handout_good!!! game theory

Microeconomic Theory: Basic Game Theory1

Juuso Valimaki

November 5, 2001

1These notes are based on to a large extent on the notes of Dirk Bergemann for a similar course at YaleUniversity. I am grateful for Dirk for the permission to use and edit those notes.

Contents

1 Game Theory: An Introduction 11.1 Game Theory and Parlor Games - a Brief History . . . . . . . . . . . . . . . . . . . 21.2 Game theory in microeconomics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Basic Elements of Noncooperative Games . . . . . . . . . . . . . . . . . . . . . . . 2

2 Normal Form Games 42.1 Pure Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Dominance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 Nash equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.3.1 Finite Normal Form Game . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.3.2 Cournot Oligopoly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.3.3 Auctions with perfect information . . . . . . . . . . . . . . . . . . . . . . . 11

2.4 Mixed Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.4.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.4.2 Construction of a mixed strategy Nash equilibrium . . . . . . . . . . . . . . 162.4.3 Interpretation of mixed strategy equilibria . . . . . . . . . . . . . . . . . . . 18

2.5 Mixed Strategies with a Continuum of Pure Strategies . . . . . . . . . . . . . . . . 182.5.1 All pay auction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.5.2 War of attrition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.5.3 Appendix: Basic probability theory . . . . . . . . . . . . . . . . . . . . . . . 26

2.6 Appendix: Dominance Solvability and Rationalizability . . . . . . . . . . . . . . . 312.6.1 Rationalizable Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.7 Appendix: Existence of Mixed Strategy Equilibria . . . . . . . . . . . . . . . . . . 34

3 Extensive Form Games 363.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.2 Definitions for Extensive Form Games . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

i

CONTENTS ii

3.2.2 Graph and Tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.2.3 Game Tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.2.4 Notion of Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.3 Subgame Perfect Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.3.1 Finite Extensive Form Game: Bank-Run . . . . . . . . . . . . . . . . . . . . 44

3.4 The Limits of Backward Induction: Additional Material . . . . . . . . . . . . . . . 453.5 Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4 Repeated Games 484.1 Finitely Repeated Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484.2 Infinitely Repeated Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.2.1 The prisoner’s dilemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514.3 Collusion among duopolists . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.3.1 Static Game . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524.3.2 Best Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524.3.3 Collusion with trigger strategy . . . . . . . . . . . . . . . . . . . . . . . . . 534.3.4 Collusion with two phase strategies . . . . . . . . . . . . . . . . . . . . . . . 53

5 Sequential Bargaining 555.1 Bargaining with Complete Information . . . . . . . . . . . . . . . . . . . . . 56

5.1.1 Extending the Basic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 595.2 Dynamic Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

5.2.1 Additional Material: Capital Accumulation . . . . . . . . . . . . . . . . . . 62

6 Games of Incomplete Information 646.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

6.1.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 646.2 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

6.2.1 Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 656.2.2 Belief . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 656.2.3 Payoffs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

6.3 Bayesian Game . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 666.3.1 Bayesian Nash Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

6.4 A Game of Incomplete Information: First Price Auction . . . . . . . . . . . . . . . 676.5 Conditional Probability and Conditional Expectation . . . . . . . . . . . . . . . . . 68

6.5.1 Disrete probabilities - finite events . . . . . . . . . . . . . . . . . . . . . . . 696.5.2 Densities - continuum of events . . . . . . . . . . . . . . . . . . . . . . . . . 70

6.6 Double Auction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

CONTENTS iii

6.6.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 726.6.2 Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 736.6.3 Equilibrium Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

7 Adverse selection (with two types) 777.1 Monopolistic Price Discrimination . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

7.1.1 First Best . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 787.1.2 Second Best: Asymmetric information . . . . . . . . . . . . . . . . . . . . . 78

8 Theoretical Complements 818.1 Mixed Strategy Bayes Nash Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . 818.2 Sender-Receiver Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

8.2.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 818.3 Perfect Bayesian Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

8.3.1 Informal Notion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 828.3.2 Formal Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

Chapter 1

Game Theory: An Introduction

Game theory is the study of multi-person decision problems. The focus of game theory is interde-pendence, situations in which an entire group of people is affected by the choices made by everyindividual within that group. As such they appear frequently in economics. Models and situationsof trading processes (auction, bargaining), labor and financial markets, oligopolistic markets andmodels of political economy all involve game theoretic reasoning. There are multi-agent decisionproblems within an organization, many person may compete for a promotion, several divisionscompete for investment 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 the other players?

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

1

CHAPTER 1. GAME THEORY: AN INTRODUCTION 2

In all game theoretic models, the basic entity is a player. In noncooperative games the individ-ual player and her actions are the primitives of the model, whereas in cooperative games coalitionof players and their joint actions are the primitives. In other words, cooperative game theory canbe interpreted as a theory that allows for binding contracts between individuals. These notes willconsider only non-cooperative game theory.

1.1 Game Theory and Parlor 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 / O. Morgenstern (1944) Theory of Games and Economic Behavior: Ax-iomatic expected utility theory, Zero-sum games, cooperative game theory.

• J. Nash (1950) Nonzero sum games and the concept of Nash equilibrium.

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

• J. Harsanyi (1967/68) games of incomplete information, Bayesian equilibrium.

1.2 Game theory in microeconomics

1. Decision theory (single agent)

2. Game theory (few agents)

3. General equilibrium theory (many agents)

1.3 Basic Elements of Noncooperative Games

A game is a formal representation of a situation in which a number of individuals interact in asetting with strategic interdependence.. The welfare of an agent depends not only on his actionbut on the action of other agents. 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, andthe payoffs or utilities.

CHAPTER 1. GAME THEORY: AN INTRODUCTION 3

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

Example 3 Meeting in New York with unknown meeting place (Non Zero Sum Game, Coordi-nation Game, Imperfect Information).

Chapter 2

Normal Form Games

We start with games that have finitely many strategies for all players and where players choosetheir strategies simultaneously at the start of the game. In these games, the solution of the gamecan 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 Alice and Bruce. Both Aliceand Bruce have a dichotomous choice to be made: either go to an art exhibition or to ballet. Thisstrategic interaction can be represented by a double matrix as follows: The first entry in each cell

Alice

BruceArt Ballet

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

Figure 2.1: Coordination

is the payoff of the “row player”. The second entry is the payoff of the “column player”. Howshould this game be played? What should each individual player do?

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

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

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

∀i, si ∈ Si, (2.2)

4

CHAPTER 2. NORMAL FORM GAMES 5

where each individual player i has a set of pure strategies Si available to him and a particularelement in the set of pure strategies is denoted by si ∈ Si. Finally there are payoff functions foreach 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 any profile we can deducethe payoffs received by the players. This representation of the game is known as normal orstrategic form of the game.

Definition 4 (Normal Form Representation) The normal form ΓN represent a game as:

ΓN = {I, {Si}i , {ui (·)}i} .

2.2 Dominance

The most primitive solution concept to games does not require knowledge of the actions takenby the other players. A dominant strategy is the best choice for a player in a game regardlessof what the others are doing. The next solution concepts assumes that rationality of the playersis common knowledge. Common knowledge of rationality is essentially a recursive notion, sothat every player is rational, every player knows that every player is rational, every player knowsthat every player knows that every player is rational..., possibly ad infinitum. Finally, the Nashequilibrium concept requires that each player’s choice be optimal given his belief about the otherplayers’ behavior, and that this belief be correct in equilibrium.

Consider next a truly famous game which figures prominently in the debate on the possibilityof cooperation in strategic situations, the prisoner’s dilemma game: Besides its somewhat puzzling

Cooperate DefectCooperate 3, 3 0, 4

Defect 4, 0 1, 1

Figure 2.2: Prisoner’s Dilemma

conclusion this game has another interesting property which comes to light when we analyze thepayoff of each individual player, for the row player and for the column player In both cases, onestrategy is better the other strategy independent of all choices made by the opponent.


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

Definition 5 (Dominant Strategy) A strategy si is a dominant strategy for player i in ΓN iffor 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 dominant strategy 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 dominant strategies for allplayers. In those games, there is little need to think about the strategic choices of others as thedominant strategy can never be beaten by another choice of strategy.

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

Example 8 Second price auction (with complete information) is an example of a game with adominant strategy equilibrium.

Definition 9 (Purely Strictly Dominated) A strategy si ∈ Si is a purely strictly dominatedstrategy for player i in ΓN if there exists a pure strategy s′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 weakly dominated strategy forplayer 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 eliminating all strictly domi-nated actions. Once again if a player knows that all the other players are rational, then he shouldnot choose a strategy in Γ′ that is never-best response to in Γ′. Continuing to argue in this wayleads to the outcome in Γ should survive an unlimited number of rounds of such elimination:

Definition 11 (Iterated Pure Strict Dominance) The process of iterated deletion of purelystrictly dominated strategies proceeds as follows: Set S0

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 dom-inated strategies.

Observe that the definition of iterated strict dominance implies that in each round all strictlydominated strategies have to be eliminated. It is conceivable to define a weaker notion of iteratedelimination, where in each step 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, 1β

(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 indicates the order in whichdominated strategies are removed. We might be tempted to suggest a similar notion of iterationfor weakly dominated strategies, however this runs into additional technical problems as orderand speed in the removal of strategies matters for the shape of the residual game as the followingexample shows: If we eliminate first β1, then α2 is weakly dominated, similarly if we eliminate

α2 β2

α1 3, 2 2, 2β1 1, 1 0, 0γ1 1, 1 0, 0

Figure 2.6: Iterated weak dominance

γ1, then β2 is weakly dominated, but if we eliminate both, then neither α2 nor β2 is strictly


L R

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

Figure 2.7: Iterated weak dominance 2

dominated anymore. A similar example is The discussion of domination demonstrated that lessrestrictive notion of play may often give good prediction.. However as the following example shows,it clearly is not enough for analyzing all strategic interactions. The next subsection develops a

L M R

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

Figure 2.8: No dominated strategies

solution concept that singles out (M,M) as the relevant solution of the above game.

2.3 Nash equilibrium

Let us now go back to the coordination game. How to solve this game. A solution should besuch that every player is happy playing his strategy and has no desire to change his strategy inresponse to the other players’ strategic choices. A start is to do well given what the strategychosen by the other player. 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 response for player i tohis rivals strategies s−i if


(s′i, s−i

)for all s′i ∈ Si. Strategy si is never a best response if there is no s−i for which 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 for every individual engagedin this play.

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


(s′i, s−i

), for all s′i ∈ Si. (2.6)

In other words, a strategy profile is a Nash equilibrium if each player’s strategy si is a bestresponse to the strategy profile s−i of all the remaining players. 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 best response mapping.Let us now consider some more examples to test our ability with these new notions.

2.3.1 Finite Normal Form Game

Consider the Hawk-Dove game where two animals are fighting over some prey. Each can behavelike a dove or like a hawk. How is this game played and what is logic behind it.

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

Figure 2.10: Hawk-Dove

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 the restrictions thatrationality imposes on the outcome. The games we are considering next are economic applications.They are distinct from our previous examples insofar as the strategy space is now a continuumof strategies. The first problem is indeed very continuous and regular optimization instrumentsapply directly. The second example has continuous but not differentiable, payoff functions andwe shall proceed directly without using the regular optimization calculus.


Figure 2.11: Reaction curves of the two firms

q1

q2

q1(q2)

q2(q1)

2.3.2 Cournot Oligopoly

Consider the following Cournot model in which the sellers offer quantities and the market priceis 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 functions of the players, whichcan be described graphically as:


for a = 1, and where 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, which reduces the system ofequation to a single equation, and a symmetric equilibrium, so that

q =a− q

2

andq =

a

3and 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 valuations for a single good

0 < v1 < v2 < .... < vn < ∞.

We shall consider two different auction mechanisms: the first and the second price auction. Theobject is given to the bidder with the highest index among all those who submit the highest bid.In the first price auction, the winner pays his own bid, in the second price auction, the winnerpays the second highest 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 property that the bidderwith the highest value always wins the object. Notice that this may not be true in the secondprice auction. However we can show that in the second price auction, the bid bi = vi weaklydominates all other strategies.

Reading: Chapter 0 in [2] is a splendid introduction into the subject. Chapter b of [8] andChapter 1-1.2 in [4] covers the material of the last two lectures as does Chapter 1-1.2 in [3].


2.4 Mixed Strategies

So far we were only concerned with pure strategies, i.e. situations where each player i picks asingle strategy si. However some games don’t have equilibria in pure strategies. In these casesit is necessary to introduces an element of “surprise” or “bluff” through considering randomizeddecisions by the players. Consider the game of “matching pennies”:

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

Figure 2.12: Matching Pennies

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

R P S

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

Figure 2.13: Rock, Paper, Scissors

2.4.1 Basics

Let us now introduce the notation for mixed strategies. Besides playing a pure strategy si eachplayer is also allowed to randomize over the set of pure strategies. Formally, we have

Definition 14 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 be played, where∑

si∈Si

σi (si) = 1

The mixed strategy can be represented as a simplex over the pure strategies:

∆ (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 be easily represented asa two-dimensional triangle sitting in a three dimensional space. σi (si) is formally a probability


mass function and σi is a probability vector. Such a randomization is called a mixed strategy σi

which is a element of set of distributions over the set of pure strategies:

σi ∈ ∆ (Si) .

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

σi ∈ Σi.

In order to respect the idea that the players are choosing their strategies simultaneously andindependently of each other, we require that the randomized strategies chosen by the differentplayers satisfy statistical independence.1 In other words, we require that the probability of a purestrategy profile 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 remaining players choose amixed 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

Opera FootballOpera 2, 1 0, 0

Football 0, 0 1, 2

Figure 2.14: Battle of Sexes1Much of what follows can be generalized to the case where this independence condition is dropped. The resulting

solution concept is then called correlated equilibrium. Since independence is a special case of such correlated

random strategies, the analysis below is a special case of correlated equilibrium. The notion of sunspots, familiar

from macroeconomic models is an example of correlated strategies. There all players observe a signal prior to the

play of the game, and condition their play on the signal even though the signal may be completely unrelated to the

underlying payoffs.


two pure strategy equilibria. To see if it has a mixed strategy equilibrium as well, calculate thepayoff to Sheila if Bruce is choosing according 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 attendingthe football game for Sheila on the x-axis and for Bruce on the y-axis. Every point of intersectionindicates a Nash equilibrium. An equilibrium mixed strategy by Sheila would be written as

σS =(

σS (O) =23, σS (F ) =

13

)(2.11)

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

σS =(

23,13

).

Before we give a systematic account of the construction of a mixed strategy equilibrium, let usextend our definitions for pure strategies to those of mixed strategies.

Definition 15 (Normal Form Representation) The normal form representation ΓN specifiesfor each player i a set of strategies Σi, with σi ∈ Σi and a payoff function ui (σ1, ..., σn) giving thevon Neumann-Morgenstern utility levels associated with the (possible random) outcome arisingfrom 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 mixed strategies, or Si ⊂ Σi, wecan 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 implies that (2.12) holds. It isthen sufficient (again, why) to show that for every σ′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, in particular 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 achieves the highest payoffui

(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 equilibria. Let

S+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 mathe-

matically referred to as the support of 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 strategy profile 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.


Sheila

BruceOpera Football

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

Figure 2.15: Battle of Sexes

2.4.2 Construction of a mixed strategy Nash equilibrium

We return to the example of the battle of sexes.Then a strategy for Sheila has the following payoff if Bruce is playing a mixed 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 supposed to use a mixedstrategy, it means that she picks either alternative with a positive probability. As the strategy issupposed to be a best-response to her, it has to be the case that the expected payoffs are identicalacross the two alternative, for otherwise Sheila would clearly be better off to use only the purestrategy which gives her the strictly higher payoff. In short, it has to 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 conditions on the mixingbehavior 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 =23. (2.15)

Thus if Sheila is to be prepared to randomize, it has be that Bruce is randomizing accordingto (6.14). Now Bruce is willing to randomize with σB = 1

3 in equilibrium if and only if heis indeed indifferent between the pure strategy alternatives. So, now we have to ask ourselves


when is this the case. Well, we have to investigate in turn how the indifference condition ofBruce determines the randomizing behavior of Sheila. (The construction of the mixed strategyequilibrium highlights the interactive decision problem, which is essentially solved as a fixed pointproblem.) A similar set of conditions as before 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 examine conditions on the mixingbehavior 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 =13. (2.17)

Thus we reached the conclusion that the following is a mixed strategy Nash equilibrium

σ∗B =(

σB (O) =13, σB (F ) =

23

)σ∗S =

(σS (O) =

23, σS (F ) =

13

).

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 mixed strategy Nash equilibrium. The only issuewhich we still have to discuss is the support of the mixed strategy equilibrium.

The following game is a helpful example for this question:

α β γ

α −1, 1 1,−1 x, x

β 1,−1 −1, 1 x, x

γ x, x x, x x, x

Figure 2.16: Augmented Matching Pennies

For x < 0, γ cannot be part of any mixed strategy equilibrium, and for x > 0, the originalmatching pennies equilibrium cannot be a mixed strategy equilibrium anymore. However for


x = 0, there can be many mixed strategy equilibria, and they differ not only in the probabilities,but in the support as well.2

2.4.3 Interpretation of mixed strategy equilibria

Finally, consider the competing interpretations of a mixed strategy Nash equilibrium:

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 randomness is introduced throughsome random private information (Harsayni’s purification argument)

Remark 20 In the battle of sexes, the mixed strategy equilibrium used private randomization byeach player.

Suppose instead they would have a common (or public) randomization device, say a coin, thenthey could realize a payoff of (3

2 , 32) by e.g. both going to opera if heads and both going to football if

tails. Observe that this payoff is far superior to the mixed strategy equilibrium payoff The notionof public randomization is used in the correlated equilibrium.

Finally we consider games where each player has a continuum of actions available. The firstone 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 investor can spend any amountin the interval [0, v]. The winner is the investor who spends the most. In the event of tie eachinvestor receives v/2. Formulate this situation as a strategic game and find its mixed strategyNash equilibria.

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

The payoff function of the very agent is therefore given by

ui (v, bi, bj) =

v − bi for bj < bi

12v − bi, for bi = bj

−bi for bi < bj

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

between β and γ.


The mixed (or pure) strategy by player i can be represented by the distribution function of thebid of i:

Fi (bi) ,

where we recall that the distribution function (or cumulative distribution function) Fi (bi) isdefining the event

Fi (bi) = Pr (b ≤ bi)

We recall the following notation:F

(b−

)= lim

a↓bF (a) ,

andF

(b+

)= lim

a↑bF (a) .

Before we can analyze and derive the equilibrium, we have to ask what is the expected payofffrom submitting a bid b1 given that player 2 uses an arbitrary distribution function F2. From thepayoff functions, we see that it can be calculated as:

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

(F2 (b2 ≤ b1)− F2 (b2 < b1))(

12v − 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

)) 12v − b1. (2.18)

This expected payoff function shows that in this auction the bidder has to find the best trade-offbetween increasing the probability of winning and the cost associated with increasing the bid.

We suppose for the moment that the distribution function F2 (·) is continuous-(ly differen-tiable) 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 maximize his payoff. In otherwords, his optimal bidding strategy is characterized by the first order conditions which representthe marginal benefit equals marginal cost condition

f2 (b1) v − 1 = 0, (2.19)

which states that the increase in probability of winning which occurs at the rate of f (b1) multipliedby the gain of v must equal the marginal cost of increasing the bid which is 1. As we attempt to


construct a mixed strategy 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) =1v. (2.20)

As we look for a symmetric equilibrium, in which agent i and j behave identically, we can omitthe subscript in (6.4) and get a uniform (constant) density

f∗ (b) =1v

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 constitutes a mixed equilibriumstrategy.

We went rather fast through the derivation of the equilibrium. Let us now complete theanalysis. Observe first that every pure strategy can be represented as a distribution function. Westart by explaining why there can’t be a pure strategy equilibrium in the all-pay auction, or inother words why there can be no atoms with probability 1.

Lemma 21 (Non-existence of pure strategy equilibrium) @ a pure strategy equilibrium inthe all pay auction.

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

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 be12v − bi,

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

v − bi − ε.

As ε → 0, and when taking the difference, we see that i would have a jump in his net utility of12v as

v − bi − ε−(

12v − bi

)=

12v − 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 the derivation above, we omitted some possibilities:(i) it could be that the distribution function of agent i is not continuous and hence has some masspoints or atoms, (ii) we have not explicitly derived the support of the distribution. We will inturn solve both problems.

A similar argument which excludes a pure strategy (and hence atoms with mass one) alsoexcludes the possibility of atoms in mixed strategy equilibrium. A strategy which is discontinuousand has an atom at b is graphically depicted below.

Lemma 22 (Non-existence of mass points in the interior) @ a mixed strategy equilibriumwith a mass point b in the interior of the support of either player’s mixed strategy ( i.e. Fi (b)−Fi (b−) > 0 for some i) in the all pay auction.

Proof. Suppose ro the contrary that the strategy of player j contains an atom so that for someb,

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 probability of winning. Butconsider 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 hence there could be atomsat the upper bound of the support of the distribution.3 It then remains to find the mixed strategy

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

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

and thus the support is

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


Figure 2.17: A distribution with an atom at b

b

F(b)

b∗

�

��

1


equilibrium which has an interval as its support. Suppose Fi (bi) is the distribution functionswhich defines the mixed strategy by player i. For her to mix over any interval

bi ∈[bi, bi

](2.23)

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

where c is the expected payoff from any mixed strategy equilibrium. Argue that c = 0, then wesolve the equation to

Fj (b) =b

v, (2.25)

which tells us how player j would have to randomize to make player i indifferent. The argumentis actually already made by the derivation of the first order conditions and the observation thatthere can’t be any atom in the interior of [0, v]. By symmetry, (2.25) then defines the equilibriumpricing strategy by each player.

Suppose we were to increase the number of bidders, then (6.10) changes under the assumptionof symmetry to

v (F (b))n−1 − b = c, (2.26)

where c is the expected payoff from any mixed strategy equilibrium. Argue again that c = 0, thenwe solve the equation to

F (b, n) =(

b

v

) 1n−1

(2.27)

and the expected payment of each individual bidder is∫ v

0bdF (b) =

∫ v

0bf (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 c per unit of timeto stay in the competition. If either of the two player drops out of the competition, then theother player gets the object immediately. This is a simple example of a timing game, wherethe strategy is defined by the time τ i at which the player drops out. There are pure strategy


Nash equilibria, which have(τ i = 0, τ j ≥ c

v

), however the only symmetric equilibrium is a mixed

strategy equilibrium, where each player must be indifferent between continuing and dropping outof 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 differential equation

dFi (t)(1− Fi (t))

=c

v.

It is very easy to solve this equation once it is observed that the left hand side 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 has a constanthazard 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 other player staying in thegame forever is ∫ ∞

0tdF (t) =

∫ ∞

0tf (t) dt =

v

c.

The properties of the exponential distribution are illustrated below for λ = 12 .

4To understand why this is called the hazard rate, think of t as the time a machine fails. Then as time t passes,

and moving from t to t + dt, one finds that the conditional probability that the machine fails in this interval,

conditional on not having failed before isf (t) dt

1− F (t).

Als recall that the conditional probability is

P (A |B ) =P (A ∩ B)

P (B)


Figure 2.18: Exponential distribution for λ = .5

t

F(t)F(t)

f (t)

12

1


2.5.3 Appendix: Basic probability theory

The general framework of probability theory involves an experiment that has various possibleoutcomes. Each distinct outcome is represented by a point in a set, the sample space. Probabilitiesare assigned to certain outcomes according to certain axioms. More precisely X is called a randomvariable, indicating that the value of X is determined by the outcome of the experiment. Thevalues X can take are denoted by x. We can then refer to events such as {X = x} or {X ≤ x}and the sample space is X .

We distinguish two cases of probability distributions: discrete and continuous.

Discrete Case

In the discrete case, the numbers of possible outcomes is either finite or countable and so we canlist them. Consider the following experiment in which the numbers x ∈ {1, 2, 3, 4} are drawn withprobabilities p (x) as indicated. Any point x with p (x) > 0 is called a mass point or an atom.The function 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 distribution function is then givenby

F (z) =∑x≤z

p (z) ,

which is pictured below:Notice that F (x) is a piecewise constant function, which is monotonically increasing 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]. The assignment of proba-bilities occurs via the density function f (x) ≥ 0, so that∫ 4

0f (x) dx = 1.

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

F(x′

)= F

(x ≤ x′

)=

∫ x′

0f (x) dx


Figure 2.19: Probability mass function for a discrete distribution

t

F(t)F(t)

1


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

F(x′′

)− F

(x′

)=

∫ x′′

x′f (x) dx

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

F (x) =

0, for x < 01−

(1− 1

4x)3

, for 0 ≤ x ≤ 41, for x > 4

(2.29)

and depicted below as F (x):We may then ask what is the probability that a number is drawn out of the interval (x, x′),

and it is given byF

(x′

)− F (x) .

The density function then identifies the rate at which probability is added to the interval by takingthe 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 the interval, but is notthe probability of a particular x, call it x̂ occurring since the probability of x̂ occurring is of course0 under a continuous distribution function, since∫ x̂

x̂f (x) dx = 0.

Thus we have to distinguish between events which have probability zero, but are possible andimpossible events. The density function associated with the distribution function F (x) is givenby

f (x) =

0, for x < 034

(1− 1

4x)2

, for 0 ≤ x ≤ 40 for x > 4

and graphically they are jointly represented by:

Mixed Case

Finally, we can of course have distribution functions with continuous and discrete parts. We sawan example of this in the case of a mixed strategy with an atom.


Figure 2.20: Cdf for a continuous distribution

t

F(t)F(t)


Figure 2.21: Cdf and density for a continuous distribution

t

F(t)F(t)

f (t)


Notice that the discrete probability mass introduces a discontinuity in the distribution functionso that

F(p−

)= lim

p′↑pF

(p′

)6= F (p) .

The point x = 2 is often called an atom as it carries strictly positive probability to the distributionfunction.

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 variablesE [x] =

∑z

zp (z)

Reading: Recall: A brief review of basic probability concepts, including sample space andevent, independent events, conditional probabilities and Bayes’ formula, discrete and continuousrandom variables, probability mass function, density and distribution function, all introduced in[12], Chapter 1 -2.4 is highly recommended. [10], chapter 3 contains a very nice discussion of theinterpretation and conceptual aspects of the equilibrium notion, especially in its mixed version.

2.6 Appendix: Dominance Solvability and Rationalizability

In this section, we elaborate on the notion of iterated deletion of dominated strategies. We definedominance in a way that allows for the use of mixed strategies as well and we also connect thisconcept to the notion of rationalizable strategies where the question is phrased in terms of findinga set of mutually compatible strategies in the sense that all such strategies are best responses tosome element in the other players’ allowed set of strategies. Nash equilibrium is a special caseof this concept where it is also required that the sets of allowed strategies are singletons for allplayers.


Definition 23 (Strictly Dominated) A strategy si ∈ Si is a strictly dominated strategy forplayer 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 or mixed strategies.Clearly, more strategies can be dominated if we allow mixed strategies to be in the set of dominatingstrategies.

Definition 25 (Never-Best response) A strategy si is never-best response if it is not a bestresponse to any belief by player i. In other words, for all σ−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-best response if and only ifit is strictly dominated.

The notion of strict dominance has hence a decision-theoretic basis that is independent of thenotion of mixed strategy.5

Now consider a game Γ′ obtained from the original game Γ after eliminating all strictly domi-nated actions. Once again if a player knows that all the other players are rational, then he shouldnot choose a strategy in Γ′ that is never-best response to in Γ′. Continuing to argue in this wayleads to the outcome in Γ should survive an unlimited number of rounds of such elimination:

Definition 27 (Iterated Strict Dominance) The process of iterated deletion of strictly dom-inated strategies proceeds as follows: Set S0

i = Si, and Σi = Σ0i . Define Sn

i 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 similarlyΣn

i = {σi ∈ Σi |σi (si) > 0 only if si ∈ Sni } .

Set

S∞i =∞⋂i=1

Si,

andΣ∞

i ={σi ∈ Σi

∣∣@σ′i ∈ Σi s.th. ui

(σ′i, s−i

)> ui (σi, s−i) , ∀s−i ∈ S∞−i

}5The above Lemma extends to more than two players if all probability distributions on S−i and not only the

ones satisfying independence are allowed.


The set S∞i is then the set of pure strategies that survive iterated deletion of strictly dominatedstrategies. The set Σ∞

i is the set of player’s i mixed strategies that survive iterated strict domi-nance.

2.6.1 Rationalizable Strategies

The notion of rationalizability begins by asking what are all the strategies that a rational playercould play? Clearly, a rational player will play a best response to his beliefs about the play of otherplayers and hence she will not use a strategy that is never a best response. Hence such strategiesmay be pruned from the original game. If rationality of all players is common knowledge, then allplayers know that their opponents will never use strategies that are never best responses. Hencea player should use only strategies that are best responses to some belief of other players’ playwhere the others are not using never best response strategies. Iteration of this reasoning leads toa definition of rationalizable strategies. Denote the convex hull of A by co (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 rationalizable strategies.

The reason for having the convex hull operator in the recursive definition is the following.Each point in the convex hull can be obtained as a randomization over undominated strategiesfor j 6= i. This randomization corresponds to i′s subjective uncertainty over j′s choice. It couldfully well be that this random element is dominated as a mixed strategy for player j. Henceco

(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 response and strictly dominated in two playergames which does not hold in three player games as the following example suggests. In the gamebelow, we have pictured the payoff to player 3, and player 3 chooses either matrix A, B, C or D.

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


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 2.22: Rationalizable vs. undominated strategies

2.7 Appendix: Existence of Mixed Strategy Equilibria

Up to this point, we have not shown that mixed strategy equilibria exist for general games. Wehave simply assumed their existence and characterized them for some games. It is important toknow that for most games, mixed strategy Nash equilibria do exist.

The first existence result was proved by Nash in his famous paper [9].

Theorem 29 (Existence in Finite Games) Finite games, i.e. games with a finite number ofplayers 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 point theorem, and it isgiven in almost all graduate level textbooks, for instance the appendix to Chapter 8 in [8] givethe proof.

Many strategic situations are more easily modeled as games with a continuum of strategiesfor each player (e.g. the Cournot quantity competition mode above). For these games, slightlydifferent existence theorems are needed. The simplest case that was also proved by Nash is thefollowing (and also covered in the same appendix of [8]).

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 iscontinuous for all i and quasiconcave in si for all i, ΓN has a pure strategy Nash equilibrium.

Observe that this theorem actually guarantees the existence of a pure strategy equilibrium.This results from the assumption of continuity and quasiconcavity. If either of these requirementsis dropped, pure strategy equilibria do not necessarily exist.

Unfirtunately, the payoffs in many games of interest are discontinuous (e.g. the all pay auctionand the war of attrition examples above show this). For these games, powerful existence resultshave been proved, but the mathematical techniques are a bit too advanced for this course. Stateof the art results are available in the recent article [11]. The basic conclusion of that paper isthat it will be very hard to find economically relevant games where existence of mixed strategyequilibria would be doubtful.


An artificial game where no equilibria (in either pure or mixed strategies) exists is obtained byconsidering a modification of the Cournot quantity game. Recall that in that game, the payoffsto 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 showthis is that all of the strategies except for qi = a

3 are deleted in finitely many rounds of iterateddeletion of strictly dominated strategies. Hence none of those strategies can be in the support ofany 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 [8], chapter 2 in [3] and to a lesser extent Chapter 1 in [4] cover thematerial of this chapter.

Chapter 3

Extensive Form Games

3.1 Introduction

This lecture covers dynamic games with complete information. As dynamic games, we understandgames which extend over many periods, either finitely many or infinitely many. The new conceptintroduced here is the notion of subgame perfect Nash equilibrium (SPE), which is closely relatedto the principle of backward induction.

The key idea is the principle of sequential rationality : equilibrium strategies should specifyoptimal behavior from any point in the game onward. The notion of subgame perfect Nashequilibrium is a strengthening of the Nash equilibrium to rule out incredible strategies. We startwith finite games, in particular games with a finite horizon.

We then consider two of the most celebrated applications of dynamic games with completeinformation: the infinite horizon bargaining game of Rubinstein 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 is better than a price war (−3,−1). Depictthe game in the normal form and the extensive form. Characterize the equilibria of the game.Distinction between subgame perfect and Nash equilibrium. Credible vs. incredible threat. Futureplay (plan) affects current action.

Example 32 Cournot Model. Stackelberg Equilibrium. Consider again the quantity settingduopoly with aggregate demand

P (q1, q2) = a− q1 − q2

and the individual profit function:

Πi (qi, qj) = (a− q1 − q2)qi

36

CHAPTER 3. EXTENSIVE FORM GAMES 37

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

qj = R (qi) =a− qi

2

Suppose player i moves first, and after observing the quantity choice of player i , player j chooseshis quantity optimal. Thus i takes into account the reaction (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 associatedNash equilibrium is called the “subgame perfect” Nash equilibrium. Comparing with the Cournotoutcome we find

qSi + qS

j ≥ qCi + qC

i =a

3+

a

3but

ΠSi > ΠC

i =a2

9, ΠS

j < ΠCj .

This example illustrates the value of commitment, which is in this game simply achieved by movingearlier than the opponent.Notice that the Cournot equilibrium is still a Nash equilibrium. But is it reasonable? One has tothink about the issue of “non credibility” and “empty threats”.

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 playercan take, (iii) rules determining who is to move and when, (iv) what players know when theymove, (v) what the outcome is as a function of the players’ actions and the players’ payoff fromeach possible outcome, (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 game tree. The languageof a tree is borrowed from graph theory, a branch of discrete mathematics and combinatorics.

A graph G is given by (V,E), where V = {v1, ..., vn} is a finite set of nodes or vertices andE = {vivj , vkvl, ..., vrvs} is a set of pairs of vertices (or 2-subsets of V), called branches or edgeswhich indicates which nodes are connected to each other. In game theory, we are mostly interestedin graphs that are directed. Edges in a directed graph are ordered pairs of nodes (v1, vj) and wesay that the edge starts at vi and ends at vj . A walk in a graph G is 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 are distinct, a walk iscalled a path. A walk v1, v2, ...., vn such that v1 = vn is called 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 other vertex 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 preceeds x in the game form,i.e. if every path reaching x goes through y. The relation ≺ totally orders the predecessors ofeach member of V . 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) = t

number of predecessors l (t) pl(t) (t) ∈ W

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

In order to define formally the extensive form game, we need the following ingredients.(i) The set of nodes V consists of initial nodes W , decision nodes X, and terminal 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). The immediate predecessornodes are p1 (x) and the immediate successor nodes of x are

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

(iii) An assignment functionα : T\W → A,

where we require that α is a one-to-one function onto the set s (x) of x. The term 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 sets if ∀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 imposed in (6.3) we definethe set of actions which can be taken at an information 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 theset of all decision nodes at which i is called to make a move. A prominent way of identifyingan information set is through the history leading to it. Hence h ∈ H is often called a particularhistory 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.

Observe that since each terminal node can only be reached through a single path, defining payoffson terminal nodes is equivalent to defining payoffs on paths of actions.

(vii) An initial assessment ρ as a probability measure over the set of initial 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 exten-sive form game.

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

Definition 37 (Almost Perfect Information) A game is of “almost” perfect information ifthe player’s choose simultaneously in evert period t, knowing all the actions chosen by everybodyat dates 0 to t− 1.

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 a single information set.


3.2.4 Notion of Strategy

A strategy is a complete contingent plan how the player will act in every possible distinguishablecircumstance in which she might be called upon to move. Consider first the following game ofperfect 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 strategy set Si, a mixed strategyfor player i,

σi : Si → [0, 1]

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

σi (si) = 1.

Definition 40 (Outcome) The outcome induced by a mixed strategy profile σ in an extensiveform game is the distribution induced by the initial assessment ρ and the mixed strategies on theterminal nodes.

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

|Si| =∏

hi∈Hi

|A (hi)|

where |S| represents in general the cardinality of a set S, i.e. the number of members in the set.If the number of actions available at each information set 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 different notion for a mixedstrategy. In the definition above, the number of components needed to specify a mixed strategyis |Si| − 1. This is often a very large number. A simpler way to define a random strategy is thefollowing. Rather than considering a grand randomization at the beginning of the game for eachplayer, we could consider a sequence of independent randomizations, one at each information setfor each of the players. This leads to the notion of a behavior strategy.


Definition 41 (Behavior Strategy) A behavior strategy of player i is a mapping

bi : Hi → ∆ (A)

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

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

|A (hi)− 1|

components must be specified. It is immediately obvious that each behavior strategy generatesa mixed strategy in the sense of the above definition. It is a much deeper insights into extensiveform games that most often it is enough to consider behavior strategies in the sense that eachoutcome induced by a mixed strategy profile is also induced by some behavior strategy profile.This is true as long as the game is one of perfect recall.

Definition 42 (Perfect Recall) An extensive form game {T,≺;A,α; I, ι;H;ui; ρ} has perfectrecall 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 to x′′.

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

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

In other words, in games of perfect recall, there is no loss of generality in restricting attentionto behavior strategies.

The following example is due to Rubinstein and Piccione (see the special issue of Games andEconomic Behavior, July 1997, on games of imperfect recall).

Example 44 (Absent Minded Driver) This is a single person game with two decision nodesfor the player. The player must decide whether to keep on driving D on the highway or whetherto exit E. If the driver chooses D twice, he ends in a bad nighborhood and gets payoff of −1. Ifhe exits at the first 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 cannot remember if he hasalready decided or not in the past. This absent mindedness is represented by a game tree where


both of the nodes are in the same information set even though the two nodes are connected by anaction by the single 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 consider the various notionsof strategies. The first step in the analysis of any games is to decide what the possible strategiesare. As usual, a strategy should be an assignment of a (possibly mixed) action to each informationset. With this definition for a strategy, it is easy to show that the optimal strategy in this gamecannot be a pure strategy. Observe also that if the driver were able to commit to a mixed strategy,then she would choose to exit with probability p = 2

3 resulting in a payoff of 29 .

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 second node in her information set. Inorder to calculate correctly the payoffs to the driver, behavior at other nodes must be kept constantwhen calculating the current payoff to exiting or driving on. Thus choosing D and then followingthe proposed strategy yields a payoff of

34(2

23− 1

3)− 1

4=

12.

Choosing E yields similarly a payoff of 12 .

The key to understanding the workings of the game lies in the interpretation of a deviationfrom the proposed equilibrium strategy. Most game theorists believe that in games of this type, adeviation should be modeled as a single deviation away from the proposed strategy. In particular,this would imply that a deviation can take place at a single node in the information set withoutnecessarily happening at the other. Piccione and Rubinstein offer a different interpretation of thegame, see the discussions in the issue of GEB as cited above.

3.3 Subgame Perfect Equilibrium

Before we can define the notion of a subgame perfect equilibrium we need to know what a subgameis. The key feature of a subgame is that, contemplated in isolation it forms its own game.

Definition 45 (Subform) A subform of an extensive form is a collection of nodes T̂ ⊆ T , to-gether with {≺;A,α; I, ι;H} all defined on the restriction to T̂ , satisfying closure under successionand 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 its successors.

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


Definition 47 (Subgame Perfect Nash Equilibrium) A profile of strategies

σ = (σ1, ..., σn)

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

Clearly, every SPE is an NE, but the converse doesn’t hold. Consider first the extensiveform games with perfect information. Chess is a finite extensive form game as is centipede gameintroduced above. What is the unique 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 connected edges which leadfrom an initial node to a final node. The equilibrium path is thus {s} and the equilibriumoutcome is the payoff realization (1, 1). Thus a description of the equilibrium strategy asks forthe behavior of every player on and off the equilibrium path.

Theorem 48 (Zermelo, Kuhn) Every finite game of perfect information ΓE has a pure strategysubgame perfect Nash equilibrium.

3.3.1 Finite Extensive Form Game: Bank-Run

Consider the following bank-run model with R > D > r > D/2. The interpretation is as follows.The bank acts as an intermediary between lender and borrower. Each lender deposit initially D.The bank also attempts to transform maturity structures. The deposits are short-term, but theloan is long-term. Suppose the deposits are applied towards an investment project which has areturn of 2r if the project is liquidated after one period and 2R if liquidated after two periods. Thereturn is 2D if the project is continued for more than two terms. If either of the depositor asksfor a return of its deposit, then the project is liquidated, and the withdrawing depositor eithergets D or his share in liquidation proceeds, whichever is larger. Denote the first stage actions byw for withdrawing and n for not withdrawing. Let the second period actions for the two playersbe W and N respectively. Thus the stage payoffs are given by the two matrices There are two

w n

w r, r D, 2r −D

n 2r −D,D second period

Figure 3.1: First period


W N

W R,R 2R−D,D

N D, 2R−D D,D

Figure 3.2: Second period

pure strategy 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 gameis a prisoner’s dilemma game and the second game is coordination game: and then Specify all

α β

α 2, 2 −1, 3β 3,−1 0, 0

Figure 3.3: First period

γ δ

γ x, x 0, 0δ 0, 0 x, x

Figure 3.4: Second period

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

3.4 The Limits of Backward Induction: Additional Material

Consider the following example of game, referred to as “Centipede Game”. There are two playerseach start with 1 dollar. They alternate saying “stop” or “continue”, starting with player 1. Ifthey say continue, 1 dollar is taken from them and 2 dollar is given to the other player. As soon aseither player says stop the game is terminated and each player receives her current pile of money.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 player 2. Instead of lookingat the ‘true’ centipede game, let us look at a smaller version 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 is a completedescription of actions at every information set (or history). Player 1 has two histories at whichhe could conceivable make a choice H1 = {∅, cc} and likewise H2 = {c, ccc′}. Thus a possiblestrategy 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 beyond what we would calla plan. We saw in the centipede game that it indeed requires a player to specify his action afterhistories that are impossible if he carries out his plan. A different interpretation of the extensiveform game may help our understanding. We may think of each player having as many agents (ormultiple selves) as information sets. The strategy of each player is to give instruction to eachagents (which he may not be able to perfectly control) as to what he should do if he is called tomake a decision. All agents of the same player have the same payoff function. A plan is then acomplete set of instruction to every agent who acts on behalf of the player.

Notice that we did not discuss notions similar to rationalizability in the context of extensiveform games. The reason for this is that making sense of what common knowledge of rationalitymeans in extensive form games is much more problematic than in normal form games. A simpleexample will illustrate.

Robert Aumann has maintained that common knowledge of rationality in games of perfectinformation implies backwards induction. Hence e.g. the centipede game should have a unique


solution where all players stop the game at all nodes. Suppose that this is indeed the case. Thenafter observing a move where the first player continues the game, the second player ought toconclude that the 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 it is not irrationalfor the first player to continue at the first node. It should also be pointed out that knowledgeand belief with probability 1 are more or less the same in normal form games, but in extensiveform games these two notions yield very different conclusions. Appropriate models of knowledgefor extensive form games form a very difficult part of interactive logic (or epistemic logic) andChapter 14 in [3] is a slightly outdated introduction into this area.

3.5 Reading

The notion of extensive form game is first (extensively) developed in [7] and nicely developed in [5]and [6]. The relation between mixed and behavior strategies is further discussed in [3], Ch.3.4.3.

Chapter 4

Repeated Games

4.1 Finitely Repeated Games

We first briefly consider games which are repeated finitely many time. Then we move on thetheory of infinitely repeated games. We first introduce some notation. Consider the prisoner’sdilemma game: and repeat the stage game T times. A pure strategy si =

(s1i , ..., s

Ti

)for a player

C D

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

Figure 4.1: Prisoner’s Dilemma

i in the repeated game is then a mapping from the history of the game Ht into the strategies inthe stage game Si:

sti : Ht−1 → Si.

A complete specification of repeated game strategy has to suggest a strategy choice after everypossible history of the game, (and not only the histories which will emerge in equilibrium). Acomplete specification can be thought of as a complete set of instructions written before playingthe game, so that no matter what happens in the game, once it started a prescription of play isfound the set of instructions. The history of the game is formed by the past observed strategychoices of the players:

Ht−1 =(s11, ..., s

1I ; ...; s

t−11 , ..., st−1

I

).

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 play begins. The payoffs forΓ (T ) are simply the sums of the payoffs from the T stage games.

48

CHAPTER 4. REPEATED GAMES 49

Definition 50 In a repeated game Γ (T ) a player’s strategy specifies the action the player willtake in each stage, for each possible history of play through the previous stage:

si ={s0i , ...., s

Ti

}where:

sti : Ht−1 → Si.

Theorem 51 If the stage game Γ has a unique Nash equilibrium then, for any finite T , therepeated game Γ (T ) has a unique subgame-perfect outcome: the Nash equilibrium of Γ is playedin every stage.

Corollary 52 Every combination of stage game Nash equilibria forms a subgame perfect equilib-rium.

Consider now the following game with multiple equilibria in the stage game Can coordination

C D E

C 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

among the players arise in a stage game?

Example 53 To understand the complexity of repeated games, even if they are repeated onlyvery few times, let us consider the prisoner’s dilemma played three times. What are all possiblehistories of a game. How many strategy choice have to be made in order for the repeated gamestrategy to be a complete specification?

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 been played once. Eachoutcome is uniquely defined by the (pure) strategy choices leading 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 isalready ∣∣H1

∣∣ = |S1| × |S2| × |S1| × |S2| = 16,

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

However the complexity of strategies grows even faster. A complete strategy for period t specifiesa particular action for every possible history leading to period t. Thus the number of contingentchoices to be made by each player in 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 instruction to act in a specificmanner in the first and second period, but needs to specify the action after each contingency.

In order to sustain cooperation two conditions need to be satisfies: (i) there are rewards fromcooperation and (ii) there are credible strategies to punish deviators. This theme of “carrotsand sticks” is pervasive in the entire theory of repeated games. As the punishment occurs in thefuture, agents must value the future enough so that lower payoffs in the future would indeed harmthe agent’s welfare sufficiently. If the discount factor represents the agent’s attitude towards thefuture, then higher discount factors make cooperation easier to achieve as lower payoffs in thefuture figure more prominently 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 repeated game in which Γ isrepeated forever and where each of the players discounts future at discount factor δ < 1.

The payoff in an infinitely repeated game with stage payoffs πt and discount factor δ < 1 isgiven by

∞∑t=0

δtπt.

Definition 55 Given the discount factor δ, the average payoff of the infinitely repeated sequenceof payoffs π0, π1, π2, ... is given by

(1− δ)∞∑

t=0

δtπt.

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


4.2.1 The prisoner’s dilemma

Let us consider again the stage game (??) which represented the prisoner’s dilemma and considerthe following strategy suggested to the players. I cooperate in the first period unconditionallyand then I continue to cooperate if my opponent cooperated in the last period. In case hedefected in the last period, I will also defect today in all future periods of the play. When canwe sustain cooperation? This strategy is often called the grim-trigger strategy, which can beformally 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 perfect equilibrium. To preventa deviation the following has to hold:

5 +β

1− β≤ 4

1− β,

or5 (1− β) + β ≤ 4,

so thatβ ≥ 1

4.

Thus if the players are sufficiently patient, cooperation can be sustained in this repeated game.The equilibrium we suggested implies cooperation forever along the equilibrium path. How-

ever, this is clearly not the only equilibrium. There are many more, and many of them of verydifferent characteristics. In the limit as δ → 1, one can show, and this is the content of thecelebrated “folk” theorem’s of the repeated games, that the entire convex hull of all individuallyrational payoffs of the stage game can be achieved as a subgame perfect equilibrium. Notice, thatthis may involve very asymmetric payoffs for the players:

4.3 Collusion among duopolists

The special feature of the prisoner’s dilemma game is that minimum payoff a player can guaranteehimself in any stage, independent of the actions chosen by all other players, is equal to the payoffin the stage game Nash equilibrium. This minimum payoff is called the reservation payoff. Itconstitutes the limit to the damage an opponent can inflict on the player. The individual rationalpayoff vi is the lowest payoff that the other players can force 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 by quantities as

p (q1, q2) = a− q1 − q2.

The Cournot stage game equilibrium is found by solving the first order conditions of the profitfunctions, where we normalize the marginal cost of production 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

2and the profits are

π(qM

)= πM =

a2

4.

4.3.2 Best Response

Suppose the two firms agree to each supply x to the market, then the profits are given by:

πi (q, q) = q (a− 2q) = ax− 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

)=

14

(3q − a)2 ≥ 0

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


4.3.3 Collusion with trigger strategy

The discrepancy between monopoly and dupoly profit suggest that the duopolists may collude inthe market to achieve higher prices. How could they do it. Consider the following strategy.

qit =

{qM

2 if ht ={(

qM

2 , qM

2

), ....,

(qM

2 , qM

2

)}or ht = ∅

qC otherwise

∣∣∣∣∣and consider whether the symmetric strategy can form a subgame perfect equilibrium? We haveto consider the alternatives available to the players. Consider an history in which

ht ={(

qM

2,qM

2

), ....,

(qM

2,qM

2

)},

then the resulting payoff would be

11− δ

12πM ≥ πd +

δ

1− δπC . (4.3)

Thus to evaluate whether the player follow the equilibrium strategy, we have to evaluate πd.

SPE

This suggests the following trigger strategy equilibrium

11− δ

12πM ≥ πd +

δ

1− δπC , (4.4)

which can be sustained for δ ≥ 917 . The question then arises whether we can do better or more

precisely whether there are punishment strategies which would act even harsher on deviators. Westill have to verify that following a deviation, i.e. all histories different from collusive histories

11− δ

πC ≥ πd +δ

1− δπC , (4.5)

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

4.3.4 Collusion with two phase strategies

We next consider a repeated game where the equilibrium strategy consists of two different phases:a maximal reward and a maximal punishment phase, a “carrot and stick” strategy. The strategymay formally be described as follows:

qit =

qM

2 if ht ={(

qM

2 , qM

2

), ....,

(qM

2 , qM

2

)}qM

2 if(qit−1, q

jt−1

)=

(q, q

)q if otherwise

∣∣∣∣∣∣∣∣


We may then write the equilibrium conditions as

11− δ

12πM ≥ πd +

Punishment︷︸︸︷δ(ax− 2x2

)+

Continuation︷︸︸︷δ2

1− δ

12πM (4.6)

andPunishment︷︸︸︷(ax− 2x2

)+

Continuation︷︸︸︷δ

1− δ

12πM ≥

Deviation︷︸︸︷(a− x)2

4+

Punishment︷︸︸︷δ(ax− 2x2

)+

Continuation︷︸︸︷δ2

1− δ

12πM (4.7)

Equation (4.6) may be rewritten as,

(1− δ) (1 + δ)1− δ

12πM ≥ πd + δ

(ax− 2x2

)and finally as:

δ

(12πM −

(ax− 2x2

))≥ πd − 1

2πM . (4.8)

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

ax− 2x2)

+ δ12πM ≥ (a− x)2

4+ δ

(ax− 2x2

). (4.9)

and which permits an intuitive interpretation in terms of today’s and tomorrow’s payoff, or alter-natively as the one shot deviation strategy. By inserting the value of πM we obtain two equalities:

δ(

12

a2

4 −(ax− 2x2

))=

(964a2 − 1

2a2

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 of punishment x = 5

12a

and the lowest level of discount factor δ = 932 for which cooperation is achievable.

Reading: [4], Ch. 2.3 is a nice introduction in the theory of repeated games. A far moreadvanced reading is [3], Ch.5. The characterization of extremal equilibria is first given in [1].

Chapter 5

Sequential Bargaining

The leading example in all of bargaining theory is the following trading problem in the presence ofbilateral monopoly power. A single seller and a single buyer meet to agree on the terms of tradefor a single unit of output. The seller incurs a cost of c to produce the good while the buyer has avalue of v for the good. If the two parties have payoff functions that are quasilinear in money andthe good, then there are positive gains from trade if and only if v > c. The two parties have alsoan opportunity to refuse any trade which yields them an outside option value of 0. Both partiesprefer (weakly) trading 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 70’s was that the determination of the price dependslargely on psychological and sociological factors determining the bargaining power of the twoparties and as a result economics had little to say on the issue. When the concept of subgameperfect equilibrium was established as the standard solution concept in games of perfect infor-mation, the bargaining issue was taken up again. Preliminary work was done on bargaining insimple two period models, but the real breakthrough was the infinite horizon bargaining modelwith alternating offers proposed by Ariel Rubinstein in [13]. There he showed that the infinitehorizon game has a unique subgame perfect equilibrium and that the division of surplus betweenthe bargainers can be explained in terms of the time discount rates of the bargainers.

It should be pointed out that other approaches to bargaining were also adopted. An entirelydifferent methodology starts by describing a set of possible outcomes X that can be obtained inthe bargaining process between the two players and an alternative d that represent the outcomein the case of no agreement. Each of the bargainers has preference relation �i defined on X ∪{d}.Let the set of possible preference relations of i be denoted by Ri. A solution to the bargainingproblem is a function that associates an outcome to each profile of preferences. In other words,the solution is a function f such that

f : R1 ×R2 → X ∪ d.

55

CHAPTER 5. SEQUENTIAL BARGAINING 56

The method of analysis in this approach is often taken to be axiomatic. The analyst decidesin the abstract a set of criteria that a good solution concept should satisfy. An example of suchcriteria could be Pareto efficiency of the solutions, i.e. if x �i x′ and x �j x′, then x′ /∈ f (�i,�j) .

Once the criteria have been established, the analyst determines the set of functions f that satisfythis criterion. Preferably this set is small (ideally a singleton) so that some clear properties ofthe solutions can be established. This approach precedes the noncooperative approach to befollowed in this handout. The first contribution to this cooperative or axiomatic approach tobargaining was made by John Nash in “The Bargaining Problem” in Econometrica 1950. Variousother solutions were proposed for two player and many player bargaining models. A good surveyof these developments can be found in chapters 8-10 in Roger Myerson’s book “Game Theory:Analysis of Conflict”. Rubinstein’s solution of the noncooperative bargaining game picks thesame solution as Nash’s original cooperative solution for that game. A recurring theme in thenoncooperative bargaining literature has been the identification of fully strategic models thatyield the classical cooperative bargaining solutions as their equilibrium outcomes.

5.1 Bargaining with Complete Information

In this section, we focus our attention on the case where two players bargain over the divisionof 1 unit of surplus. (Sometimes more colorful language is used and it is said that the playersbargain over their shares of a pie that has size of 1). We start by considering the following simpleextensive form to the bargaining procedure. There are just two stages in the game. Player 1proposes a sharing rule for the surplus we assume that all real numbers between 0 and 1 representfeasible shares. Denote the shares going to 1 and 2 by x and 1−x respectively. Player 2 then seesthe proposal and decides whether to accept it or not. If the proposal is accepted, the outcome inthe game is a vector (x, 1− x) for the two players, if the offer is rejected, the outcome is (0, 0) . Itis easy to see that the only subgame perfect equilibrium in this simple game results in a proposalx = 1 which is accepted in equilibrium by player 2. In this case, we say that player 1 has all thebargaining power because she is in the position of making a take-it-or-leave-it offer to player 2and as a result, player 1 gets the entire surplus in the game.

Consider next the version of the game that extends over 2 periods. The first period is identicalto the one described in the paragraph above except that if player 2 refuses the first period offer,the game moves into the second period. The second period is identical to the first, but now player2 makes the offer and 1 decides whether to accept or not. At the end of the second stage, thepayoffs are realized (obviously if player 2 accepts the first offer, then the payoffs are realized inperiod 1). If the players are completely patient, i.e. their discount factor is 1, then it is clear thatthe subgame perfect equilibria in the game have all the same outcomes. Since player 2 can refusethe first period 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 could say that player 2 has allthe bargaining power since she is the one in a position to make the last take it or leave it offer.The situation changes somewhat if the players are less than perfectly patient. Assume now thatplayer i has a discount factor δi between the periods in the game. If player 2 refuses the first offerand gets all of the surplus in the second period, her payoff from this strategy is δ2. Hence shewill accept any first period offer x > δ2 and refuse any offer x < δ2. At x = δ2, she is indifferentbetween accepting and rejecting the offer, but it is again easily established that in the uniquesubgame perfect equilibrium of the game, she will accept the offer x = δ2 and the outcome of thegame is (1− δ2, δ2) . Notice that the introduction of time discounting lessens the advantage frombeing in the position to make the take it or leave it offer since the value of that eventual surplusis diminished.

If there were 3 periods in the game, the unique equilibrium in the subgame that starts followinga rejection of the first offer would be the solution of the 2 period game (by uniqueness of thesubgame perfect equilibrium in the 2 period game with discounting). Hence player 2 could securea payoff of 1− δ1 from the second period onwards. As a result, player 1 must offer to keep exactly1− 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 of the equilibrium in a game with T periods.

Proposition 56 Let xT denote the unique subgame perfect equilibrium share of player 1 in thebargaining 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, player 2 must offer a share ofδ1x

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

T)

to player 2. Hence player 1 must offer a shareδ2

(1− δ1x

T)

to player 2 in period T +2 proving the first claim. To see that in equilibrium, the firstperiod offer must be accepted, assume to the contrary and denote the equilibrium continuationpayoff vector by (v1, v2) after the refusal by player 2. Because of time discounting and feasibilityv1+v2 < 1. But then it is possible for 1 to offer x′ such that x′ > v1 and (1− x′) > v2 contradictingthe optimality of the original offer.

The proposition gives the equilibrium shares as a solution to a difference equation. The mostinteresting question about the shares is what happens as the 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 monotonically decreasing inδ and that (by using L’Hopitals rule) limδ→1 x∗ (δ) = 1

2 . Hence the unique outcome in finite butlong bargaining games with equally patient players is the equal division of surplus.


Notice that finite games of this type always put special emphasis on the effects of last stages.By the process of backwards induction, these effects have implications on the earlier stages in thegame as well. If the bargaining can effectively take as much time as necessary, it is questionableif it is appropriate to use a model which features endgame effects. The simplest way aroundthese effects is to assume that the bargaining process may take infinitely many periods. A simpleargument based on the continuity of the payoff functions establishes that making the offers fromthe finite game in all 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 are no other subgameperfect equilibria in the game. In order to establish this, we start by defining the strategies a bitmore carefully.

The analysis here is taken from Fudenberg and Tirole, Chapter 4. The infinite horizon alter-nating offer bargaining game is a game with perfect information, i.e. at all stages, a single playermoves and all of the previous moves are observed by all players. In odd periods, player 1 makesthe offer and player 2 either accepts (thereby ending the game) or rejects and the game moves tothe next period. In even periods, the roles are reverse. Denote by ht the vector of all past offersand acceptance decisions. A strategy for player i 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 the beginning of thesubgame indexed by ht, every strategy that assigns positive probability to at

i is strictly dominated.Iterated deletion of conditionally dominated strategies removes in each round all conditionallydominated strategies in all subgames given that the opponent’s strategies have survived previousdeletions. Observe that this process never deletes subgame perfect equilibrium strategies. Wewill show that the alternating offer bargaining game has a unique strategy profile that survivesiterated deletion of conditionally dominated strategies.

In what follows, we take each offer to be an offer for player 1’s share in the game. Observe firstthat it is conditionally dominated for one to refuse any offer from 2 that gives 1 a share exceedingδ1 and similarly 2 must accept any share for 1 that is below 1 − δ2. At the second round, 2 willnever offer more than δ1 and 2 will reject all offers above 1−δ2 (1− δ1) . Also 1 never offers below1− δ2 and never accepts offers below δ1 (1− δ2) .

Suppose then that after k iterations of the deletion process, 1 accepts all offers above xk and2 accepts all offers below yk where yk < xk. Then after one more round of deletions, 2 neveroffers more than xk and rejects all offers above 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) + δ1δ2xk. 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 never reached. The second possibility is that 2 accepts somefuture 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 this is at most δ21x

k. Thepayoff from the second possibility is the largest of the three. Hence 1 must accept all offers abovexk+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

and accepts 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 only in finitely many pieces,then any sharing of the surplus can be realized in a subgame perfect equilibrium as δ → 1. Again,the interpretation of this result relies on the view one takes on modeling. If it is thought that thefact that monetary sums can be divided only into pounds and pennies is important in the mind ofthe bargainers, then this result is probably troublesome. If on the other hand it is thought thatthis is not that important for the bargainers, then it is probably best to use a model where theindivisibility is not present.

A more troublesome observation is that the result is somewhat sensitive to the extensive formof the bargaining game. If the protocol of offers and rejections is modified, then other outcomescan be supported in equilibrium as well. Finally, the extensions to 3 or more player bargaininghave been unsuccessful. The uniqueness seems to be beyond the reach of the theory. Probablythe most successful models in this class have been the ones where a random proposer suggests asharing of the surplus in each period and the acceptance is decided by majority vote. Models alongthese lines have been developed by Baron and Ferejohn in the literature on political economy.

5.2 Dynamic Programming

How to extend the notion of backwards induction to an infinite time horizon. Consider first aschematic decision problem in a decision tree and observe what the process of backward reductiondoes to reduce the complexity of the decision problem. It represents and reduces subtrees through


values, where the values represent the maximal value one can attain in any subtree. When doesthis reduction lead to the optimal decision through the entire tree: if and only if we replace eachsubtree with the optimal value.

Theorem 57 Every finite decision tree (with finitely many periods and finitely may choices) hasan optimal solution which can be obtained by backwards induction.

We can extend this theorem certainly to an interactive decision problem, provided that itmaintains the structure of single person decision tree, namely perfect information.

Theorem 58 (Zermelo, Kuhn) Every finite game of perfect information ΓE has a pure strategysubgame perfect Nash equilibrium.

The theorem applies to many parlor games, in particular to chess. Thus in principle we canapply backward induction to solve for a subgame perfect Nash equilibrium in chess. How does acomputer program approach the problem of evaluating a move? By approximating the value ofa move through a forward looking procedure which assesses the value of the position in three orfour moves from then on. We may apply the same idea of approximation to every problem withan infinite horizon. In fact, we can obtain in many situations an exact approximation.

We now return to the bargaining problem as defined above. Two players, 1 and 2 bargain overa prize of value 1.. The first player starts by making an offer to the second player who can eitheraccept or reject the offer. If he rejects the offer, then in the next period he has the right to makean offer which can again be accepted or rejected. The process of making offers then starts 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 and the even more surprisingsimplicity of the argument is given first. We can adopt the definition of stationarity also forequilibrium strategies.

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

}I,∞i=1,t=0

is stationary ifsti = si, ∀i,∀t.

Theorem 60 The unique subgame perfect equilibrium to the alternate move bargaining game isgiven 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 ifv < δ/1− δ.

Notice that you have to specify the strategy for every period in the infinite time horizon model.The proof proceeds in two parts. First we show that a stationary equilibrium exists, and then weshow that it is the unique subgame perfect equilibrium, stationary or not. To do the later partof 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 shows that an equilibriumexists and that it is unique.

Proof. We start with the stationary equilibrium and use directly the payoffs and then lateron derive the strategies which support these payoff. Suppose then that there is a stationaryequilibrium. Then the following has to be true for the person who accepts today:

v = δ (1− v) ,

which leaves us withv =

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 2can expect in the subgame which begins after she refused the offer of player 1. By backwardsinduction the payoff of player 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 in the first period. Butwhat about making an offer that is rejected in the first period? As the second player gets at leastv in the next period, this cannot give the player more than δ (1− v) as he has to wait until thenext period. Hence

v̄ ≤ 1− δv.

Note that this implies thatv̄ ≤ 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. Let us denote this payoffby v∗. Since v∗ = 1− δv∗, we find that player 1 must earn

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 in period t is αkt. Thecapital 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 is given by

c (it) =12i2t .

The value of the investment problem can therefore be written as

V (k0) , max{it}∞t=0

{ ∞∑t=0

δt

(αkt −

12i2t

)}and likewise for all future periods

V (kτ ) , max{it}∞t=τ

{ ∞∑t=0

δ(t−τ)

(αkt −

12i2t

)}

Thus if we knew the values in the future say V (k1), we could certainly find the optimal policyfor period 0, as we could formulate the following problem

V (k0) , maxi0

{αk0 −

12i20 + 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 all t = 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− δ

However, it still remains to verify what the optimal solution is. But we can do this now byanalyzing the first order condition of the Bellman equation

δα− λk∗

1− δ− λk∗ = 0

Thus we get

k∗ =αδ

λ.

Chapter 6

Games of Incomplete Information

6.1 Introduction

This is a very modest introduction into the theory of games with incomplete information orBayesian games. We start by defining incomplete information, then suggest how to model incom-plete information (as a move by nature) to transform it into a game of imperfect information. Wethen introduce the associated equilibrium concept of Bayesian Nash equilibrium.

6.1.1 Example

Harsanyi’s insight is illustrated by the following example. Suppose payoffs of a two player twoaction game are either:

α β

α 1,1 0,0

β 0,1 1,0

orα β

α 1,0 0,1

β 0,0 1,1

i.e. either player II has dominant strategy to play H or a dominant strategy to play T . Supposethat II knows his own payoffs but player I thinks there is probability p that payoffs are given bythe first matrix, probability 1− p that they are given by the second matrix. Say that player II isof type 1 if payoffs are given by the first matrix, type 2 if payoffs are given by the second matrix.Clearly equilibrium must have: II plays H if type 1, T if type 2; I plays H if p > 1

2 , T if p < 12 .

But how to analyze this problem in general? All payoff uncertainty can be captured by a singlemove of “nature” at the beginning of the game.

64

CHAPTER 6. GAMES OF INCOMPLETE INFORMATION 65

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 toprobability distribution p (.). Each player i observes his own type ti and chooses an action ai.We also refer to ti as the private information of agent i. Now player i receives payoffs gi (a, t).By introducing fictional moves by nature, first suggested by Harsanyi (1967) we transform agame of incomplete information into a game of imperfect information. In the transformed gameone player’s incomplete information about the other player’s type becomes imperfect informationabout nature’s moves.

6.2.1 Strategy

A pure strategy is a mapping si : Ti → Ai. Write Si for the set of such strategies, and letS = S1 × ..× SI . A mixed strategy can also depend on his private information:

σi : Ti → ∆(Si).

6.2.2 Belief

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

i=1 there is someobjective distribution function

p (t1, ..., tI)

where ti ∈ Ti, which is called the prior distribution or prior belief. The conditional 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 types give one’s owntype ti. This expression assumes that Ti are finite. Similar expressions hold for infinite Ti. If theplayer’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 then he could calculatethe expected payoff from his decision given the conditional probability 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 game the player with themove does not know the complete history of the game thus far. Notice that with this specification,the only proper subgame of a game of incomplete information is the whole game. Any Nashequilibrium is then subgame perfect and the subgame perfection has no discriminating power.

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 equilibrium and the Nashequilibrium. In the former the payoff function may depend on the type. Moreover the payoff iscalculated as an expected payoff given his own information, with the conditional distribution overall other agent’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 the valuation of the objectis private information. Each buyer i = 1, 2 has a valuation vi ∼ U ([0, 1]) and submits a bid bi forthe object. The valuations are assumed to be statistically independent. If bidder i submits thehigher of 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)}. The optimal strategy isfound for each player by solving the following problem

maxbi

(vi − bi) Pr (bi > bj (vj)) +12

(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 thatPr (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 henceci = cj =

12,

and consequentlyai = aj = 0.

Hence the equilibrium strategy is to systematically underbid

bi (vi) =12vi.

6.5 Conditional Probability and Conditional Expectation

Before we analyze the next game, the double auction, recall some basic concepts in probabilitytheory, namely conditional probability and conditional expectation.


6.5.1 Disrete probabilities - finite events

We start with a finite number of events, and then consider a continuum of events Consider thefollowing example. Suppose a buyer is faced with a lottery of prices

0 < p1 < ... < pi < ... < pn < ∞

where each price pi has an unconditional probability 0 < αi < 1, and∑n

i=1 αi = 1, by which theprice pi is chosen. Suppose the buyer is willing to accept all prices, then the expected price to bepaid by the buyer is

E [p] =n∑

i=1

αipi, (6.7)

which is the unconditional expectation of the price to be paid. Suppose next, that the buyeradopts a rule to accepts all price higher or equal to pk and reject all other prices. (Recall this isjust an example.) Then the unconditional 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 entered in the formationof the expectation. Suppose we represent the acceptance rule of the buyer by the index k belowthe 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 different policies conditional on

accepting a price and paying it, in other words, we ask for the conditional expectation:

Ek [p |p ≥ pk ]

We consider now exclusively the events, i.e. realization of prices p, where p ≥ pk for some k. Theconditional probability of a price pi being quoted conditional 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 writemore 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 then price, is then nothingelse but taken the expectation with respect to the conditional probabilities rather than with theunconditional probabilities, so that

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 accepting the price. You maynow 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 of prices where theprices are distributed according to a distribution function.

6.5.2 Densities - continuum of events

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

p ∈[p, p̄

]where the distribution of prices is given by a (continuous) distribution function F (p) and anassociated density function f (p). Again suppose the buyer is willing to accept all prices, then theexpected price to be paid by the buyer is

E [p] =∫ p̄

ppf (p) dp =

∫ p̄

ppdF (p) (6.13)

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

Pr (A |B ) =Pr (A ∩ B)

Pr (B)(6.11)


which is the unconditional expectation of the price to be paid. Notice that the summation is nowreplaced with the integral operation. Suppose next, that the buyer adopts a rule to accept allprices higher or equal to p̂ and reject all other prices. (Recall this is just an example.) Then theunconditional expectation of the price to be paid is

Ep̂ [p] =∫ P̄

p̂pdF (p) (6.14)

Again, the expectation in (6.14) is still unconditional since no new information entered in theformation of the expectation and we represent the acceptance rule of the buyer by the index p̂

below the expectations operator E [·], as in Ep̂ [·]. The expected payment is then

Ep̂ [p] =∫ p̂

p0dF (p) +

∫ p̄

p̂pdF (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

ppdF (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 different policies conditional on

accepting a price and paying it, in other words, we ask for the conditional expectation:

Ep̂ [p |p ≥ p̂ ]

We consider now exclusively the events, i.e. realization of prices p, where p ≥ p̂ for some p̂. Noticethat we have now a continuum of events, each of which has probability zero, but all of which haveassociated densities. We then switch to densities. The conditional densities of a price p beingquoted conditional 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 this follows from

{p} ∩ {p ≥ p̂} = {p̂}

as long as p ≥ p̂, and otherwise the intersection is of course the empty set Notice that, as beforethe conditional densities integrate up to 1 since∫ p̂

p̂f (p |p ≥ p̂) dp =

∫ p̂

p̂

f (p)1− F (p̂)

dp =1

1− F (p̂)

∫ p̂

p̂f (p) dp = 1, (6.17)


where the last equality of course follows from2:∫ p̂

p̂f (p) dp = 1− F (p̂) .

Finally, the conditional expectation of the prices, conditioned on accepting then price, is thennothing else but taken the expectation with respect to the conditional probabilities rather thanwith the unconditional probabilities, so that

Ep̂ [p |p ≥ p̂ ] =∫ p̂

p̂pf (p |p ≥ p̂) dp =

∫ p̂

p̂p

f (p)1− F (p̂)

dp =1

1− F (p̂)

∫ p̂

p̂pf (p) dp.

Again you may verify that the expected price or average price paid, conditional on paying isincreasing 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 courselimp̂→p̄

Ep̂ [p] = 0. (6.19)

You may wish to verify (6.18) and (6.19) to realize the difference between expected value (price)and conditional expected value (price) in the second example.

6.6 Double Auction

6.6.1 Model

We present here another important trading situation where both sides of the market have privateinformation. The trading game is also called a double auction. We assume that the seller’svaluation is

vs ∼ F (vs) = U ([0, 1]) , (6.20)

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

vb ∼ F (vb) = U ([0, 1]) . (6.21)

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

2 (ps + pb) , if ps ≤ pb. There is no trade if ps > pb. If thereis no trade both, buyer and seller receive zero 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

2Recall the similarity between the second to last term in (6.17) to the term appearing when computing the

expected value of the bid ps (vs).


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 byp∗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 thatEp∗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 equilibrium through (6.23) and(6.24) is that the buyer has to find an optimal bid for every value vb while he is uncertainty aboutthe ask (p∗s) of the seller. From the buyer’s point of view, he is uncertain about p∗s as the ask,p∗s = p∗s (vs) will depend 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 the seller will adopt theequilibrium strategy: p∗s : Vs → [0, 1], but not knowing the realization of vs and hence not knowingthe 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 have to able to evaluatethe net utility of the buyer for every possible bid he could make, given a particular value vb heassigns to the object,

Ep∗s [ub (vb, pb, p∗s)] =

∫{p∗s |p∗s≤pb }

[vb −

12

(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 the equilibrium ask, p∗s,of the seller will depend on the realization of the seller’s value vs.Thus we can equivalently, theexpectation in (6.25) with respect to vs, or

Evs [ub (vb, pb, p∗s (vs))] =

∫{vs|p∗s(vs)≤pb }

[vb −

12

(pb + p∗s (vs))]

dF (vs) (6.26)

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

Evs [ub (vb, pb, p∗s (vs))] =

∫{vs|p∗s(vs)≤pb }

[vb −

12

(pb + p∗s (vs))]

dvs. (6.27)

However, as we don’t know the exact relationship between vs and p∗s, we can’t take the expectationyet. Similar to the auction example, we therefore make the following guess, which we shall verifyeventually about the relationship of 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 −

12

(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 −

12

(pb + as + csvs)]

dvs

Integrating yields [vbvs −

12

(pbvs + asvs +

12csv

2s

)] pb−ascs

0

orEvs [ub (vb, pb, p

∗s (vs))] =

pb − as

cs

(vb −

12

(pb + as +

12cs

pb − as

cs

))(6.30)


The expected utility for the buyer is now expressed as a function of vb, pb and the constants inthe strategy of the seller, but independent of p∗s or vs. Simplifiying, we get

ub (vb, pb, as, cs) =

probability of trade︷︸︸︷pb − as

cs×

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

12

(32pb +

12as

)), (6.31)

which essentially represents the trade-off for the buyer. A higher bid pb increases the probabilityof trade but decreases the revenue conditional on trade. The first order conditions are

∂ub (vb, pb, as, cs)∂pb

= vb −(

3pb + as

4

)− 3

4(pb − as) = 0

orp∗b (vb) =

2vb

3+

as

3. (6.32)

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

ps−abcb

(12

(ps + ab + cbvb)− vs

)dvb

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

us (vs, ps, ab, cb) =[12

(psvb + abvb +

12cbv

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︷︸︸︷14

(3ps + cb + ab − 4vs)

where the trade-off is now such that a higher ask leads to lower probability of trade, but a highernet utility conditional on trade. Taking the first order conditions, leads us to

∂us (vs, ps, ab, cb)∂ps

= 0

or equivalently

p∗s (vs) =13cb +

13ab +

23vs (6.33)

From the solution of the two pricing policies, we can now solve for ab and as as well as

13cb +

13ab = as

as

3= ab


and henceas =

14, ab =

112

and thus the complete resolution is

p∗b (vb) =112

+23vb

andp∗s (vs) =

14

+23vs

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

p∗b (vb) ≥ p∗s (vs) ⇔112

+23vb ≥

14

+23vs

⇔ vb − vs ≥14

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 ≥ x

0 otherwiseps =

{x if vs ≤ x

1 otherwise.

Readings. Chapter 3 in [4], [3] Chapter 6, [8], Chapter 8.E.

Chapter 7

Adverse selection (with two types)

Often also called self-selection, or “screening”. In insurance economics, if a insurance companyoffers a tariff tailored to the average population, the tariff will only be accepted by those withhigher than average risk.

7.1 Monopolistic Price Discrimination

A simple model of wine merchant and wine buyer, who could either have a coarse or a sophisticatedtaste, which is unobservable to the merchant. What qualities should the merchant offer and atwhat 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 the transfer (price)buyer i has to pay for the quality qi. The taste parameters θi 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 pay is given by

p = Pr (θi = θh)

We also observe that the difference in utility for the high and low valuation buyer for any givenquality q

u (θh, q)− u (θl, q)

77

CHAPTER 7. ADVERSE SELECTION (WITH TWO TYPES) 78

is increasing in q. (This is know as the Spence-Mirrlees sorting condition.). If the taste parameterθi were a continuous variable, the sorting condition could be written in terms of the second crossderivative:

∂2u (θ, q)∂θ∂q

> 0,

which states that taste θ and quality q are complements. The profit for the seller 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 types have different prefer-ences, they should consume different qualities. The social surplus for each type can be maximizedseparately 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 monopolist can perfectly dis-criminate between the types (first degree price discrimination) or if there is perfect competition.The two outcomes differ only in terms of the distribution of the social surplus. With a perfectlydiscriminating monopolist, 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 setprices 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 verified immediately that perfectdiscrimination 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 constraint are IRl and ICh,whereas the remaining constraints are not binding. We then solve for tl and th, which in turnallows us to solve for qh, and leaves us with an unconstrained problem for ql.Thus we want toshow

(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 constraintsand 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 thatq̂h = 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 constant terms and we cansimplify 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 (with the possibility ofcomplete 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 seller to depress thequality supply to the low end of the market.

The material is also covered in [14], 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 ∈ suppσ∗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 write that:

σ∗−i (s−i|t−i) =∏j 6=i

σ∗j (sj |tj) .

8.2 Sender-Receiver Games

8.2.1 Model

A signaling games is a dynamic game of incomplete information involving two players: a sender (S)and receiver (R). The sender is the informed party, the receiver the uniformed party. (Describe

81

CHAPTER 8. THEORETICAL COMPLEMENTS 82

the game in its extensive form). The timing is as follows:

1. nature draws a type ti ∈ T = {t1, ..., tI} according to a probability distribution 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 have a belief about whichnode in the information set has been reached by the play of the game. For a non singletoninformation set, a belief is a probability distribution over the nodes in the information set; for asingleton information set, the player’s belief puts probability one on the single decision node.

Condition 68 Given their beliefs, the players’ strategies must be sequentially rational. That is,at each information set the action taken by the player with the move (and the player’s subsequentstrategy) must be optimal given the player’s belief at that information set and the other players’subsequent strategies (where a “subsequent strategy” is a complete plan of action covering everycontingency that might arise after the given information set has been reached).

Definition 69 For a given equilibrium in a given extensive-form game, an information set is onthe equilibrium path if it will be reached with positive probability if the game is played accordingto the equilibrium strategies, and is off the equilibrium path if it is certain not to be reached ifthe game is played according 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 determined by Bayes’ ruleand the players’ equilibrium strategies.

Condition 71 At information sets off the equilibrium path beliefs are determined by Bayes’ ruleand the players’ equilibrium strategies where possible.

Definition 72 A perfect Bayesian equilibrium consists of strategies and beliefs satisfying Require-ments 1 through 4.


8.3.2 Formal Definition

Condition 73 After observing any message mj from M , the Receiver must have a belief aboutwhich types could have sent mj . Denote this belief by the probability distribution µ (ti |mj ), whereµ (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 maximize the Receiver’s expectedutility, given the belief µ (ti |mj ) about which 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’s utility, given theReceiver’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’sbelief at the information set corresponding to mj must follow from Bayes’ rule and the Sender’sstrategy:

µ (ti |mj ) =p (ti)∑

ti∈T

p (ti). (8.4)

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

Definition 76 A pure-strategy perfect Bayesian equilibrium in a signaling game is a pair ofstrategies m∗ (ti) and a∗ (mj) and a belief µ (ti |mj ) satisfying signaling requirements (73)-(75).

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

Definition 77 (PBE) A pure strategy Perfect Bayesian Equilibrium is a set 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 maxuS (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 the payoffs in [4] have aproblem, they don’t allow for a separating equilibrium, modify)

Definition 80 (Equilibrium Domination) Given a PBE, the message mj 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 equilibriumpath and mj equilibrium dominated for type ti, then

µ (ti |mj ) = 0.

This is possible provided that mj is not equilibrium dominated for all types in T .Readings: [8] Chapter 9.C, [3] Chapter 8.

Bibliography

[1] D. Abreu. Extremal equilibria of oligopolistic supergames. Journal of Economic Theory,39:191–225, 1986.

[2] K. Binmore. Fun and Games. D.C. Heath, Lexington, 1992.

[3] D. Fudenberg and J. Tirole. Game Theory. MIT Press, Cambridge, 1991.

[4] R. Gibbons. Game Theory for Applied Economists. Princeton University Press, Princeton,1992.

[5] D. M. Kreps. A Course in Microeconomic Theory. Princeton University Press, Princeton,1990.

[6] D.M. Kreps and R. Wilson. Sequential equilibria. Econometrica, 50:863–894, 1982.

[7] H. Kuhn. Extensive games and the problem of information. In H. Kuhn and A. Tucker,editors, Contributions to the Theory of Games, pages 193–216. Princeton University Press,Princeton, 1953.

[8] A. Mas-Collel, M.D. Whinston, and J.R. Green. Microeconomic Theory. Oxford UniversityPress, Oxford, 1995.

[9] J. Nash. Equilibrium points in n-person games. Proceedings of the National Academy ofSciences, 36:48–49, 1950.

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

[11] P. Reny. On t he existence of pure and mixed strategy nash equilibria in discontinuous games.Econometrica, 67:1029–1056, 1999.

[12] S.M. Ross. Probability Models. Academic Press, San Diego, 1993.

[13] A. Rubinstein. Perfect equilibrium in a bargaining model. Econometrica, 50:97–109, 1982.

[14] B. Salanie. The Economics of Contracts. MIT Press, Cambridge, 1997.

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