation easier to achieve as lower payoffs in the future figure more promi-nently in the agent’s calculation.


4.2 Infinitely Repeated Games

Next we consider games which are repeated infinitely often.

Definition 54 Given a stage game Γ, let Γ (δ) denote the infinitely repeatedgame in which Γ is repeated forever and where each of the players discountsfuture at discount factor δ < 1.

The payoff in an infinitely repeated game with stage payoffs πt and dis-count factor δ < 1 is given by

∞∑t=0

δtπt.

Definition 55 Given the discount factor δ, the average payoff of the in-finitely repeated sequence of payoffs π0, π1, π2, ... is given by

(1− δ)∞∑

t=0

δtπt.

Notice that in infinitely repeated games every subgame is identical to thegame itself.

4.2.1 The prisoner’s dilemma

Let us consider again the stage game in figure 4.1 which represented the pris-oner’s dilemma and consider the following strategy suggested to the players.I cooperate in the first period unconditionally and then I continue to cooper-ate if my opponent cooperated in the last period. In case he defected in thelast period, I will also defect today in all future periods of the play. Whencan we sustain cooperation? This strategy is often called the grim-triggerstrategy, which can be formally described by

sit =

{C if Ht−1 = {CC, ...., CC} or H−1 = ∅D if otherwise

We now want to show that this set of strategies form a subgame perfectequilibrium. To prevent a deviation the following has to hold:

5 +β

1− β≤ 4

1− β,


or5 (1− β) + β ≤ 4,

so that

β ≥ 1

4.

Thus if the players are sufficiently patient, cooperation can be sustained inthis repeated game.

The equilibrium we suggested implies cooperation forever along the equi-librium path. However, this is clearly not the only equilibrium. There aremany more, and many of them of very different characteristics. In the limitas δ → 1, one can show, and this is the content of the celebrated “folk”theorem of the repeated games, that the entire convex hull of all individu-ally rational payoffs of the stage game can be achieved as a subgame perfectequilibrium. Notice, that this may involve very asymmetric payoffs for theplayers.

4.3 Collusion among duopolists

The special feature of the prisoner’s dilemma game is that the minimum pay-off a player can guarantee himself in any stage, independent of the actionschosen by all other players, is equal to the payoff in the stage game Nashequilibrium. This minimum payoff is called the reservation payoff. It con-stitutes the limit to the damage an opponent can inflict on the player. Theindividual rational payoff vi is the lowest payoff that the other players canforce upon player i:

vi = mins−i∈S−i

maxsi∈Si

ui (si, s−i) .

4.3.1 Static Game

Consider the following Cournot oligopoly model with prices determined byquantities as

p (q1, q2) = a− q1 − q2.

The Cournot stage game equilibrium is found by solving the first order con-ditions of the profit functions, where we normalize the marginal cost of pro-duction to zero,

πi (q1, q2) = qi (a− q1 − q2) .


We saw earlier that the Cournot outcome is

qCi =

a

3,

and the associated profits are:

πi

(qC1 , qC

2

)=

a2

9.

In contrast, the strategy of the monopolist would be to maximize

π (q) = q (a− q) ,

which would result in the first order conditions

a− 2q = 0

and henceqM =

a

2

and the profits are

π(qM)

= πM =a2

4.

4.3.2 Best Response

Suppose the two firms agree to each supply q to the market, then the profitsare given by:

πi (q, q) = q (a− 2q) = aq − 2q2.

By the best response function, the optimal response to any particular q is

q∗ (q) =a− q

2(4.1)

and the resulting profits are

π1 (q∗ (q) , q) =(a− q)2

4. (4.2)

where:(a− q)2

4−(aq − 2q2

)=

1

4(3q − a)2 ≥ 0

and with equality at q = a3, which is the Cournot output.


4.3.3 Collusion with trigger strategies

The discrepancy between monopoly and duopoly profit suggests that theduopolists may collude in the market to achieve higher prices. How couldthey do it? Consider the following strategy.

qit =

{qM

2if ht =

{(qM

2, qM

2

), ....,

(qM

2, qM

2

)}or ht = ∅

qC otherwise

∣∣∣∣∣and consider whether the symmetric strategy can form a subgame perfectequilibrium. We have to consider the alternatives available to the players.Consider a history in which

ht =

{(qM

2,qM

2

), ....,

(qM

2,qM

2

)},

then the resulting payoff would be

1

1− δ

1

2πM ≥ πd +

δ

1− δπC . (4.3)

Thus to evaluate whether the player follows the equilibrium strategy, we haveto evaluate πd.

SPE

This suggests the following trigger strategy equilibrium:

1

1− δ

1

2πM ≥ πd +

δ

1− δπC , (4.4)

which can be sustained for δ ≥ 917

. The question then arises whether wecan do better; or more precisely whether there are punishment strategieswhich would act even more harshly on deviators. We still have to verify thatfollowing a deviation i.e., all histories different from collusive histories,

1

1− δπC ≥ πd +

δ

1− δπC ; (4.5)

but since qi = qC , (4.4) is trivially satisfied.


4.3.4 Collusion with two phase strategies

We next consider a repeated game where the equilibrium strategy consists oftwo different phases: a maximal reward and a maximal punishment phase,a “carrot and stick” strategy. The strategy may formally be described asfollows:

qit =

qM

2if ht =

{(qM

2, qM

2

), ....,

(qM

2, qM

2

)}qM

2if

(qit−1, q

jt−1

)=(q, q)

q if otherwise

∣∣∣∣∣∣∣Let x denote the level of output in the punishment phase (to be determinedas part of the equilibrium). We may then write the equilibrium conditionsas:

1

1− δ

1

2πM ≥ πd +

Punishment︷︸︸︷δ(ax− 2x2

)+

Continuation︷︸︸︷δ2

1− δ

1

2πM (4.6)

and

Punishment︷︸︸︷(ax− 2x2

)+

Continuation︷︸︸︷δ

1− δ

1

2πM ≥

Deviation︷︸︸︷(a− x)2

4+

Punishment︷︸︸︷δ(ax− 2x2

)+

Continuation︷︸︸︷δ2

1− δ

1

2πM (4.7)

Equation (4.6) may be rewritten as,

(1− δ) (1 + δ)

1− δ

1

2πM ≥ πd + δ

(ax− 2x2

)and finally as:

δ

(1

2πM −

(ax− 2x2

))≥ πd − 1

2πM . (4.8)

The inequality (4.8) suggests that the relative gains from deviating in con-trast to collusion today must be smaller than the relative losses tomorrowinduced by a deviation today and again in contrast to continued collusion.Similarly for the punishment to be credible, starting from (4.7), we get

(ax− 2x2

)+ δ

1

2πM ≥ (a− x)2

4+ δ

(ax− 2x2

). (4.9)

and which permits an intuitive interpretation in terms of today’s and tomor-row’s payoff, or alternatively as the one shot deviation strategy. By inserting


the value of πM we obtain two equalities:

δ(

12

a2

4− (ax− 2x2)

)=(

964

a2 − 12

a2

4

)δ(

12

a2

4− (ax− 2x2)

)= (a−x)2

4− (ax− 2x2)

which we can solve and obtain the following results as the harshest level ofpunishment x = 5

12a and the lowest level of discount factor δ = 9

32for which

cooperation is achievable.Reading: Gibbons (1992), Ch. 2.3 is a nice introduction in the theory

of repeated games. A far more advanced reading is FT, Ch.5. The charac-terization of extremal equilibria is first given in Abreu (1986).

Chapter 5

Sequential Bargaining

The leading example in all of bargaining theory is the following trading prob-lem in the presence of bilateral monopoly power. A single seller and a singlebuyer meet to agree on the terms of trade for a single unit of output. Theseller incurs a cost of c to produce the good while the buyer has a value ofv for the good. If the two parties have payoff functions that are quasilinearin money and the good, then there are positive gains from trade if and onlyif v > c. The two parties have also an opportunity to refuse any trade whichyields them an outside option value of 0. Both parties prefer (weakly) trad-ing at price p to not trading at all if p satisfies v ≥ p ≥ c. Which (if any) ofsuch prices will be agreed upon when the two parties negotiate?

The conventional wisdom up to late 1970s was that the determinationof the price depends largely on psychological and sociological factors deter-mining the bargaining power of the two parties and as a result economicshad little to say on the issue. When the concept of subgame perfect equi-librium was established as the standard solution concept in games of perfectinformation, the bargaining issue was taken up again. Preliminary work wasdone on bargaining in simple two period models, but the real breakthroughwas the infinite horizon bargaining model with alternating offers proposedby Ariel Rubinstein in 1982. There he showed that the infinite horizon gamehas a unique subgame perfect equilibrium and that the division of surplusbetween the bargainers can be explained in terms of the time discount ratesof the bargainers.

It should be pointed out that other approaches to bargaining were alsoadopted. An entirely different methodology starts by describing a set ofpossible outcomes X that can be obtained in the bargaining process between

57

CHAPTER 5. SEQUENTIAL BARGAINING 58

the two players and an alternative d that represent the outcome in the case ofno agreement. Each of the bargainers has preference relation �i defined onX ∪{d}. Let the set of possible preference relations of i be denoted by Ri. Asolution to the bargaining problem is a function that associates an outcometo each profile of preferences. In other words, the solution is a function fsuch that

f : R1 ×R2 → X ∪ d.

The method of analysis in this approach is often taken to be axiomatic. Theanalyst decides in the abstract a set of criteria that a good solution conceptshould satisfy. An example of a criterion could be Pareto efficiency of thesolutions, i.e. if x �i x′ and x �j x′, then x′ /∈ f (�i,�j) . Once the cri-teria have been established, the analyst determines the set of functions fthat satisfy these criteria. Preferably this set is small (ideally a singleton)so that some clear properties of the solutions can be established. This ap-proach precedes the noncooperative approach to be followed in this handout.The first contribution to this cooperative or axiomatic approach to bargain-ing was made by John Nash in “The Bargaining Problem” in Econometrica1950. Various other solutions were proposed for two player and many playerbargaining models. A good survey of these developments can be found inchapters 8-10 in Roger Myerson’s book “Game Theory: Analysis of Conflict”.Rubinstein’s solution of the noncooperative bargaining game picks the samesolution as Nash’s original cooperative solution for that game. A recurringtheme in the noncooperative bargaining literature has been the identifica-tion of fully strategic models that yield the classical cooperative bargainingsolutions as their equilibrium outcomes.

5.1 Bargaining with Complete Information

In this section, we focus our attention on the case where two players bargainover the division of 1 unit of surplus. We start by considering the followingsimple extensive form to the bargaining procedure. There are just two stagesin the game. Player 1 proposes a sharing rule for the surplus we assumethat all real numbers between 0 and 1 represent feasible shares. Denote theshares going to 1 and 2 by x and 1 − x respectively. Player 2 then sees theproposal and decides whether to accept it or not. If the proposal is accepted,the outcome in the game is a vector (x, 1− x) for the two players, if the offeris rejected, the outcome is (0, 0) . It is easy to see that the only subgame


perfect equilibrium in this simple game results in a proposal x = 1 which isaccepted in equilibrium by player 2. In this case, we say that player 1 hasall the bargaining power because she is in the position of making a take-it-or-leave-it offer to player 2 and as a result, player 1 gets the entire surplusin the game.

Consider next the version of the game that extends over 2 periods. Thefirst period is identical to the one described in the paragraph above exceptthat if player 2 refuses the first period offer, the game moves into the secondperiod. The second period is identical to the first, but now player 2 makes theoffer and 1 decides whether to accept or not. At the end of the second stage,the payoffs are realized (obviously if player 2 accepts the first offer, then thepayoffs are realized in period 1). If the players are completely patient, i.e.their discount factor is 1, then it is clear that the subgame perfect equilibriain the game have all the same outcomes. Since player 2 can refuse the firstperiod offer which put her in the position of player 1 in the paragraph above,it must be that the equilibrium shares in the game are (0, 1) . Hence we couldsay that player 2 has all the bargaining power since she is the one in a positionto make the last take it or leave it offer. The situation changes somewhat ifthe players are less than perfectly patient. Assume now that player i has adiscount factor δi between the periods in the game. If player 2 refuses thefirst offer and gets all of the surplus in the second period, her payoff from thisstrategy is δ2. Hence she will accept any first period offer x > δ2 and refuseany offer x < δ2. At x = δ2, she is indifferent between accepting and rejectingthe offer, but it is again easily established that in the unique subgame perfectequilibrium of the game, she will accept the offer x = δ2 and the outcomeof the game is (1− δ2, δ2) . Notice that the introduction of time discountinglessens the advantage from being in the position to make the take it or leaveit offer since the value of that eventual surplus is diminished.

If there were 3 periods in the game, the unique equilibrium in the subgamethat starts following a rejection of the first offer would be the solution of the2 period game (by uniqueness of the subgame perfect equilibrium in the 2period game with discounting). Hence player 2 could secure a payoff of 1−δ1

from the second period onwards. As a result, player 1 must offer to keepexactly 1− δ2 (1− δ1) to himself in the first period and the outcome is thus(1− δ2 (1− δ1) , δ2 (1− δ1)) . At this point, we are ready to guess the form ofthe equilibrium in a game with T periods.

Proposition 56 Let xT denote the unique subgame perfect equilibrium share


of player 1 in the bargaining game of length T. Then we have

xT+2 = 1− δ2

(1− δ1x

T).

Furthermore, the offer of xT is accepted immediately.

Proof. Consider the game with T + 2 periods. In period T + 1, player2 must offer a share of δ1x

T to player 1. This results in surplus(1− δ1x

T)

to player 2. Hence player 1 must offer a share δ2

(1− δ1x

T)

to player 2 inperiod T+2 proving the first claim. To see that in equilibrium, the first periodoffer must be accepted, assume to the contrary and denote the equilibriumcontinuation payoff vector by (v1, v2) after the refusal by player 2. Becauseof time discounting and feasibility v1 + v2 < 1. But then it is possible for 1to offer x′ such that x′ > v1 and (1− x′) > v2 contradicting the optimalityof the original offer.

The proposition gives the equilibrium shares as a solution to a differenceequation. The most interesting question about the shares is what happens asthe horizon becomes arbitrarily long, i.e. as T → ∞. Let x∗ = limT→∞ xT .Then it is easy to see that

x∗ =1− δ2

1− δ1δ2

.

Observe that in the case of identical discount factors, this share is monoton-ically decreasing in δ and that (by using L’Hopitals rule) limδ→1 x∗ (δ) = 1

2.

Hence the unique outcome in finite but long bargaining games with equallypatient players is the equal division of surplus.

Notice that finite games of this type always put special emphasis on theeffects of last stages. By the process of backwards induction, these effectshave implications on the earlier stages in the game as well. If the bargainingcan effectively take as much time as necessary, it is questionable if it isappropriate to use a model which features endgame effects. The simplestway around these effects is to assume that the bargaining process may takeinfinitely many periods. A simple argument based on the continuity of thepayoff functions establishes that making the offers from the finite game inall periods of the infinite game continues to be an equilibrium in the infinitegame. The more difficult claim to establish is the one showing that there areno other subgame perfect equilibria in the game. In order to establish this,we start by defining the strategies a bit more carefully.


The analysis here is taken from Fudenberg and Tirole, Chapter 4. Theinfinite horizon alternating offer bargaining game is a game with perfectinformation, i.e. at all stages, a single player moves and all of the previousmoves are observed by all players. In odd periods, player 1 makes the offerand player 2 either accepts (thereby ending the game) or rejects and the gamemoves to the next period. In even periods, the roles are reverse. Denote byht the vector of all past offers and acceptance decisions. A strategy for playeri is then a sequence of functions:

st1 : ht → [0, 1] for t odd,

st1 : ht × [0, 1] → {A, R} for t even.

The strategies of player 2 are similarly defined.We say that an action at

i is conditionally dominated at history ht if in thebeginning of the subgame indexed by ht, every strategy that assigns positiveprobability to at

i is strictly dominated. Iterated deletion of conditionally dom-inated strategies removes in each round all conditionally dominated strategiesin all subgames given that the opponent’s strategies have survived previousdeletions. Observe that this process never deletes subgame perfect equilib-rium strategies. We will show that the alternating offer bargaining gamehas a unique strategy profile that survives iterated deletion of conditionallydominated strategies.

In what follows, we take each offer to be an offer for player 1’s share inthe game. Observe first that it is conditionally dominated for one to refuseany offer from 2 that gives 1 a share exceeding δ1 and similarly 2 must acceptany share for 1 that is below 1− δ2. At the second round, 2 will never offermore than δ1 and 2 will reject all offers above 1 − δ2 (1− δ1) . Also 1 neveroffers below 1− δ2 and never accepts offers below δ1 (1− δ2) .

Suppose then that after k iterations of the deletion process, 1 accepts alloffers above xk and 2 accepts all offers below yk where yk < xk. Then afterone more round of deletions, 2 never offers more than xk and rejects all offersabove 1 − δ2

(1− xk

). Player 1 never offers below yk and rejects all offers

below δ1yk.

At the next round, player 1 must accept all offers above xk+1 ≡ δ1 (1− δ2)+δ1δ2x

k. To see this, consider the implications of refusing an offer by player 2.This has three possible implications. First, it may be that agreement is neverreached. The second possibility is that 2 accepts some future offer from 1.This has a current value of at most δ1

(1− δ2

(1− xk

)). The third possibility


is that player 1 accepts one of player 2’s future offers. The payoff from thisis at most δ2

1xk. The payoff from the second possibility is the largest of the

three. Hence 1 must accept all offers above xk+1.A similar argument shows that player 2 must accept all offers below

yk+1 ≡ 1− δ2 + δ1δ2yk.

The sequences xk and yk are monotonic sequences with limits

x∞ =δ1 (1− δ2)

1− δ1δ2

, y∞ =1− δ2

1− δ1δ2

.

Hence player 2 rejects any offer where 1’s share is above y∞ = 1−δ21−δ1δ2

andaccepts all shares below y∞. Hence the unique outcome is again for 1 to

propose the share(

1−δ21−δ1δ2

, 1− 1−δ21−δ1δ2

)and for 2 to accept immediately.

5.1.1 Extending the Basic Model

It is a relatively easy exercise to show that if the surplus is divisible onlyin finitely many pieces, then any sharing of the surplus can be realized in asubgame perfect equilibrium as δ → 1. Again, the interpretation of this resultrelies on the view one takes on modeling. If it is thought that the fact thatmonetary sums can be divided only into pounds and pennies is important inthe mind of the bargainers, then this result is probably troublesome. If on theother hand it is thought that this is not that important for the bargainers,then it is probably best to use a model where the indivisibility is not present.

A more troublesome observation is that the result is somewhat sensitiveto the extensive form of the bargaining game. If the protocol of offers andrejections is modified, then other outcomes can be supported in equilibriumas well. Finally, the extensions to 3 or more player bargaining have beenunsuccessful. The uniqueness seems to be beyond the reach of the theory.Probably the most successful models in this class have been the ones wherea random proposer suggests a sharing of the surplus in each period and theacceptance is decided by majority vote. Models along these lines have beendeveloped by Baron and Ferejohn in the literature on political economy.

5.2 Dynamic Programming

How can the notion of backwards induction be extended to an infinite timehorizon? Consider first a schematic decision problem in a decision tree and


observe what the process of backward reduction does to reduce the com-plexity of the decision problem. It represents and reduces subtrees throughvalues, where the values represent the maximal value one can attain in anysubtree. When does this reduction lead to the optimal decision through theentire tree? If and only if we replace each subtree with the optimal value.

Theorem 57 Every finite decision tree (with finitely many periods and fi-nitely may choices) has an optimal solution which can be obtained by back-wards induction.

We can extend this theorem certainly to an interactive decision problem,provided that it maintains the structure of single person decision tree, namelyperfect information.

Theorem 58 (Zermelo, Kuhn) Every finite game of perfect informationΓE has a pure strategy subgame perfect Nash equilibrium.

The theorem applies to many parlour games, in particular to chess. Thusin principle we can apply backward induction to solve for a subgame perfectNash equilibrium in chess. How does a computer program approach theproblem of evaluating a move? By approximating the value of a move througha forward looking procedure which assesses the value of the position in threeor four moves from then on. We may apply the same idea of approximationto every problem with an infinite horizon. In fact, we can obtain in manysituations an exact approximation.

We now return to the bargaining problem as defined above. Two players,1 and 2 bargain over a prize of value 1.. The first player starts by makingan offer to the second player who can either accept or reject the offer. If herejects the offer, then in the next period he has the right to make an offerwhich can again be accepted or rejected. The process of making offers thenstarts again. The agents are impatient and have common discount factor of

δ = 1/ (1 + r) < 1.

Consider first the game with an infinite horizon. The surprising result andthe even more surprising simplicity of the argument is given first. We canadopt the definition of stationarity also for equilibrium strategies.


Definition 59 (Stationary strategies) A set of strategies:{st

i

}I,∞i=1,t=0

is stationary ifst

i = si, ∀i,∀t.

Theorem 60 The unique subgame perfect equilibrium to the alternate movebargaining game is given by for the offering party i to suggest

v =δ

1 + δ

in every period and for the receiving party i to always accept if v ≥ δ/ (1 + δ)and reject if v < δ/1− δ.

Notice that you have to specify the strategy for every period in the infinitetime horizon model. The proof proceeds in two parts. First we show that astationary equilibrium exists, and then we show that it is the unique subgameperfect equilibrium, stationary or not. To do the later part of the argument,first upper and lower bounds are obtained for each player’s equilibrium payoff.Then it is shown that the upper and lower bounds are equal, which showsthat an equilibrium exists and that it is unique.

Proof. We start with the stationary equilibrium and use directly thepayoffs and then later on derive the strategies which support these payoff.Suppose then that there is a stationary equilibrium. Then the following hasto be true for the person who accepts today:

v = δ (1− v) ,

which leaves us with

v =δ

1 + δ.

Next we show that this is indeed the unique subgame perfect equilibrium.Let us start in the first period. Let v̄ be the largest payoff that player 1

gets in any SPE. By the symmetry, or stationarity of the bargaining game,this is also the largest sum seller 2 can expect in the subgame which beginsafter she refused the offer of player 1. By backwards induction the payoff ofplayer 1 cannot be less than

v = 1− δv̄,


or1 = v + δv̄. (5.1)

Next we claim that v̄ cannot be larger than 1− δv if the offer is accepted inthe first period. But what about making an offer that is rejected in the firstperiod? As the second player gets at least v in the next period, this cannotgive the player more than δ (1− v) as he has to wait until the next period.Hence

v̄ ≤ 1− δv.

Note that this implies that

v̄ ≤ 1− δv = v + δv̄ − δv, (5.2)

where the last equality has been obtained through (5.1). But rewriting (5.2)we obtain

v̄ (1− δ) ≤ v (1− δ) ,

which be the definition of the upper and lower bounds imply that v̄ = v. Letus denote this payoff by v∗. Since v∗ = 1 − δv∗, we find that player 1 mustearn

v∗ =1

1 + δ,

and player 2 must earn

δv∗ =δ

1 + δ.

The derivation of the equilibrium strategies follows directly from here.

5.2.1 Additional Material: Capital Accumulation

Consider the following investment problem. The return of a capital kt inperiod t is αkt. The capital stock depreciates in every period by λ, so that

kt+1 = (1− λ) kt + it

and can be replenished by an investment level it. The cost of investment isgiven by

c (it) =1

2i2t .


The value of the investment problem can therefore be written as

V (k0) , max{it}∞t=0

{∞∑

t=0

δt

(αkt −

1

2i2t

)}and likewise for all future periods

V (kτ ) , max{it}∞t=τ

{∞∑

t=0

δ(t−τ)

(αkt −

1

2i2t

)}Thus if we knew the values in the future say V (k1), we could certainly findthe optimal policy for period 0, as we could formulate the following problem

V (k0) , maxi0

{αk0 −

1

2i20 + δV ((1− λ) k0 + i0)

}(5.3)

Supposing that V is differentiable, we would get as a first order condition

δV ′ ((1− λ) k0 + i0)− i0 = 0.

Consider now a stationary solution.

Definition 61 (Stationary) A policy (or strategy) is stationary if for allt = 0, 1, ...,∞:

kt = k

In other words, the strategy is time invariant and clearly we have that

it = i = λk for all t.

Suppose there exists a k∗ such that

V (k∗) =∞∑

t=0

δt

(αk∗ − 1

2(λk∗)2

)or

V (k∗) =αk∗ − 1

2(λk∗)2

1− δHowever, it still remains to verify what the optimal solution is. But we cando this now by analyzing the first order condition of the Bellman equation

δα− λk∗

1− δ− λk∗ = 0

Thus we get

k∗ =αδ

λ.

Chapter 6

Games of IncompleteInformation

6.1 Introduction

This is a very modest introduction into the theory of games with incompleteinformation or Bayesian games . We start by defining incomplete information,then suggest how to model incomplete information (as a move by nature) totransform it into a game of imperfect information. We then introduce theassociated equilibrium concept of Bayesian Nash equilibrium.

6.1.1 Example

Harsanyi’s insight is illustrated by the following example. Suppose payoffsof a two player two action game are either:

α βα 1,1 0,0β 0,1 1,0

orα β

α 1,0 0,1β 0,0 1,1

i.e. either player II has dominant strategy to play H or a dominant strategyto play T . Suppose that II knows his own payoffs but player I thinks there

67

CHAPTER 6. GAMES OF INCOMPLETE INFORMATION 68

is probability p that payoffs are given by the first matrix, probability 1 − pthat they are given by the second matrix. Say that player II is of type 1 ifpayoffs are given by the first matrix, type 2 if payoffs are given by the secondmatrix. Clearly equilibrium must have: II plays H if type 1, T if type 2; Iplays H if p > 1

2, T if p < 1

2. But how to analyze this problem in general?

All payoff uncertainty can be captured by a single move of “nature” at thebeginning of the game.

6.2 Basics

A (finite) static incomplete information game consists of

• Players 1, ..., I

• Actions sets A1, ..., AI

• Sets of types T1, ..., TI

• A probability distribution over types p ∈ ∆ (T ), where T = T1× ...×TI

• Payoff functions g1, ..., gI , each gi : A× T → R

Interpretation. Nature chooses a profile of players types, t ≡ (t1, ..., tI) ∈T according to probability distribution p (.). Each player i observes his owntype ti and chooses an action ai. We also refer to ti as the private infor-mation of agent i. Now player i receives payoffs gi (a, t). By introducingfictional moves by nature, first suggested by Harsanyi (1967) we transforma game of incomplete information into a game of imperfect information. Inthe transformed game one player’s incomplete information about the otherplayer’s type becomes imperfect information about nature’s moves.

6.2.1 Strategy

A pure strategy is a mapping si : Ti → Ai. Write Si for the set of suchstrategies, and let S = S1× ..×SI . A mixed strategy can also depend on hisprivate information:

σi : Ti → ∆(Si).


6.2.2 Belief

A player’s “type” ti is any private information relevant to the player’s decisionmaking. The private information may refer to his payoff function, his beliefabout other agent’s payoff function, the likelihood of some relevant event,etc. We assume that for the players types {ti}I

i=1 there is some objectivedistribution function

p (t1, ..., tI)

where ti ∈ Ti, which is called the prior distribution or prior belief. Theconditional probability

p(t′−i |ti

)=

p(t′−i, ti

)∑t−i∈T−i

p (t−i, ti)

by Bayes rule denotes the conditional probability over the other player’s typesgive one’s own type ti. This expression assumes that Ti are finite. Similarexpressions hold for infinite Ti. If the player’s types are independent then

p (t−i |ti ) = pi (t−i) , ∀ti ∈ Ti.

6.2.3 Payoffs

The description of the Bayesian game is completed by the payoff function

gi (a1, ..., aI ; t1, ..., tI) .

If player i knew the strategy of the other players as a function of his type thenhe could calculate the expected payoff from his decision given the conditionalprobability p (t−i |ti ).

6.3 Bayesian Game

Definition 62 A Bayesian game in normal form is given by

ΓN ={I, (Ai)

Ii=1 , (Ti)

Ii=1 , p (·) , (gi (·; ·))I

i=1

}.

The timing of a static Bayesian game is then as follows:

1. (a) nature draws a type vector t = (t1, ..., tn) , ti ∈ Ti,


(b) nature reveals ti to player i but not to any other player,

(c) players choose simultaneously actions si ∈ Si,

(d) payoffs gi (a1, ..., aI ; t1, ..., tI) are received.

Again, by imperfect information we mean that at some move in the gamethe player with the move does not know the complete history of the gamethus far. Notice that with this specification, the only proper subgame of agame of incomplete information is the whole game. Any Nash equilibriumis then subgame perfect and the subgame perfection has no discriminatingpower.

6.3.1 Bayesian Nash Equilibrium

Player i’s payoff function of the incomplete information game, ui : S → R, is

ui (s) ,∑t∈T

p (t) gi (s (t) , t)

where s = (s1, ..., sI) and s (t) = (s1 (t1) , ..., sI (tI)). Recall:

(Old) Definition: Strategy profile s∗ is a pure strategy Nash equilibrium if

ui

(s∗i , s

∗−i

)≥ ui

(si, s

∗−i

)for all si ∈ Si and i = 1, .., I

This can be re-written in the context of incomplete information as:

∑t∈T

p (t) gi

((s∗i (ti) , s∗−i (t−i)

), t)≥∑t∈T

p (t) gi

((si (ti) , s∗−i (t−i)

), t)

for all si ∈ Si and i = 1, .., I(6.1)

Writing p (ti) =∑t′−i

p(ti, t

′−i

)and p (t−i|ti) ≡ p(ti,t−i)

p(ti), this can be re-written

as:

∑t−i∈T−i

p (t−i|ti) gi

((s∗i (ti) , s∗−i (t−i)

), t)≥

∑t−i∈T−i

p (t−i|ti) gi

((ai, s

∗−i (t−i)

), t)

for all ti ∈ Ti, ai ∈ Ai and i = 1, ..., I(6.2)


Definition 63 A pure strategy Bayesian Nash equilibrium of

Γ = {A1, ..., AI ; T1, ..., TI ; u1, ..., uI ; p}

is a strategy profile s∗ = (s∗1, ..., s∗I) such that, for all i, for all ti ∈ T , condition

(6.8) is satisfied.

Notice the differences in the definition between the Bayesian Nash equi-librium and the Nash equilibrium. In the former the payoff function maydepend on the type. Moreover the payoff is calculated as an expected payoffgiven his own information, with the conditional distribution over all otheragent’s type.

6.4 A Game of Incomplete Information: First

Price Auction

Suppose there are two bidders for an object sold by an auctioneer and thevaluation of the object is private information. Each buyer i = 1, 2 has avaluation vi ∼ U ([0, 1]) and submits a bid bi for the object. The valuationsare assumed to be statistically independent. If bidder i submits the higherof the bids, buyer i receives vi − bi and the seller receives bi.

ui (vi) =

vi − bi, if bi ≥ b−i,

12(vi − bi) , if bi = b−i,

0, if bi < b−i.

A Bayesian Nash equilibrium is then a strategy pair {b1 (v1) , b2 (v2)}. Theoptimal strategy is found for each player by solving the following problem

maxbi

(vi − bi) Pr (bi > bj (vj)) +1

2(vi − bi) Pr (bi = bj (vj)) . (6.3)

We simplify the analysis by looking at linear equilibria:

bi = ai + civi.

Hence we can write the objective function in (6.3) as

maxbi

(vi − bi) Pr (bi ≥ aj + cjvj)


Notice that

Pr (bi ≥ aj + cjvj) = Pr

(vj ≤

bi − aj

cj

), (6.4)

and by the uniformity this results in

maxbi

(vi − bi)bi − aj

cj

.

The first-order conditions for this problem result in

bi (vi) = (vi + aj) /2. (6.5)

Similarly and by symmetry we have

bj (vj) = (vj + ai) /2. (6.6)

But in equilibrium we must have

(vi + aj) /2 = ai + civi, ∀vi,

and(vj + ai) /2 = aj + cjvj, ∀vj,

and hence

ci = cj =1

2,

and consequentlyai = aj = 0.

Hence the equilibrium strategy is to systematically underbid

bi (vi) =1

2vi.

6.5 Conditional Probability and Conditional

Expectation

Before we analyze the next game, the double auction, recall some basic con-cepts in probability theory, namely conditional probability and conditionalexpectation.


6.5.1 Discrete probabilities - finite events

We start with a finite number of events, and then consider a continuum ofevents Consider the following example. Suppose a buyer is faced with alottery of prices

0 < p1 < ... < pi < ... < pn < ∞

where each price pi has an unconditional probability 0 < αi < 1, and∑ni=1 αi = 1, by which the price pi is chosen. Suppose the buyer is will-

ing to accept all prices, then the expected price to be paid by the buyeris

E [p] =n∑

i=1

αipi, (6.7)

which is the unconditional expectation of the price to be paid. Supposenext, that the buyer adopts a rule to accepts all price higher or equal topk and reject all other prices. (Recall this is just an example.) Then theunconditional expectation of the price to be paid is

E [p] =n∑

i=k

αipi, (6.8)

The expectation in (6.8) is still unconditional since no new information en-tered in the formation of the expectation. Suppose we represent the accep-tance rule of the buyer by the index k below the expectations operator E [·],as in Ek [·]. The expected payment is then

Ek [p] =k−1∑i=1

αi0 +n∑

i=k

αipi,

since all price strictly lower than pk are rejected. Clearly

E1 [p] > Ek [p]

as

E1 [p]− Ek [p] =n∑

i=1

αipi −n∑

i=k

αipi =k−1∑i=1

αipi > 0. (6.9)

since the buyer is willing to pay pi in more circumstances in (6.7) and (6.8).


Consider now the average price which is paid under the two differentpolicies conditional on accepting a price and paying it, in other words, weask for the conditional expectation:

Ek [p |p ≥ pk ]

We consider now exclusively the events, i.e. realization of prices p, wherep ≥ pk for some k. The conditional probability of a price pi being quotedconditional on pi ≥ pk is then given by

Pr (pi |pi ≥ pk ) =Pr (pi)

Pr (pi ≥ pk), (6.10)

sincePr ({pi} ∩ {pi ≥ pk}) = Pr (pi)

if indeed pi satisfies pi ≥ pk, which is the nothing else than Bayes rule.1

Notice that we can write more explicitly as:

Pr (pi |pi ≥ pk ) =αi∑n

j=k αj

, (6.12)

and thatn∑

i=k

(pi |pi ≥ pk ) = 1.

The conditional expectation of the prices, conditioned on accepting thenprice, is then nothing else but taken the expectation with respect to theconditional probabilities rather than with the unconditional probabilities, sothat

Ek [p |p ≥ pk ] =n∑

i=k

αi∑nj=k αj

pi,

where Ek [p |p ≥ pk ] is now the expected price paid conditional on acceptingthe price. You may now verify that the expected price or average price paid,conditional on paying is increasing in k, as we might have expected it.

Next we generalize the notion of conditional expectation to continuum ofprices where the prices are distributed according to a distribution function.

1Recall that Bayes rule says for any two events, A and B, that

Pr (A |B ) =Pr (A ∩B)

Pr (B)(6.11)


6.5.2 Densities - continuum of events

You will see that the generalization is entirely obvious, but it may help tofamiliarize yourself with the continuous case. Suppose a buyer is faced witha lottery of prices

p ∈[p, p̄]

where the distribution of prices is given by a (continuous) distribution func-tion F (p) and an associated density function f (p). Again suppose the buyeris willing to accept all prices, then the expected price to be paid by the buyeris

E [p] =

∫ p̄

p

pf (p) dp =

∫ p̄

p

pdF (p) (6.13)

which is the unconditional expectation of the price to be paid. Notice that thesummation is now replaced with the integral operation. Suppose next, thatthe buyer adopts a rule to accept all prices higher or equal to p̂ and rejectall other prices. (Recall this is just an example.) Then the unconditionalexpectation of the price to be paid is

Ep̂ [p] =

∫ p̄

p̂

pdF (p) (6.14)

Again, the expectation in (6.14) is still unconditional since no new infor-mation entered in the formation of the expectation and we represent theacceptance rule of the buyer by the index p̂ below the expectations operatorE [·], as in Ep̂ [·]. The expected payment is then

Ep̂ [p] =

∫ p̂

p

0dF (p) +

∫ p̄

p̂

pdF (p)

since all price strictly lower than p̂ are rejected. Clearly

E [p] > Ep̂ [p] , for all p̂ > p.

as

E [p]− Ep̂ [p] =

∫ p̂

p

pdF (p) > 0. (6.15)

since the buyer is willing to pay p in more circumstances in (6.13) and (6.14).


Consider now the average price which is paid under the two differentpolicies conditional on accepting a price and paying it, in other words, weask for the conditional expectation:

Ep̂ [p |p ≥ p̂ ]

We consider now exclusively the events, i.e. realization of prices p, wherep ≥ p̂ for some p̂. Notice that we have now a continuum of events, each ofwhich has probability zero, but all of which have associated densities. Wethen switch to densities. The conditional densities of a price p being quotedconditional on p ≥ p̂ is then given by

f (p |p ≥ p̂) =f (p)

1− F (p̂). (6.16)

Compare this expression to (6.10) and (6.12) above and notice that thisfollows from

{p} ∩ {p ≥ p̂} = {p̂}

as long as p ≥ p̂, and otherwise the intersection is of course the empty setNotice that, as before the conditional densities integrate up to 1 since∫ p̄

p̂

f (p |p ≥ p̂) dp =

∫ p̄

p̂

f (p)

1− F (p̂)dp =

1

1− F (p̂)

∫ p̄

p̂

f (p) dp = 1, (6.17)

where the last equality of course follows from2:∫ p̄

p̂

f (p) dp = 1− F (p̂) .

Finally, the conditional expectation of the prices, conditioned on acceptingthen price, is then nothing else but taken the expectation with respect to theconditional probabilities rather than with the unconditional probabilities, sothat

Ep̂ [p |p ≥ p̂ ] =

∫ p̂

p̂

pf (p |p ≥ p̂) dp =

∫ p̂

p̂

pf (p)

1− F (p̂)dp =

1

1− F (p̂)

∫ p̂

p̂

pf (p) dp.

2Recall the similarity between the second to last term in (6.17) to the term appearingwhen computing the expected value of the bid ps (vs).


Again you may verify that the expected price or average price paid, condi-tional on paying is increasing in p̂, as we might have expected it. In particular,

limp̂→p̄

Ep̂ [p |p ≥ p̂ ] = p̄ (6.18)

where in contrast, we have of course

limp̂→p̄

Ep̂ [p] = 0. (6.19)

You may wish to verify (6.18) and (6.19) to realize the difference betweenexpected value (price) and conditional expected value (price) in the secondexample.

6.6 Double Auction

6.6.1 Model

We present here another important trading situation where both sides of themarket have private information. The trading game is also called a doubleauction. We assume that the seller’s valuation is

vs ∼ F (vs) = U ([0, 1]) , (6.20)

which we may think of the cost of producing one unit of the good. Buyer’svaluation

vb ∼ F (vb) = U ([0, 1]) . (6.21)

The valuations vs and vb are independent. If sale occurs at p, buyer receivesvb − p and seller receives p− vs. Trade occurs at p = 1

2(ps + pb) , if ps ≤ pb.

There is no trade if ps > pb. If there is no trade both, buyer and seller receivezero payoff. Given vb, pb and ps, the buyer’s payoff is:

ub (vb, pb, ps) =

{vb − 1

2(ps + pb) if pb ≥ ps

0 if pb < ps

and similar for the seller

us (vs, pb, ps) =

{12(ps + pb)− vs if pb ≥ ps

0 if pb < ps


6.6.2 Equilibrium

A strategy for each agent is then a mapping

pi : Vi → R. (6.22)

Denote an equilibrium strategy by

p∗i (vi)

We can then define a Bayes-Nash equilibrium as follows.

Definition 64 A Bayes-Nash equilibrium is given by

{p∗b (vb) , p∗s (vs)}

such that

Ep∗s [ub (vb, p∗b (vb) , p∗s)] ≥ Ep∗s [ub (vb, p

′b, p

∗s)] , ∀vb, p

′b, (6.23)

andEp∗b

[us (vs, p∗s (vs) , p∗b)] ≥ Ep∗b

[us (vs, p′s, p

∗b)] , ∀vs, p

′s. (6.24)

Remark 65 The Bayes part in the definition of the Bayes-Nash equilibriumthrough (6.23) and (6.24) is that the buyer has to find an optimal bid forevery value vb while he is uncertainty about the ask (p∗s) of the seller. Fromthe buyer’s point of view, he is uncertain about p∗s as the ask, p∗s = p∗s (vs) willdepend on the value the seller assigns to the object, but as this is the privateinformation of the seller, the buyer has to make his bid only knowing that theseller will adopt the equilibrium strategy: p∗s : Vs → [0, 1], but not knowingthe realization of vs and hence not knowing the realization of p∗s = p∗s (vs).

6.6.3 Equilibrium Analysis

To find the best strategy for the buyer, and likewise for the seller, we haveto able to evaluate the net utility of the buyer for every possible bid he couldmake, given a particular value vb he assigns to the object,

Ep∗s [ub (vb, pb, p∗s)] =

∫{p∗s |p∗s≤pb }

[vb −

1

2(pb + p∗s)

]dG∗ (p∗s) + (6.25)


+

∫{p∗s |p∗s>pb }

0dG∗ (p∗s)

whereG∗ (x) = Pr (p∗s ≤ x)

is the distribution of the equilibrium asks of the seller. Notice now that theequilibrium ask, p∗s, of the seller will depend on the realization of the seller’svalue vs.Thus we can equivalently, the expectation in (6.25) with respect tovs, or

Evs [ub (vb, pb, p∗s (vs))] =

∫{vs|p∗s(vs)≤pb }

[vb −

1

2(pb + p∗s (vs))

]dF (vs) (6.26)

The advantage of the later operation is that we have direct information aboutthe distribution of values, whereas we have only indirect information aboutthe distribution of the equilibrium asks. Using (6.20), we get


∫{vs|p∗s(vs)≤pb }

[vb −

1

2(pb + p∗s (vs))

]dvs. (6.27)

However, as we don’t know the exact relationship between vs and p∗s, we can’ttake the expectation yet. Similar to the auction example, we therefore makethe following guess, which we shall verify eventually about the relationshipof vs and p∗s:

p∗s (vs) = as + csvs. (6.28)

Inserting (6.28) into (6.27) leads us to∫{vs|p∗s(vs)≤pb }

[vb −

1

2(pb + as + csvs)

]dvs. (6.29)

The final question then is what is the set of vs over which we have to integrate,or

{vs |p∗s (vs) ≤ pb} = {vs |as + csvs ≤ pb}

=

{vs

∣∣∣∣vs ≤pb − as

cs

}and hence (6.29) is ∫ pb−as

cs

0

[vb −

1

2(pb + as + csvs)

]dvs


Integrating yields [vbvs −

1

2

(pbvs + asvs +

1

2csv

2s

)] pb−ascs

0

or


pb − as

cs

(vb −

1

2

(pb + as +

1

2cs

pb − as

cs

))(6.30)

The expected utility for the buyer is now expressed as a function of vb, pb

and the constants in the strategy of the seller, but independent of p∗s or vs.Simplifying, we get

ub (vb, pb, as, cs) =

probability of trade︷︸︸︷pb − as

cs

×

expected net utility conditional on trade︷︸︸︷(vb −

1

2

(3

2pb +

1

2as

)), (6.31)

which essentially represents the trade-off for the buyer. A higher bid pb

increases the probability of trade but decreases the revenue conditional ontrade. The first order conditions are

∂ub (vb, pb, as, cs)

∂pb

= vb −(

3pb + as

4

)− 3

4(pb − as) = 0

or

p∗b (vb) =2vb

3+

as

3. (6.32)

We can perform a similar steps of arguments for the seller to obtain∫ 1

ps−abcb

(1

2(ps + ab + cbvb)− vs

)dvb

and we get the sellers payoff as a function of vs, ps, ab, cb:

us (vs, ps, ab, cb) =

[1

2

(psvb + abvb +

1

2cbv

2b

)− vsvb

]1

ps−abcb

and when evaluated it give us

us (vs, ps, ab, cb) =

probability of trade︷︸︸︷cb − ps + ab

cb

×

expected net utility conditional on trade︷︸︸︷1

4(3ps + cb + ab − 4vs)


where the trade-off is now such that a higher ask leads to lower probabilityof trade, but a higher net utility conditional on trade. Taking the first orderconditions, leads us to

∂us (vs, ps, ab, cb)

∂ps

= 0

or equivalently

p∗s (vs) =1

3cb +

1

3ab +

2

3vs (6.33)

¿From the solution of the two pricing policies, we can now solve for ab

and as as well as

1

3cb +

1

3ab = as

as

3= ab

and hence

as =1

4, ab =

1

12

and thus the complete resolution is

p∗b (vb) =1

12+

2

3vb

and

p∗s (vs) =1

4+

2

3vs

Next we can evaluate the efficiency of the equilibrium. Since trade occurs if

p∗b (vb) ≥ p∗s (vs) ⇔1

12+

2

3vb ≥

1

4+

2

3vs

⇔ vb − vs ≥1

4

trade occurs if the gains from trade exceed 14. The following strategies also

form an equilibrium for any x ∈ [0, 1] .

pb =

{x if vb ≥ x0 otherwise

ps =

{x if vs ≤ x1 otherwise

.

Readings. Chapter 3 in Gibbons (1992), FT Chapter 6, MWG, Chapter8.E.

Chapter 7

Adverse selection (with twotypes)

Often also called self-selection, or “screening”. In insurance economics, if ainsurance company offers a tariff tailored to the average population, the tariffwill only be accepted by those with higher than average risk.

7.1 Monopolistic Price Discrimination

A simple model of wine merchant and wine buyer, who could either have acoarse or a sophisticated taste, which is unobservable to the merchant. Whatqualities should the merchant offer and at what price?

The model is given by the utility function of the buyer, which is

v (θi, qi, ti) = u (θi, qi)− ti = θiqi − ti, i ∈ {l, h} (7.1)

where θi represent the marginal willingness to pay for quality qi and ti is thetransfer (price) buyer i has to pay for the quality qi. The taste parametersθi satisfies

0 < θl < θh < ∞. (7.2)

The cost of producing quality q ≥ 0 is given by

c (q) ≥ 0, c′ (q) > 0, c′′ (q) > 0. (7.3)

The ex-ante (prior) probability that the buyer has a high willingness to payis given by

p = Pr (θi = θh)

82

CHAPTER 7. ADVERSE SELECTION (WITH TWO TYPES) 83

We also observe that the difference in utility for the high and low valuationbuyer for any given quality q

u (θh, q)− u (θl, q)

is increasing in q. (This is know as the Spence-Mirrlees sorting condition.).If the taste parameter θi were a continuous variable, the sorting conditioncould be written in terms of the second cross derivative:

∂2u (θ, q)

∂θ∂q> 0,

which states that taste θ and quality q are complements. The profit for theseller from a bundle (q, t) is given by

π (t, q) = t− c (q)

7.1.1 First Best

Consider first the nature of the socially optimal solution. As different typeshave different preferences, they should consume different qualities. The socialsurplus for each type can be maximized separately by solving

maxqi

{θiqi − c (qi)}

and the first order conditions yield:

qi = q∗i ⇐⇒ c′ (q∗i ) = θi ⇒ q∗l < q∗h.

The efficient solution is the equilibrium outcome if either the monopolistcan perfectly discriminate between the types (first degree price discrimina-tion) or if there is perfect competition. The two outcomes differ only in termsof the distribution of the social surplus. With a perfectly discriminating mo-nopolist, the monopolist sets

ti = θiqi (7.4)

and then solves for each type separately:

max{ti,qi}

π (ti, qi) ⇐⇒ max{ti,qi}

{θiqi − c (qi)} ,

using (7.4). Likewise with perfect competition, the sellers will break even,get zero profit and set prices at

ti = c (q∗i )

in which case the buyer will get all the surplus.


7.1.2 Second Best: Asymmetric information

Consider next the situation under asymmetric information. It is verifiedimmediately that perfect discrimination is now impossible as

θhq∗l − t∗l = (θh − θl) q∗l > 0 = θ∗hq

∗h − th (7.5)

but sorting is possible. The problem for the monopolist is now

max{tl,ql,th,qh}

(1− π) tl − c (ql) + π (th − (c (qh))) (7.6)

subject to the individual rationality constraint for every type

θiqi − ti = 0 (IRi) (7.7)

and the incentive compatibility constraint

θiqi − ti = θiqj − tj (ICi) (7.8)

The question is then how to separate. We will show that the binding con-straint are IRl and ICh, whereas the remaining constraints are not binding.We then solve for tl and th, which in turn allows us to solve for qh, and leavesus with an unconstrained problem for ql.Thus we want to show

(i) IRl binding, (ii) ICh binding, (iii) q̂h ≥ q̂l (iv) q̂h = q∗h (7.9)

Consider (i). We argue by contradiction. As

θhqh − th =ICh

θhql − tl =θh>θl

θlql − tl (7.10)

suppose that θlql − tl > 0, then we could increase tl, th by a equal amount,satisfy all the constraints and increase the profits of the seller. Contradiction.

Consider (ii) Suppose not, then as

θhqh − th > θhql− tl =θh>θl

θlql − tl(IRl)

= 0 (7.11)

and thus th could be increased, again increasing the profit of the seller.(iii) Adding up the incentive constraints gives us (ICl) + (ICl)

θh (qh − ql) = θl (qh − ql) (7.12)


and since:θh > θl ⇒ q̂h − q̂l = 0. (7.13)

Next we show that ICl can be neglected as

th − tl = θh (qh − ql) = θl (qh − ql) . (7.14)

This allows to say that the equilibrium transfers are going to be

tl = θlql (7.15)

andth − tL = θh (qh − qL) ⇒ th = θh (qh − qL) + θlql.

Using the transfer, it is immediate that

q̂h = q∗h

and we can solve for the last remaining variable, q̂l.

maxql

{(1− p) (θlql− (c (ql)) + p (θh (q∗h − qL) + θlql − c (q∗h)))}

but as q∗h is just as constant, the optimal solution is independent of constantterms and we can simplify the expression to:

maxql

{(1− p) (θlql − c (ql))− p (θh − θl) ql}

Dividing by (1− p) we get

maxql

{θlql − c (ql)−

p

1− p(θh − θl) ql

}for which the first order conditions are

θl − c′ (ql)−p

1− p(θh − θl) ql = 0

This immediately implies that the solution q̂l:

⇐⇒ c′ (q̂l) < θl ⇐⇒ q̂l < q∗l

and the quality supply to the low valuation buyer is inefficiently low (withthe possibility of complete exclusion).

Consider next the information rent for the high valuation buyer, it is

I (ql) = (θh − θl) ql

and therefore the rent is increasing in ql which is the motivation for the sellerto depress the quality supply to the low end of the market.

The material is also covered in Salanie (1997), Chapter 2.

Chapter 8

Theoretical Complements

8.1 Mixed Strategy Bayes Nash Equilibrium

Definition 66 A Bayesian Nash equilibrium of

Γ = {S1, ..., SI ; T1, ..., TI ; u1, ..., uI ; p}

is a strategy profile σ∗ = (σ∗1, ..., σ∗I ) such that, for all i, for all ti ∈ T and all

si ∈ supp σ∗i (ti):

Bayes︷︸︸︷∑t−i∈T−i

pi (t−i|ti)

mixed strategy Nash︷︸︸︷ ∑s−i∈S−i

ui ((s′i, s−i) ; (ti, t−i)) σ∗−i (s−i|t−i)

≥

Bayes︷︸︸︷∑t−i∈T−i

pi (t−i|ti)

mixed strategy Nash︷︸︸︷ ∑s−i∈S−i

ui ((s′i, s−i) ; (ti, t−i)) σ∗−i (s−i|t−i)

, ∀s′i

where σ∗−i (s−i|t−i) ≡(σ∗1 (s1|t1) , ..., σ∗i−1 (si−1|ti−1) , σ∗i+1 (si+1|ti+1) , ..., σ∗n (sn|tn)

).

Since the strategy choices are independent across players, we can writethat:

σ∗−i (s−i|t−i) =∏j 6=i

σ∗j (sj|tj) .

86

CHAPTER 8. THEORETICAL COMPLEMENTS 87

8.2 Sender-Receiver Games

8.2.1 Model

A signaling games is a dynamic game of incomplete information involving twoplayers: a sender (S) and receiver (R). The sender is the informed party,the receiver the uniformed party. (Describe the game in its extensive form).The timing is as follows:

1. nature draws a type ti ∈ T = {t1, ..., tI} according to a probabilitydistribution p (ti) ≥ 0,

∑i p (ti) = 1,

2. sender observes ti and chooses a message mj ∈ M = {m1, ...,mJ},

3. receiver observes mj (but not ti) and chooses an action ak ∈ A ={a1, ..., aK},

4. payoffs are given by uS (ti, mj, ak) and uR (ti, mj, ak).

8.3 Perfect Bayesian Equilibrium

8.3.1 Informal Notion

The notion of a perfect Bayesian equilibrium is then given by:

Condition 67 At each information set, the player with the move must havea belief about which node in the information set has been reached by the playof the game. For a non singleton information set, a belief is a probabilitydistribution over the nodes in the information set; for a singleton informationset, the player’s belief puts probability one on the single decision node.

Condition 68 Given their beliefs, the players’ strategies must be sequen-tially rational. That is, at each information set the action taken by the playerwith the move (and the player’s subsequent strategy) must be optimal giventhe player’s belief at that information set and the other players’ subsequentstrategies (where a “subsequent strategy” is a complete plan of action cov-ering every contingency that might arise after the given information set hasbeen reached).


Definition 69 For a given equilibrium in a given extensive-form game, aninformation set is on the equilibrium path if it will be reached with positiveprobability if the game is played according to the equilibrium strategies, and isoff the equilibrium path if it is certain not to be reached if the game is playedaccording to the equilibrium strategies (where “equilibrium” can mean Nash,subgame-perfect, Bayesian, or perfect Bayesian equilibrium).

Condition 70 At information sets on the equilibrium path, beliefs are de-termined by Bayes’ rule and the players’ equilibrium strategies.

Condition 71 At information sets off the equilibrium path beliefs are deter-mined by Bayes’ rule and the players’ equilibrium strategies where possible.

Definition 72 A perfect Bayesian equilibrium consists of strategies and be-liefs satisfying Requirements 1 through 4.

8.3.2 Formal Definition

Condition 73 After observing any message mj from M , the Receiver musthave a belief about which types could have sent mj. Denote this belief by theprobability distribution µ (ti |mj ), where µ (ti |mj ) ≥ 0 for each ti in T and∑

ti∈T

µ (ti |mj ) = 1 (8.1)

Condition 74

1. For each mj in M, the Receiver’s action a∗ (mj) must maximizethe Receiver’s expected utility, given the belief µ (ti |mj ) aboutwhich types could have sent mj. That is, a∗ (mj) solves

maxak∈A

∑µ (ti |mj )

ti∈T

UR (ti, mj, ak) . (8.2)

2. For each ti in T, Sender’s message m∗ (ti) must maximize the Sender’sutility, given the Receiver’s strategy a∗ (mj) . That is, m∗ (ti) solves:

maxmj∈M

Us (ti, mj, a∗ (mj)) (8.3)


Condition 75 For each mj in M , if exists ti in T such that m∗ (ti) = mj,then the Receiver’s belief at the information set corresponding to mj mustfollow from Bayes’ rule and the Sender’s strategy:

µ (ti |mj ) =p (ti)∑

ti∈T

p (ti). (8.4)

We can summarize these results to get the definition of a Perfect BayesianEquilibrium:

Definition 76 A pure-strategy perfect Bayesian equilibrium in a signalinggame is a pair of strategies m∗ (ti) and a∗ (mj) and a belief µ (ti |mj ) satis-fying signaling requirements (73)-(75).

Alternatively, we can state it more succinctly as saying that:

Definition 77 (PBE) A pure strategy Perfect Bayesian Equilibrium is aset of strategies {m∗ (ti) , a∗ (mj)} and posterior beliefs p∗ (ti |mj ) such that:

1. ∀ti, ∃p∗ (ti |mj ) , s.th. p∗ (ti |mj ) ≥ 0, and∑

ti∈T p∗ (ti |mj ) = 1;

2. ∀mj, a∗ (mj) ∈ arg max∑

ti∈T uR (ti, mj, ak) p∗ (ti |mj ) ;

3. ∀ti, m∗ (ti) ∈ arg max uS (ti, mj, a∗ (mj))

4. ∀mj if ∃ti ∈ T s.th. at m∗ (ti) = mj, then:

p (ti |mj ) =p (ti)∑

{t′i|m∗(t′i)=mj } p (t′i).

We can now introduce additional examples to refine the equilibrium set.

Example 78 Job Market Signalling

Example 79 Cho-Kreps (Beer-Quiche) - Entry Deterrence. Remark thepayoffs in Gibbons (1992) have a problem, they don’t allow for a separat-ing equilibrium, modify)


Definition 80 (Equilibrium Domination) Given a PBE, the messagemj is equilibrium-dominated for type ti, if

U∗S (ti) > max

ak∈AUS (ti, mj, ak) .

Definition 81 (Intuitive Criterion) If the information set following mj

is off the equilibrium path and mj equilibrium dominated for type ti, then

µ (ti |mj ) = 0.

This is possible provided that mj is not equilibrium dominated for alltypes in T .

Readings: MWG Chapter 9.C, FT Chapter 8.


8.4 References

D. Abreu. Extremal equilibria of oligopolistic supergames. Journal of Eco-nomic Theory, 39:191–225, 1986.

K. Binmore. Fun and Games. D.C. Heath, Lexington, 1992.D. Fudenberg and J. Tirole. Game Theory. MIT Press, Cambridge, 1991.R. Gibbons. Game Theory for Applied Economists. Princeton University

Press, Princeton, 1992.D. M. Kreps. A Course in Microeconomic Theory. Princeton University

Press, Princeton, 1990.D.M. Kreps and R. Wilson. Sequential equilibria. Econometrica, 50:863–

894, 1982.H. Kuhn. Extensive games and the problem of information. In H. Kuhn

and A. Tucker, editors, Contributions to the Theory of Games, pp. 193–216.Princeton University Press, Princeton, 1953.

A. Mas-Collel, M.D. Whinston, and J.R. Green. Microeconomic Theory.Oxford University Press, Oxford, 1995.

J. Nash. Equilibrium points in n-person games. Proceedings of the Na-tional Academy of Sciences, 36:48–49, 1950.

M.J. Osborne and A. Rubinstein. A Course in Game Theory. MIT Press,Cambridge, 1994.

P. Reny. On the existence of pure and mixed strategy Nash equilibria indiscontinuous games. Econometrica, 67:1029–1056, 1999.

S.M. Ross. Probability Models. Academic Press, San Diego, 1993.A. Rubinstein. Perfect equilibrium in a bargaining model. Econometrica,

50:97–109, 1982.B. Salanie. The Economics of Contracts. MIT Press, Cambridge, 1997.

Mason. Game Theory

Documents

k0 i0

vi aj

normal form representation

vj ai

pb ps

ps pb

perfect bayesian equilibrium

subgame perfect equilibria