3. Game Theory - Economic Theory (Prof. Schmidt) - LMU

3. Game Theory 

Klaus M. Schmidt 

LMU Munich 

Micro (Research), Winter 2011/12 

Klaus M. Schmidt (LMU Munich) 3. Game Theory Micro (Research), Winter 2011/12 1 / 168

Introduction 

Game theory considers interpersonal or interactive decision making: Two 

or more “players” interact with each other and the outcome depends on all of 

their decisions. 

When a player chooses his action he must form beliefs about what his 

opponents are going to do and how his decision may affect the decision of the 

other players. This is called strategic interaction. 

Discussion: Is there strategic interaction 

on perfectly competitive markets? 

on oligopolistic markets? 

on a monopolistic market? 

c○ 2011 Klaus M. Schmidt 


Other Applications of Game Theory 

Bargaining: buyer and seller, unions and employers, political parties, etc. 

Auctions: Optimal strategies in auctions, optimal design of auctions. 

Cooperation in small groups: Provision of public goods, coordination, etc. 

Conflicts: strategic interaction in strikes, in war, in tariff wars, etc. 

Behavior within and between organisations: Motivation of workers, 

control of managers, etc. 

Difference between Game Theory and Contract Theory 

Game theory takes the strategic environment as exogenously given. 

Contract theory (mechanism design, implementation theory) designs 

strategic environment to induce desired behavior. 


Historical Notes 

Game Theory started out as a branch of mathematics (Zermelo 1913, 

Borel 1920s) 

Analysis of parlor games: Chess, Poker 

Focus on zero-sum games 

John von Neumann and Oskar Morgenstern (1944), “The Theory of 

Games and Economic Behavior” 

Cooperative vs. non-cooperative game theory 

John Nash (1951): Nash equilibrium 

Reinhard Selten (1967): Subgame perfect equilibrium 

John Harsanyi (1969): Bayesian Nash equilibrium 

since 1980s widespread applications in all fields of economics 

since 1994 many Nobel prizes for game theorists 


Interpretation of Game Theory 

Traditional game theory assumes that all players are perfectly rational and 

able to compute and coordinate on an equilibrium of the game. This 

assumption is much stronger than in theories of individual decision making. 

Normative Interpretation: Game theory asks what a perfectly rational player 

should do. 

What does rationality mean when players interact strategically? 

What are rational expectations? 

What is an equilibrium? 


Positive Interpretation: Game theory tries to explain and to predict behavior 

of real people. 

Real people are only boundedly rational, i.e. they try to be rational but fail 

occasionally. 

If people experience a certain strategic situation repeatedly, they 

eventually learn to behave rationally and to play the equilibrium of the 

game. 

Deviations from rational behavior are random and cancel out in the 

aggregate. 

Experimental economics studies how real people actually behave. 

Behavioral economics tries to model this behavior. 

The reference point is always the prediction of traditional game theory. 


Literature: 

Most of this chapter is based on: 

MWG, Chapters 7 - 9 

This is required reading! 

In addition, it may be useful to consider another textbook covering this 

material, e.g. 

Kreps (1990) Chapters 11-14 

Gibbons (1992) Chapters 1-4 

Fudenberg and Tirole (1991), Chapters 1-3, 6, 8. 


3.1 Static Games of Complete Information 

We start out with games where all players simultaneously have to take a 

decision and all players know the structure of the game. To analyze such a 

game we have to know: 

1) The set of players 

I = {1, 2, . . . , n} 

2) The set of strategies each player can choose from 

Si, i ∈ I, with si ∈ Si 

A profile of strategies for all players is a vector 

Sometimes we will write: 

s = (s1, s2, . . . , sn) ∈ × n i=1Si . 

s = (si, s−i) ∈ (Si, S−i) , 

where s−i is the vector of strategies of all other players (except player i) 


3) The payoff function of each player 

ui : S1 × S2 × · · · × Sn → R , 

mapping all possible strategy profiles into payoffs for all players. 

Definition 3.1 (Normal Form) 

The normal form of a game G = [I; {Si}; {ui}] consists of 

1) the set of players, I = {1, . . . , n}, 

2) the strategy sets of all players {Si} = {S1, . . . , Sn}, 

3) the payoff functions of all players {ui} = {u1, . . . , un}. 


The normal form of a two-player game can be expressed and analyzed in a 

bimatrix: 

2 

❅ 

❅ 

1 

❅ 

C 

D 

C D 

4, 4 0, 5 

5, 0 1, 1 

FIG. 3.1: Normal Form of the Prisoner’s Dilemma Game in a Bimatrix 


Remarks: 

What is meant by “simultaneous move”? 

Payoffs are assumed to be von Neumann-Morgenstern utilities. What 

does this imply if there is uncertainty? 

Keep in mind that in the real world payoffs are often monetary payoffs. 

What does it imply to treat monetary payoffs as von 

Neumann-Morgenstern utilities? 

We assume not only that all players know the structure of the game. We 

assume in addition that the structure of the game is “common 

knowledge”. What does this mean? Why is this important? 


Dominance 

Definition 3.2 (Strictly Dominated Strategy) 

A strategy ˆsi of player i is strictly dominated if there exists another strategy 

˜si ∈ Si such that ˜si yields a strictly greater payoff than ˆsi no matter what 

strategies are chosen by his opponents: 

ui(ˆsi, s−i) < ui(˜si, s−i) ∀s−i ∈ S−i . 

Are there any beliefs about what the other players will do under which a 

rational player would ever choose a strictly dominated strategy? 


Definition 3.3 (Dominant Strategy) 

Strategy s∗ i of player i is a dominant strategy if it strictly dominates all other 

strategies of player i, i.e. a dominant strategy is a strictly best response 

against all other strategy vectors s−i: 

ui(s ∗ i , s−i) > ui(si, s−i) ∀si ∈ Si \ {s ∗ i }, s−i ∈ S−i. 

Are there any beliefs about what the other players will do under which a 

rational player will not play a dominant strategy? 


Proposition 3.1 (Equilibrium in Dominant Strategies) 

If in G = [I, {Si}; {ui}] each player i has a dominant strategy, and if each 

player is rational, then the unique equilibrium is that all players use their 

dominant strategies. 

Remarks: 

How to prove this? 

Does this proposition require the assumption of common knowledge? 

What does this proposition imply for the prisoner’s dilemma game? 

What would you conclude from the observation that many real people do 

not play the dominant strategy in the prisoner’s dilemma game? 


Mixed Strategies 

So far we considered only pure strategies. However, a player can also 

randomize over his available strategies and play a mixed strategy. 

Definition 3.4 (Mixed Strategy) 

Consider a normal form game G with finite strategy sets Si = {si1, . . . , siKi }. A 

mixed strategy for player i is a probability distribution σi = (σi1, . . . , σiKi ) over 

Si, with 0 ≤ σik ≤ 1 for k = 1, . . . , Ki and σi1 + . . . + σiKi = 1. 

Remarks: 

1) A pure strategy can be seen as the extreme case of a mixed strategy: All 

probability mass is put on one strategy. 

2) The support of a mixed strategy is the set of all actions that are chosen 

with strictly positive probability. 

3) The set of all mixed strategies of player i is denoted by ∆(Si). 

4) A mixed strategy σi is a best response to the (possibly mixed) strategy 

profile σ−i of all other players if and only if each pure strategy in the 

support of σi is a best response to σ−i. Why? 


4) A mixed strategy σi can strictly dominate a pure strategy si even if none 

of the pure strategies in the support of σi dominate si. Example: 

2 

❅ 

❅ 

1 

❅ 

U 

M 

D 

ℓ r 

3, - 0, - 

0, - 3, - 

1, - 1, - 

FIG. 3.2: A Mixed Strategy Strictly Dominates a Pure Strategy 

D is neither dominated by U nor by M, but the mixed strategy ( 1 1 

2 , 2 , 0) 

strictly dominates D! 


Best Response 

Definition 3.5 (Best Response) 

In game G = [I, {∆(Si)}; {ui}] 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 ∈ ∆(Si). Strategy σi is never a best response if there is no σ−i for 

which σi is a best response. 

Remarks: 

1. A “best response” is often called a “best reply”. This is the same thing. 

2. If a rational player believes that his opponents play σ−i he wants to play a 

best response against σ−i. 

3. A pure strategy that is not a best response against any pure strategy may 

still be a best response against a mixed strategy. Example: 


2 

❅ 

❅ 

1 

❅ 

U 

M 

D 

L R 

3, - 0, - 

0, - 3, - 

2, - 2, - 

FIG. 3.3: Best Response Against a Mixed Strategy 

D is not a best response against L and not a best response against R. 

However, if player 2 chooses L with probability 1 

1 

2 and R with probability 2 , 

then D is a best response of player 1. 


Iterated Elimination of Strictly Dominated Strategies 

Which strategies can be eliminated in the following game? 

2 

❅ 

❅ 

1 

❅ 

U 

D 

L M R 

1, 0 1, 2 0, 1 

0, 3 0, 1 2, 0 

FIG. 3.4: Iterated Elimination of Strictly Dominated Strategies 


Remarks: 

IESDS requires not only that players are rational, but also that rationality 

is common knowledge. Why? 

The sequence of elimination of strictly dominated strategies does not 

affect the result. 

You may also eliminate strategies that are strictly dominated by a mixed 

strategy. 

But: You are not allowed to eliminate weakly dominated strategies! 

In some games IESDS yields a unique equilibrium outcome, e.g. in the 

Cournot game or the Beauty Contest. 


Rationalizability 

If a rational player believes that his opponents choose strategy profile σ−i then 

he should choose a strategy σi that is a best response to σ−i. 

This implies that if σi is never a best response, i.e., if there is no strategy σ−i 

such that σi is a best response to it, then a rational player should not choose 

σi. We say that σi is not rationalizable. 

Strategies that are not rationalizable can be eliminated, because all players 

know that these strategies will not be used. ⇒ Iterated elimination of 

strategies that are never best responses. 

Definition 3.6 (Rationalizability) 

In game G = [I, {∆(Si)}; {ui}] the strategies in ∆(Si) that survive the iterated 

elimination of strategies that are never best responses are known as player i’s 

rationalizable strategies. 


Remarks: 

1. Every strategy that is strictly dominated is not rationalizable. 

2. The set of rationalizable strategies must be a subset of the set of 

strategies that survive IESDS. Why? 

3. It turns out that in two player games the set of strategies that survive 

IESDS and the set of rationalizable strategies coincide. 

4. However, in games with more than two players there can be strategies 

that are never a best response and yet are not strictly dominated. 

5. If a strategy σi is “rationalizable” then player i can tell a coherent story 

why he uses σi. Example: 


2 

1 ❅ 

❅ 

a1 

a2 

a3 

a4 

b1 b2 b3 b4 

0, 7 2, 5 7, 0 0, 1 

5, 2 3, 3 5, 2 0, 1 

7, 0 2, 5 0, 7 0, 1 

0, 0 0, -2 0, 0 10, -3 

FIG. 3.5: Rationalizable Strategies 


4 and a4 can be eliminated. 

All other strategies are rationalizable. Consider e.g. a1. Player 1 can 

rationalize a1 by pointing to the following chain of arguments: 

(a1, b3, a3, b1, a1, b3, a3, b1, . . .). 

How can player 2 rationalize b2? 

Is it possible to rationalize a4? 

Suppose B’s payoff at (a4, b4) is not −3 but +1. Is it now possible to 

rationalize a4? 


Nash Equilibrium 

In many games there are many rationalizable strategies. Which strategies 

should we expect to be played? 

John Nash’s answer to this question is that strategies should form an 

equilibrium in the sense that each player plays a best response to the 

strategies chosen by all other players, so nobody has an incentive to deviate 

from this strategy profile. 

Definition 3.7 (Nash Equilibrium) 

A strategy profile s = (s1, . . . , sI) is a Nash Equilibrium (NE) of game 

G = [I, {Si}, {ui(·)}] if for every i ∈ {1, . . . , I} 

for all s ′ i 

∈ Si. 

ui(si, s−i) ≥ ui(s ′ i , s−i) 


Remarks: 

1. Nash Equilibrium requires that the strategy of each player is a best 

response against the actual strategies chosen by all other players. In 

contrast, rationalizability only requires that a strategy is a best response 

to a profile of rationalizable strategies of his opponents. 

2. Thus, Nash Equlibrium requires not only that players play a best 

response against the strategies of their opponents. It also requires that 

players form correct expectations about the strategies that the other 

players are going to choose. It is the second requirement that is 

particularly demanding. 

3. We only require a weak inequality. If a strict inequality holds we have a 

“Strict Nash Equilibrium”. However, in many games a Strict Nash 

Equilibrium does not exist. 

4. Nash Equilibrium strategies must be rationalizable, they cannot be strictly 

dominated, but they may be weakly dominated. Why? 

5. We only require that unilateral deviations from the proposed strategy 

profile are not profitable. We do not require that a coordinated deviation 

of several players is unprofitable. This is why this is called 

“non-cooperative” game theory. 


How to find a Nash Equilibrium 

In a two-player game with a finite number of strategies it is easy to find all 

Nash equilibria: 

Consider each strategy of player 2 and mark the best responses of player 

1 by underlining his highest payoffs in column. 

Consider each strategy of player 1 and mark the best responses of player 

2 by underlining her highest payoffs in each row. 

Those cells in which both payoffs are underlined constitute a pair of 

mutually best responses, i.e. a Nash Equilibrium. 

With more than two players you have to systematically go through all 

possibilities. If the game is symmetric, try to exploit this in order to reduce the 

number of cases. 

If strategy spaces are continuous, look at the first order conditions in order to 

find “best response correspondences”. Don’t forget to check the second order 

conditions. 


2 

1 ❅ 

❅ 

a1 

a2 

a3 

a4 

b1 b2 b3 b4 

0, 7 2, 5 7, 0 0, 1 

5, 2 3, 3 5, 2 0, 1 

7, 0 2, 5 0, 7 0, 1 

0, 0 0, -2 0, 0 10, -3 

FIG. 3.6: Nash Equilibrium 


2 

❅ 

❅ Marienplatz University 

1 

❅ 

Marienplatz 

University 

10, 10 0, 0 

0, 0 10, 10 

FIG. 3.7: Meet you in Munich 


Why should we expect real players to play a Nash Equilibrium? 

1. NE as an implication of rationality and common knowledge of 

rationality 

This argument is not correct! A rational player must play a rationalizable 

strategy, but rationality does not imply that his forecast of what other 

players are going to do is correct. 

2. NE as an implication of consistency 

If rationality is common knowledge, each player knows that each player 

will play a best response to what he beliefs what the other players will do. 

Only NE are strategy combinations at which all players play best 

responses to their beliefs and all beliefs are correct. If any other strategy 

combination was played, then at least one player must make a mistake by 

either choosing the wrong strategy given his belief or by having a belief 

that is mistaken. Thus, if there is a unique NE, then this is the only 

consistent way how the game can be played. But if there are multiple NE, 

players somehow have to coordinate on which equilibrium to play. 


3. NE as a self-enforcing agreement 

Suppose that players can talk about which strategies to play before the 

game starts. Would they ever agree to play a strategy profile that is not a 

NE? If there are multiple NE and players agree to play one of them, then 

nobody has an incentive to deviate from this agreement. Agreeing to play 

a NE is a self-enforcing agreement. 

4. NE as a focal point 

Even if there are multiple equilibria it is sometimes “obvious” which 

equilibrium will be played. Reasons for “obviousness” can be efficiency, 

symmetry, or a joint cultural background. See Schelling (1960) for many 

fun examples. 

5. NE as a stable social norm or convention 

There are many social conventions or norms that select one out of many 

possible NE, e.g. “Drive/Walk on the right side of the street/sidewalk”, 

“Ladies first”, “share equally”, etc. 


6. NE as a stable outcome of a learning or an evolutionary process 

If players play a game very often they will eventually learn to play a Nash 

Equilibrium. Similarly, evolution will shape “strategies” of plants and 

animals such that they are best responses to each other. To make these 

arguments precise we have to specify the learning/evolutionary process 

explicitly. But many of these processes converge to NE. 

Whether players play a Nash equilibriium is an empirical question. The 

experimental evidence suggests that real players are more likely to play a 

Nash Equilibrium the more experience they have with the game (Discussion: 

beauty contest). Furthermore, it helps if the Nash equilibrium is unique or if 

there is one NE that is the obvious focal point. 

If you play a game against somebody who is inexperienced and whom you 

expect not to play his NE stragegy, then you should typically not choose your 

NE stragegy either! 


Proposition 3.2 

If the strategy profile (s∗ 1 , . . . , s∗ n) is a NE, then all s∗ i , i = 1, . . . , n, must 

survive the iterated elimination of strictly dominated strategies. 

Proof:(by contradiction) Suppose that some s∗ i do not survive IESDS. Let j be 

the first player whose equilibrium strategy s∗ i is going to be eliminated. Then 

there exists a strategy ˆsj such that 

uj(s ∗ j , s−j) < uj(ˆsj, s−j) 

for all s−j that have not yet been eliminated. In particular we must have 

uj(s ∗ j , s ∗ −j) < uj(ˆsj, s ∗ −j). 

But this is a contradiction to the assumption that (s ∗ 1 , . . . , s∗ n) is a NE. Q.E.D. 



If G is a finite game and IESDS yields a unique strategy profile (s ∗ 1 , . . . , s∗ n) 

then this strategy profile is the unique NE of the game. 

Proof: Suppose that (s ∗ 1 , . . . , s∗ n) ist not a NE. Then there exists a player i and 

a strategy ˆsi such that 

ui(s ∗ i , s ∗ −i) < ui(ˆsi, s ∗ −i). 

This ˆsi has been eliminated in the process of IESDS. Thus, there exists an s ′ i 

such that ui(ˆsi, s ∗ −i ) < ui(s ′ i , s∗ −i ). 

Either s ′ i = s∗ i , then we have found a contradiction already. 

Or s ′ i = s∗ i , but then there exists a strategy s′′ 

i that dominates s ′ i on some 

earlier stage of IESDS. 

◮ Either s ′′ 

i = s ∗ i ⇒ contradiction 

◮ Or s ′′ 

i = s ∗ i ⇒ s ′′′ 

i 

Because the set of strategies is finite we must eventually get at a 

contradiction. 

Uniqueness is implied by Proposition 3.2. Why? Q.E.D. 


Mixed Strategy Equlibria 

2 

❅ 

❅ 

1 

❅ 

Heads 

Tails 

What is the NE of this game? 

Heads Tails 

1, -1 -1, 1 

-1, 1 1, -1 

FIG. 3.8: Matching Pennies 


Definition 3.8 (Mixed Strategy Equilibrium) 

A mixed strategy profile σ = (σ1, . . . , σI) is a NE of game 

G = [I, {∆(Si)}, {ui(·)}] if for every i = 1, . . . , I, 

for all σ ′ i ∈ ∆(Si). 

ui(σi, σ−i) ≥ ui(σ ′ i , σ−i) 

Remark: If a player chooses a mixed strategy σi in equilibrium, then all pure 

strategies si in the support of σi must yield the same expected payoff which is 

greater or equal than the expected payoff of any strategy that is not in the 

support of σi. Why? 


In Matching Pennies the unique NE is that both players choose “Heads” with 

probability 1/2. 


Proof of Proposition 3.4: 

Let p ∈ [0, 1] denote the probability with which player 1 chooses “Heads” and 

q ∈ [0, 1] the probability that player 2 chooses “Heads”. Suppose player 1 

chooses “Heads”. Then his expected payoff is: 

u1(H) = q · 1 − (1 − q) · 1 = 2q − 1 

Suppose player 1 chooses “Tails”. Then his expected payoff is: 

⇒ Player 1 should choose “Heads” iff 

u1(T ) = −q · 1 + (1 − q) · 1 = 1 − 2q 

2q − 1 ≥ 1 − 2q ⇔ q ≥ 1 

2 

If q = 1 

2 player 1 is just indifferent between his two strategies. 

Similarly we get for player 2: She should choose “Heads” iff p ≤ 1 

2 and she is 

just indifferent between “Heads” and “Tails” if p = 1 

2 . Q.E.D. 


q 

.... .. .. 

.. . 

.. 

FIG. 3.9: Best Response Correspondences for Matching Pennies 

There is a unique intersection at p = q = 1 

2 . This is a pair of mutual best 

responses. 

Klaus M. Schmidt (LMU Munich) 3. Game Theory Micro (Research), Winter 2011/12 38 / 168 

p

Finding all Nash Equilibria 

2 

❅1 

❅ 

Bo 

Ba 

Bo Ba 

X1, X2 

0, 0 

0, 0 Y1, Y2 

p 

.. . 

.. .. .. 

q 

FIG. 3.10: All NE in the “Battle of the Sexes” 


Let p denote the probability that player 1 chooses “Boxing” and q the 

probability that player 2 chooses “Ballet”. 

Similarly: 

u1(Bo) = q · X1 + (1 − q) · 0 = qX1 

u1(Ba) = q · 0 + (1 − q) · Y1 = (1 − q)Y1 

u1(Bo) ≥ u1(Ba) ⇔ qX1 ≥ (1 − q)Y1 ⇔ q ≥ 

u2(Bo) ≥ u2(Ba) ⇔ pX2 ≥ (1 − p)Y2 ⇔ p ≥ 

Hence, there are three NE: 

two pure strategy equlibria: (0,0) and (1,1) 

one mixed strategy equilibrium: 

Y2 

X2+Y2 , 

Y1 

X1+Y1 

 

. 

Y1 

X1 + Y1 

Y2 

X2 + Y2 


Remarks: 

1) The two pure strategy equilibria are also mixed strategy equilibria with 

degenerate probabilities. 

2) In the mixed strategy equilibrium player 1 chooses “Boxing” with 

probability p∗ = Y2 and player 2 chooses “Boxing’ with probability 

X2+Y2 

q∗ = Y1 

X1+Y1 . Note that p∗ is independent of the payoffs X1, Y1 of player 1. 

If X1 increases, player 1 will not go to “Boxing” with a higher probability in 

equilibrium. Why not? 

p ∗ depends only on the payoffs X2, Y2 of player 2. Each player chooses 

his probabilities so as to keep the other player indifferent! 

3) It is possible to show that if generic games have a finite number of 

equilibria, then this number must be odd! Thus, if you find an even 

number of NE you must be missing at least one! 


Interpretation of Mixed Strategy Equilibria 

In a mixed strategy equilibrium a player is indifferent between all pure 

strategies in the support of his mixed equilibrium strategy. The probabilities 

are chosen so as to keep the other player indifferent. Why should a player 

take the trouble and do this? 

1) Experienced players do randomize properly (e.g. in soccer or baseball). 

2) If a player does not randomize properly, his opponent will eventually 

exploit him (if there is repeated interaction). 

3) Play “Stone, Paper, Scissors” with a friend many times in a row. Figure 

out by introspection whether you randomize. 

4) Harsanyi offered an interesting interpretation of a mixed strategy 

equilibrium. He assumes that there is incomplete information about the 

payoff function of each player. Each player chooses a pure strategy that 

depends on his “type”. The mixed strategy equilibrium is the limit of a 

sequence of pure strategy equilibria with incomplete information when 

the incompleteness of information goes to 0. We will discuss this in more 

detail when we get to games with incomplete information. 


Existence of Nash equilibria 

John Nash (1951) has shown that a Nash equilibrium exists under fairly weak 

conditions. 

Proposition 3.5 (Existence I) 

Every normal form game with a finite number of players and finite strategy 

spaces has at least one NE, possibly in mixed strategies. 

Proposition 3.6 (Existence II) 

Consider a normal form game G = [I, {Si}, {ui(·)}] with strategy sets Si that 

are non-empty, compact and subsets of some Euclidean space R M . 

(a) If ui(·) is continuous in s for all i = 1, . . . , I then a NE in pure or mixed 

strategies exists. 

(b) If in addition strategy spaces Si are convex and payoff functions ui(·) are 

quasiconcave in s for all i = 1, . . . then there exists a pure strategy NE. 


Remarks: 

1) Nash equilibria exist for a large class of games. 

2) Even if payoff functions are discontinuous there do exist mixed strategy 

NE in many cases (see Dasgubpta and Maskin, 1986). 

3) Existence is important: 

a) If we don’t know whether a NE exists, it does not make sense to characterize 

the properties of all NE. 

b) If NE did not exist in naturally defined games, then something was wrong 

with our notion of rationality and equilibrium. How should a rational individual 

behave, if there is no consistent way how to do so?W 


Outline of the proof of Proposition 3.5: 

The proof uses a fixed point argument. To illustrate this type of argument 

consider Brower’s fixed point theorem first: 

Proposition 3.7 (Brouwer) 

Let A ⊂ R N be a non empty, compact and convex set and f : A → A be a 

continuous function mapping A into A. Then f has a fixed point, i.e. there 

exists an x such that f (x) = x. 


.. 

.. 

.. 

 

.. 

. 

 

 

 

 

 

 

 

 

 

 

1 

.. 

... 

.. 

. 

0 

1 

 

 

 

 

 

 

 

 

 

1 

0 

1 

FIG. 3.11: Brouwer’s Fixed Point Theorem 


Definition 3.9 (Upper Hemicontinuity) 

Let A ⊂ R N and the compact set Y ⊂ R K . A correspondence f : A → Y is 

upper hemicontinuous (uhc) if it has a closed graph. 

Proposition 3.8 (Kakutani) 

Supose that A ⊂ R N is a nonempty, compact and convex set and f : A → A is 

an upper hemicontinuous correspondence mapping A to A with the property 

that the set f (x) ⊂ A is nonempty and convex for every x ∈ A ist. Then f (·) 

has a fixed point, i.e., there exists an x ∈ A such that x ∈ f (x). 


.. 

.. 

.. 

 

.. 

. 

 

 

 

 

 

 

 

 

 

 

1 

.. 

... 

.. 

. 

0 

1 

 

 

 

 

 

 

 

 

 

1 

0 

1 

FIG. 3.12: Kakutani’s Fixed Point Theorem 


We now outline the proof of Proposition 3.5. 

Let σi be a mixed strategy of player i and Σi the set of all mixed strategies. 

Note that Σi is a nonempty, compact and convex subset of R K , where K is the 

number of pure strategies of player i. 

Define the best response correspondence 

for player i: 

Bi : Σ−i → Σi 

Bi(σ−i) = arg max ui(σi, σ−i). 

σi 

It is possible to show that this correspondence is uhc and that Bi(σ−i) is 

nonempty and convex for all σ−i ∈ Σ−i. 


Consider now the correspondence B : Σ → Σ defined by: 

B(σ1, . . . , σn) = B1(σ−1) × · · · × Bn(σ−n) 

This correspondence assigns to every possible strategy profile the best 

responses to this strategy profile. A fixed point of this correspondence is a 

strategy profile with the property that every strategy in this profile is a best 

response to all other strategies in this profile, i.e., it is a Nash Equilibrium. 

Thus, we only have to show that the correspondence B has a fixed point. We 

can do this by using Kakutani’s fixed point theorem. The conditions of 

Kakutani are satisfied 

⇒ a fixed point exists 

⇒ a Nash Equilibrium exists. Q.E.D. 


3.2 Dynamic Games of of Complete Information 

So far we looked at games where all players choose their strategies 

simultaneously and act only once. These games are best described by their 

normal form. 

Now we want to look at games that have a more complicated time and 

information structure. In a dynamic game players may act sequentially, they 

may act several times, and their information may depend on what has 

happened in the past. In order to capture this, we have to introduce the 

extensive form of a game. 


Definition 3.10 (Extensive Form) 

The extensive form of a game describes: 

(1) the set of players, {1, . . . , I}; 

(2a) at which point in time which player is called to move; 

(2b) which actions are feasible for the player when he is called to 

move; 

(2c) what a player knows about the previous history of the game 

when he is called to move; 

(3) the payoff of each player as a function of all possible final 

histories of the game. 

The definition of the extensive form is similar to the definition of the normal 

form, but the strategy spaces can be much more complicated. 


In many cases the extensive form of a game can be depicted nicely by a 

game tree. 

A game tree consists of a set of ordered nodes that are linked to each other. 

There are 

Decision nodes, at which exactly one player has to choose an action out 

of set of feasible actions. Each action leads to new decision or end node. 

End nodes, at which the game ends and payoffs are made. 

The game tree begins with exactly one initial node. 

Each node (except for the initial node) has exactly one predecessor. 

Each decision node has at least one successor. An end node does not 

have any successors. 

A predecessor of node x cannot also be a successor of node x. 

For each decision node or end node there exists a unique sequence of 

decision nodes linking it to the initial node. Such a sequence of decision 

nodes is called the history of the game played up to this node. 


player 2 

. . 

player 1 

. 

 

action 1 action 2 

. . 

player 2 

action 3 action 4 action 3 action 4 

a 

b 

c 

d 

e 

f 

FIG. 3.13: A Game Tree 

g 

h 


Exogenous Uncertainty 

In many games there is some exogenous uncertainty. For example, an entrant 

has to decide whether to enter the market or not. If he enters there are two 

possibilities. With 50 percent probability demand is strong and with 50 percent 

probability demand is weak. 

This can be modeled by introducing Nature as an additional, non-strategic 

player. When nature is called to move she determines the state of the world 

with the exogenously given probabilities. 


0 

2 

Entrant 

. 

. 

N E 

1 

1 

Monopolist 

2 

. . 

2 

Monopolist 

. . 

. . 

Nature 

f n f n 

0 

0 

3 

3 

−1 

−1 

 

1 

1 

FIG. 3.14: A Market Entry Game with Exogenous Uncertainty 


Information Sets 

It may happen that a player does not observe all previous moves (of other 

players and/or of nature). This can be described by using information sets. 

Definition 3.11 (Information Set) 

An information set for a player is a collection of decision nodes satisfying: 

(i) the player has to move at every node in the information set, and 

(ii) the player does not know which point of the information set he has 

reached. 

This definition requires that the set of feasible actions the player can choose 

from is the same at each node of his information set. Why? 

The definition also implies that each decision node belongs to exactly one 

information set. Why? 

If the decision maker knows at which decision node in the game tree he is, 

then this decision node belongs to an information set that is a singleton. 


Examples: 

1 

1 

2 

. 

. 

D C 

5 

0 

1 

. 

 

D C 

0 

5 

. . 

2 

D C 

 

4 

4 

FIG. 3.15: Prisoner’s Dilemma in Extensive Form 


1 

. 

 

2 

. 

. 

2 

. . 

1 

. . 

1 

. . 

1 

. 

1 

. . 

FIG. 3.16A: Possible and Impossible Information Sets 


1 

. 

 

2 

. 

. 

2 

. . 

1 

. . 

1 

. . 

1 

. 

1 

. . 

FIG. 3.16B: Possible and Impossible Information Sets 


1 

. 

 

2 

. 

. 

2 

. . 

1 

. . 

1 

. . 

1 

. 

1 

. . 

FIG. 3.16C: Possible and Impossible Information Sets 


Games of Perfect and Imperfect Information 

Definition 3.12 (Perfect Information) 

A game is one of perfect information if each information set contains a 

single decision node. Otherwise it is a game of imperfect information. 

Note: 

We will see shortly that finite games of perfect information are particularly 

simple to solve. 

If players move simultaneously then the game has imperfect information. 

If a player does not observe the entire previous history of play the game 

is also of imperfect information. 


Strategies in Extensive Form Games 

In a static game there is no difference between an action and a strategy. 

In a dynamic game, however, a strategy can be much more complex than an 

action. 

Definition 3.13 (Strategy) 

Let Hi denote the collection of player i’s information sets, A the set of possible 

actions in the game, and C(Hi) ⊂ A the set of actions possible at information 

set H ∈ Hi. A strategy of player i is a function si : H → A such that 

si(H) ∈ C(H) for all H ∈ Hi. 

In words: A strategy is a complete contingent plan of actions, i.e., for every 

possible contingence (every possible information) a strategy must specify 

what the player is going to do. 

Note: It is extremely important that the strategy is completely specified. For 

example, it is often the case that some information sets cannot be reached if 

the player follows a particular strategy. Nevertheless, his strategy has to 

specify what the agent would do if these information sets were reached! 


Examples: 

3 

1 

2 

. 

. 

ℓ r 

1 

2 

1 

. 

 

L R 

2 

1 

. . 

2 

ℓ r 

 

0 

0 

FIG. 3.17: Strategies in a Game with Perfect Information 

Player 1 has 2 strategies in this game: L, R 

Player 2 has 4 strategies: ℓℓ, ℓr, rℓ, rr 


2 

❅ 

❅ 

1 

❅ 

L 

R 

ℓℓ ℓr rℓ rr 

3, 1 3, 1 1, 2 1, 2 

2, 1 0, 0 2, 1 0, 0 

FIG. 3.18: Normal Form of this Game 


1 

. 

 

L R 

2 

. 

. 

2 

. . 

ℓ r 

ℓ r 

1 

. . 

1 

. . 

1 

. 

1 

. . 

L ′ R ′ L ′ R ′ L ′′ R ′′ L ′′ R ′′ 

FIG. 3.19: Strategies in a Game with Imperfect Information 

Strategies of player 1: 

Strategies of player 2: 


Remarks: 

Every game in extensive form can be transformed into a game in normal 

form. 

But: For a given game in normal form there may be several games in 

extensive form. Example? 

Which form of a game contains more information? 

If a game has several stages, the number of strategies grows very rapidly. 

Consider for example the following game with perfect information: 

◮ At stage 1 player 1 chooses between L and R 

◮ At stage 2 player 2 chooses between l and r 

◮ At stage 3 player 1 chooses between L and R 

◮ at stage 4 player 2 chooses between l and r 

How many stragegies does player 1 have? 

How many strategies does player 2 have? 

What if we add identical stages 5 and 6? 


Backward Induction 

 

0 

2 

Entrant 

. 

 

N E 

f 

−1 

−1 

. . 

Incumbent 

a 

 

2 

1 

FIG. 3.20: A Market Entry Game 

The entrant decides whether to enter (E) the market or not (N). 

If there is entry the incumbent decides whether to fight (f ) or to 

accommodate entry(a). 


A sequential game with finitely many stages is solved by backward 

induction: 

Incumbent: Given that the entrant entered the market, what is the optimal 

strategy for the incumbent? Accommodate. 

Entrant: The entrant anticipates that if he enters the incumbent will 

accommodate. Therefore, she should enter. 

Backward Induction and Nash Equilibrium 

The result of backward induction is a Nash Equilibrium: 

Given that the entrant enters it is optimal for the incumbent to 

accommodate. 

Given that the incumbent accommodates, it is optimal for the entrant to 

enter. 


I 

❅ 

❅ 

E 

❅ 

N 

E 

f 

a 

0, 2 0, 2 

-1, -1 2, 1 

FIG. 3.21: Normal Form of the Entry Game 

The analysis of the normal form shows that there are two Nash equilibria: 

(E, a) 

(N, f ) 


Remarks: 

(N, f ) is a NE: Given that the incumbent fights it is optimal for the entrant 

to stay out. Given that there is no entry it is optimal for the incumbent to 

threaten to fight. 

If there was entry it would be suboptimal for the incumbent to carry out 

his threat of fighting. However, this does not happen in equilibrium (it is a 

zero probability event). Therefore, the strategy “fight” is optimal. 

However, the second NE is not convincing. It relies on a threat that is not 

credible if the incumbent is called to carry it out. 

Note that the normal form of the game does not show whether a threat is 

credible or not. However, if we look at the extensive form and solve the 

game by backward induction then all incredible threats are automatically 

eliminated. 


Equilibrium Refinements 

In dynamic games there are often many NE. However, we have seen that not 

all NE are equally convincing. Some NE are supported by threats that are not 

credible, while other NE do not require such threats. 

In the following we want to refine the notion of Nash Equilibrium. We will 

impose additional desirable conditions that a “good” NE should satisfy. These 

“good” NE are given special names, such as “Subgame Perfect Nash 

Equilibrium”, “Perfect Equilibrium”, “Perfect Bayesian Equilibrium” etc. 

Note: 

Even if a NE is not convincing, it is still a NE. 

We have to be careful not to require too much. If we impose too strong 

conditions, then it may be that many games do not have a NE satisfying 

this requirement. 

Example: A natural and plausible requirement is that a player should not 

use a weakly dominated strategy. However, this condition turns out to be 

so strong that in some games the only NE is in weakly dominated 

strategies. If we would impose this condition, no equilibrium would be left. 


Subgame Perfect Equilibria 

Definition 3.14 (Subgame) 

A subgame of an extensive form game is a subset of the game with the 

following properties: 

a) It begins with an information set containing a single decision node, 

contains all the decision nodes that are successors (both immediate and 

later) of this node, and contains only these nodes 

b) If decision node x is in the subgame, then every decision node x ′ that is 

in the same information set as x is also in the subgame (i.e., there are no 

broken information sets). 

Note: 

The game is a whole is a subgame of itself. Therefore, every game has at 

least one subgame. 

Whether a dynamic game has only one or several subgames depends on 

the information structure. 

A subgame is a game in its own right that can be analyzed independently 

of the rest of the game. 


1 

. 

 

L R 

2 

. 

. 

2 

. . 

ℓ r 

ℓ r 

1 

. . 

1 

. . 

1 

. 

1 

. . 

L ′ R ′ L ′ R ′ L ′′ R ′′ L ′′ R ′′ 

FIG. 3.22: Subgames 


The following definition is due to Reinhard Selten (1965). 

Definition 3.15 (Subgame Perfect Nash Equilibrium) 

A Nash Equilibrium of an extensive form game is called a Subgame Perfect 

Nash Equilibrium (SPNE) if it induces a Nash Equilibrium in every subgame of 

the game. 

Remarks: 

Subgame Perfection is more general than backward induction. It can also 

be applied if the game has simultaneous moves at some stages or if the 

game has infinitely many stages. 

If the only subgame of a game is the game itself, then every NE is a 

SPNE. Why? 

A “proper subgame” is a subgame that is not identical with the game 

itself. 

A SPNE induces a SPNE in every subgame of the game. Why? 


3 

1 

2 

. 

✈. 

ℓ r 

1 

2 

1 

. 

✈ 

L R 

2 

1 

. ✈. 

2 

ℓ r 

 

0 

0 

FIG. 3.23: Subgame Perfect Nash Equilibrium 


Remarks: 

The equilibrium path is (R, l) 

But: The equilibrium has to specify also what happens off the equilibrium 

path. Therefore, the SPNE is (R, rl). 

There exists a second NE: (L, rr). But this NE is not subgame perfect. It 

is based on the incredible threat that player 2 chooses r if player 1 

chooses R. 

SPNE are well defined for games with perfect and with imperfect 

information. However, if the game has imperfect information, it may be 

that it does not have any proper subgame. 


Proposition 3.9 (Existence and Uniqueness) 

(a) Every finite game of perfect information has at least one pure strategy 

SPNE. 

(b) If no player has the same payoffs at any two terminal nodes, then there is 

a unique SPNE. 


Proof: The proof of part (a) is by construction and uses the generalized 

backward induction procedure: 

(1) Start at the end of the game tree and identify the NE for each of the final 

subgames. 

(2) Select one NE in each of the final subgames and replace the subgame by 

the payoffs that result when this NE is played. 

(3) Repeat steps (1) and (2) for the reduced game and continue this 

procedure until every move in the game is determined. This collection of 

moves at all information sets constitutes a profile of SPNE strategies. 

If at no step of this procedure multiple equilibria are encountered the SPNE is 

unique. If the game has perfect information and no player has the same 

payoffs at any two terminal nodes this is indeed the case, which proves part 

(b). 


Remarks: 

1. If there are multiple equilibria in a subgame then the set of all subgame 

equilibria can be found by using the procedure of generalized backwards 

induction with every possible combination of equilibria. 

2. Part (b) of the proposition can be strengthened. If no player has the same 

payoffs at any two terminal nodes of a finite game of perfect information, 

then this game has a unique NE. This is Zermelo’s (1913) Theorem. 

3. Chess is a finite game with perfect information. Moreover, chess is a 

zero-sum game. It can be shown that all Nash equilibria of a zero-sum 

game yield the same equilibrium payoffs for all players. Thus, it must be 

the case that in all Nash equilibria of chess either white wins or black 

wins or there is a remi. Furthermore, the solution to the game can be 

found by backward induction. Thus, from a game theoretic perspective 

chess is trivial. 


Example: A Game with Imperfect Information 

Consider the following game with two periods, in which two players play a 

prisoner’s dilemma game in period 1 and a coordination game with 0 < x < y 

in period 2: 

Period 1: 

B 

❅ 

❅ cooperate defect 

A 

❅ 

cooperate 

defect 

2, 2 -1, 3 

3, -1 0, 0 


Period 2: 

B 

❅ 

❅ 

A 

❅ 

L 

R 

ℓ r 

x, x 0, 0 

0, 0 

y, y 

FIG. 3.24: SPE of a Game of Imperfect Information 

What are the SPNE of this game? 

Under which assumptions on x und y is it possible to sustain a SPNE in 

which both players cooperate in the first period? 


Subgame Perfection and Rationality 

Subgame perfection requires that it is common knowledge that all players are 

rational. This can raise difficult problems as in the following game: 

1 

. 

. 

L R 

ℓ 

2 

. 

 

r 

1 

L 

. 

 

′ R ′ 

 

2 

0 

 

1 

1 

3 

0 

 

0 

2 

FIG. 3.25: Rationality and Backward Induction 


Backward induction implies that player 1 has to choose L and the game ends. 

Why? 

Suppose now that player 1 plays R. How should player 2 interpret this move? 

A rational player 1 would have chosen L. 

Doesn’t player 1 demonstrate that he is irrational by choosing R? 

But if player 1 is irrational, he may also choose R ′ at the last stage of the 

game. 

But then it would be optimal for player 2 to choose r. 

But if player 2 does so, a rational player 1 should anticipate this and 

choose R. 


In order to deal with this problem, Selten (1975) introduced the notion of 

Trembling-hand Perfect Nash Equilibrium or simply Perfect Nash 

Equilibrium (PNE). The idea behind this concept is as follows: 

1) It is common knowledge that all players are rational. 

2) But: All players make mistakes: 

◮ With probability 1 − ɛ they choose the strategy that they want to choose. 

◮ With probability ɛ > 0 they “tremble” and choose some other (randomly 

chosen) strategy. 

3) If a player observes that another player does not choose the equilibrium 

strategy he concludes that this player must have trembled. 

4) If trembles are uncorrelated over time, a deviation from the equilibrium 

strategy does not induce the other players to change their equilibrium 

strategies. 


To properly define a Perfect Nash Equilibrium for sequential games Selten 

introduces the notion of an “agent normal form” game. In the agent normal 

form of a game a player is replaced by a different agent in each information 

set at which he is called to move. All of his agents have exactly the same 

payoff function. The agent normal form makes sure that a player cannot 

correlate his trembles. 

Next, we need the notion of a perturbed game. In a perturbed game each 

agent of each player is constrained to choose each possible strategy with a 

probability of at least ɛ(si) > 0. 

An ɛ-constrained equilibrium is a totally mixed strategy profile σ ɛ such that, 

for each player i, σ ɛ i solves maxσi ui(σi, σ−i) subject to σi(si) ≥ ɛ(si) for all si 

und for some {ɛ(si)}si ∈Si , i∈I with 0 < ɛ(si) < ɛ. 


Definition 3.16 ((Trembling-hand) Perfect Nash Equilibrium) 

A Nash equilibrium is (trembling-hand) perfect if it is any limit of 

ɛ-constrained equilibria σ ɛ as ɛ goes to 0. 

Note that any trembling hand-perfect equilibrium is subgame perfect. The 

reason is that each subgame is reached with strictly positive probability, and 

so strategies have to be best responses in all subgames. 

Note also that is often very difficult to verify whether a strategy profile is a 

trembling-hand perfect equilibrium. Therefore, we will restrict attention 

Subgame Perfect Nash Equilibria in the following. 


The One-shot Deviation Principle 

If a game has many stages it may become very difficult to check whether an 

equilibrium candidate is indeed an equilibrium. The reason is that if a player 

moves at several stages, then his set of possible strategies becomes very 

large very quickly. In principle all of these strategies have to be checked in 

order to see whether there is a profitable deviation. However, the following 

proposition tells us that this task can be simplified dramatically. 

Proposition 3.10 (One-shot Deviation Principle) 

A strategy profile s ∗ is a SPNE if and only if there does not exist any player i 

and any strategy ˜si that differs from s ∗ i in one period t and after one history h t 

only and is strictly better than s ∗ i if the subgame after h t is reached. 


Remarks: 

1) Why is this condition a necessary condition for a SPNE? 

2) It is less obvious that this condition is also sufficient. Even if there does 

not exist a profitable strategy ˜si differing from s∗ i at only one information 

set, there could still be a profitable strategy ˜si deviating at multiple 

information sets from s∗ i . 

3) The one-shot deviation principle makes life much easier. Now we only 

have to check whether there is any information set at which a player has 

an incentive to deviate. We do not have to consider entire strategies that 

differ at several information sets from the proposed equilibrium strategy. 

4) We do the proof for finite games only. However, it also applies to games 

with an infinite horizon if players discount future payoffs. 


Proof: The proof is by contradiction. Consider a strategy profile s∗ that 

satisfies the one-shot deviation condition, but that is not a SPNE. Then there 

exists a strategy ˜si for some player i that deviates from s∗ i at at least two 

information sets and that yields a strictly higher payoff for player i. 

Consider the last information set at which there is a deviation from s∗ i . If ˜si 

does not yield a strict improvement for player i if this information set is 

reached, replace this part of ˜si by the corresponding part of s∗ i and go to the 

next “last” information set at which ˜si and s∗ i differ until you have found an 

information set at which ˜si yields a strictly higher payoff then s ∗ i if this 

information set is reached. Call the history that leads to this information set ˜ h ˜t . 

Consider now a strategy that ˆsi that coincides with s ∗ i in all information sets 

except for those following after ˜ h ˜t where it coincides with ˜si. However, ˜si 

coincides with s ∗ i in all information sets except for the one at ˜ h ˜t . Hence, ˆsi 

deviates from s ∗ i only at ˜ h ˜t . By the construction of ˆsi it must be the case that ˆsi 

yields a strictly higher payoff than s ∗ i if ˜ h ˜t is reached. This is a contradiction to 

our assumption that s ∗ satisfies the one-shot deviation condition. Q.E.D. 


Infinite Games 

A War of Attrition: 

Consider two players who fight for a prize. In each period t, t = 1, 2, . . ., each 

player decides whether to fight or to give up: 

If both players fight, both lose one unit of utility and the game moves on 

to the next period. 

If one player gives up and the other one fights, the fighter wins and gets 

the prize of v > 1 while the loser gets 0 and the game ends. 

Payoffs: Let ˆt denote the period in which the loser gives up. 

Loser: 

Winner: 

uL(ˆt) = −(1 + δ + · · · + δ ˆt−1 ) · 1 = − 1 − δ ˆt 

1 − δ 

ug(ˆt) = −(1 + δ + · · · + δ ˆt−1 ) · 1 + δ ˆt v = − 1 − δ ˆt 

1 − δ + δˆt v 


Is there a symmetric, stationary SPNE? 

A symmetric equilibrium must be in mixed strategies. Suppose that both 

players give up each period with probability p and fight with probability 1 − p. 

These strategies are an equilibrium if in each period both players are 

indifferent whether to fight or to give up. Thus, we must have in every period t: 

0 = p v + (1 − p) · [−1 + δ · 0] 


Interpretation: 

All losses that have been accumulated up to period t are sunk costs that 

do not affect future play. Thus, we only have to consider future payoffs. 

If player i gives up in period t he gets 0 right away. 

If player i fights there are two possibilities: 

◮ With probability p his opponent gives up and he gets v. 

◮ With probability 1 − p his opponent fights in which case he loses 1 this 

period. In the next period he is again indifferent between fighting and giving 

up, so his continuation payoff is exactly 0. 

Note that we only have to check that no player has an incentive to deviate 

in any one period. By the One-shot Deviation Principle we do not have to 

check whether deviations in multiple periods are profitable. 

Solving for p yields: 

p ∗ = 1 

1 + v 


Remarks: 

1) The higher the prize v, the smaller is the probability of giving up in every 

period. 

2) The outcome is inefficient because the players fight with positive 

probability. There is also a positive probability that the costs of fighting 

exceed the value of the prize. 

3) There are many other SPNE. For example: Player 1 fights in every period 

and player 2 gives up in every period is a SPNE. Why? 

4) But: If the situation is completely symmetric, a symmetric equilibrium may 

be more convincing. 

5) The war of attrition game is used by biologists to explain animal behavior 

(Maynard Smith 1974). Interpretation? 


Repeated Games 

Let G denote a finite game in normal or extensive form. Then G T is a 

repeated game in extensive form in which the stage game G is played T 

times in a row, with T ∈ IN ∪ {∞}. 

Examples: 

Two players play a repeated prisoner’s dilemma game. 

Two oligopolists play a repeated Cournot game, etc. 

Remarks: 

1) Repeated games are a special case of dynamic games. 

2) But, because of their special structure they have some interesting 

properties that do not hold for all dynamic games. 

3) Be careful: The following games are not repeated games: 

◮ The war of attrition game 

◮ A bargaining game in which the players take turns in making offers 

◮ Investment games, resource extraction games, etc. 

Why not? 


Finitely Repeated Games 

Suppose that the prisoners’ dilemma game is played by two players two times 

in a row: 

ℓ r 

L 

R 

1, 1 5, 0 

0, 5 4, 4 

FIG. 3.26: Repeated Prisoner’s Dilemma Game 

The payoffs of the repeated game are simply the sums of the payoffs of the 

two stage games. 


Analysis of the Game 

Period 2: No matter what has happened in period 1, the prisoner’s dilemma 

game has a unique Nash Equilibrium (L, ℓ) that must be played in period 2 in 

every SPNE. 

Period 1: What happens in period 1 has no impact on what is going to 

happen in period 2. Therfore, we can simply add the payoffs (1,1) from period 

t to the payoffs in period 1. 

L 

R 

ℓ r 

2, 2 6, 1 

1, 6 5, 5 

FIG. 3.27: Reduced Normal Form of the Repeated Prisoner’s Dilemma 


Conclusion: 

Both players should choose “left” in period 1. 

The unique SPNE is: 

◮ Equilibrium strategy of player 1: (L1, L2L2L2L2) 

◮ Equilibrium strategy of player 2: (ℓ1, ℓ2ℓ2ℓ2ℓ2) 


If the stage game G has a unique NE, then the finitely repeated game G T , 

T < ∞, has a unique SPNE that is simply the T -fold repetition of the NE of 

the stage game. 

Proof: Follows immediately from the principles of backward induction and 

subgame perfection as in the repeated prisoner’s dilemma game. Q.E.D. 

Exercise: Show that the equilibrium described for the repeated prisoner’s 

dilemma is not just the unique SPNE but also the unique NE. 


Consider now the following stage game that is played twice in a row: 

2 

❅ 

❅ 

1 

❅ 

L 

M 

R 

ℓ m r 

1, 1 5, 0 0, 0 

0, 5 4, 4 0, 0 

0, 0 0, 0 3, 3 

FIG. 3.28: Multiple Nash Equilibria in the Stage Game 


How many pure strategies does every player have in the repeated game? 

The stage game has two NE in pure strategies: (L, ℓ) and (R, r). 


It is a SPNE to play the same NE of the stage game in every period. 

Proof: Subgame perfection requires that a NE is played in every subgame. In 

the last period this is obviously the case. Because play in the second last 

period does not affect what happens in the last period, we also have a NE in 

the subgame starting in the second to last period, and so on. Q.E.D. 

But: There are many additional SPNE. Example: 

Period 1: Both players play (M, m) in period 1. 

Period 2: 

If both players choose these actions in period 1 they play (R, r) 

in period 2. 

If at least one player deviated from (M, m), they play (L, ℓ) in 

period 2. 


Remarks: 

This equilibrium yields payoffs (7, 7) that are higher than the payoffs 

(6, 6) that the players would get if they played (R, r) twice. 

In this equilibrium cooperation in period 1 is sustained by the threat to 

play the “bad” NE (L, ℓ). 

This threat is subgame perfect. But is it really credible? What if players 

can renegotiate? 


Infinitely Repeated Games 

In finitely repeated games the “last period effect” plays a crucial role. If there 

is a unique NE in the stage game, we know what we are going to do in the last 

period. But then we also know what we will do in the second to last period, 

and the game unravels. Thus, in any finite repetition of the prisoner’s dilemma 

game the unique SPNE is that all players always defect. This seems very 

implausible. 

Experiments have shown that it is indeed often the case that cooperation 

breaks down in the last period. Nevertheless, players manage to cooperate in 

most previous periods. 

Axelrod experiments. 

If players do not solve the game by backward induction it may be better to 

describe a repeated relationship as an infinitely repeated game, even if there 

are no infinitely repeated games in the real world. 


Payoffs 

If a player would maximize the sum of his payoffs in an infinitely repeated 

game his utility would always be infinite and his maximization problem would 

not be well defined. 

We assume that players have a discount factor δ < 1 and maximize the 

discounted sum of future payoffs. Possible Interpretations: 

δ = 1 

1+r , where r > 0 is the interest rate, 

δ is the probability that there is a next period, 

or δ reflects a combination of the two. 

Furthermore, we normalize all payoffs in the repeated game by multiplying 

them with (1 − δ) in order to make them comparable to the payoffs of the 

one-shot game. Why can we do this without loss of generality? 


Definition 3.17 

An infinitely repeated stage game G with discount factor δ is denoted by 

G ∞ (δ). The payoff of player i in G ∞ (δ) is given by 

vi = (1 − δ) 

∞ 

δ t−1 ui(a t i , at −i ). 

t=1 

Example: If player i gets payoff 4 in every period, his payoff in the repeated 

game is 

∞ 

vi = (1 − δ) δ t−1 4 = (1 − δ) 1 

4 = 4. 

1 − δ 

t=1 



If δ is sufficiently close to 1, there exists a SPNE in the infinitely repeated 

prisoner’s dilemma game in which both players cooperate in all periods on the 

equilibrium path. 

Proof: Consider the following symmetric pair of strategies for players i and j: 

“Play ‘cooperate’ in period 1 and in all following periods as long as 

both players played ‘cooperate’ in all previous periods. However, if at 

least one player deviated in any previous period, then choose ‘defect’ 

forever after. 

Consider the payoffs in Fig. 3.26: 

If both players follow their equilibrium stragegies, each of them gets 

(1 − δ) 

∞ 

t=1 

δ t−1 4 = (1 − δ) 1 

4 = 4 . 

1 − δ 


If one player deviates in this period, his payoff is 

 

∞ 

(1 − δ) 5 + δ t−1 

1 = (1 − δ) 5 + δ 

 

= 5 − 4δ . 

1 − δ 

t=2 

A deviation is profitable if and only if 4 < 5 − 4δ ⇔ δ < 1 

4 . 

Thus, for δ ≥ 1 

4 the above strategies form a NE. 

We still have to show that these strategies are also a SPNE. 

We have shown already that a deviation after a history, in which all 

players always cooperated is not profitable if δ ≥ 1 

4 . 

Suppose now that we are off the equilibrium path, i.e. at least one player 

has deviated at least once in the past. In this case a deviation does not 

pay either. Both players are supposed to defect forever after. But this is 

just the repetition of the one-shot NE, which is always subgame perfect. 

Q.E.D. 


Remarks: 

These strategies are called “grim strategies” or “trigger strategies”. They 

have the drawback that a mistake of one the players triggers a 

catastrophe for everybody (“doomsday machine”). 

It is possible to sustain cooperation with other punishment strategies, e.g. 

with “perfect tit-for-tat”: 

“Play ‘cooperate’ in period 1 and whenever the outcome of the 

last period was either (‘cooperate’, ‘cooperate’) or (‘defect’, 

‘defect’). Play ‘defect’ if the outcome of the last period was 

(‘cooperate’, ‘defect’) or (‘defect’, ‘cooperate’)” 

Exercise 1: Show that it is a SPNE if both players play “perfect tit-for-tat” 

if δ is sufficiently close to 1. Use the one-shot-deviation principle. You 

have to check four cases. 

Exercise 2: Show that it is not a SPNE if both players play “tit-for-tat” 

(“Cooperate in period 1. In all following periods play the strategy used by 

your opponent in the previous period.”) 


The proposition shows that the efficient outcome can be implemented 

(i.e. is a SPNE outcome) if players play an infinitely repeated game and if 

the discount factor is sufficiently close to one. However, there are many 

other SPNE outcomes. The “Folk-Theorem” shows that any payoff vector 

that gives each player at least his minmax payoff can be sustained as a 

SPNE outcome. 

Discussion. 


3.3 Static Games with Incomplete Information 

So far we assumed that there is common knowledge about the game itself, 

i.e. 

all players know the set of players, 

all players know what the strategy sets of all players are, 

all players know what the payoff functions of all players are, 

and everybody knows that everybody knows that ... 

How can we model games with incomplete information, e.g. games 

in which a player does not know which actions are feasible for his 

opponent, 

or what the payoff function of his opponent is, 

or what information his opponent has when he is called to move? 


Harsanyi (1967/68) suggested how to transform a game with incomplete 

information into a game with imperfect information that we know how to deal 

with. 

He suggested that all private information of a player is summarized in his 

“type”. Thus, for each player i with private information we have to specify his 

type space Ti, i.e. what possible types ti ∈ Ti he may have. 

The utility function of a player may now depend on his type, i.e. 

ui = ui(ai, a−i, ti) , 

or, slightly more formally: ui : Ai × A−i × Ti → R. 


Examples: 

Player i has private information about his payoffs, e.g. about his cost to 

do something, his willingness to pay for something, etc. 

Player i has private information about his strategy set. Let Ai denote the 

set of all actions that are feasible. If player i cannot use some action 

âi ∈ Ai, then he is of a type ˆti ∈ Ti for whom the cost of taking action âi is 

infinitely high, i.e. 

ui(âi, a−i,ˆti) = −∞ . 

Thus, action ai is strictly dominated and will never be taken. 

Player i may be an “irrational” player who always chooses some action āi 

no matter what the payoff consequences are. Then the player is of type 

¯ti ∈ Ti with utility function 

 

1 if ai = āi 

ui(ai, a−i,¯ti) = 

0 if ai = āi 

For this type it is a dominant strategy to play āi. 


The structure of the game is now as follows: 

At stage 0 nature draws the types of all players according to some joint 

cumulative distribution function F(t), t ∈ T , with t = (t1, t2, . . . , tI) and 

T = T1 × T2 × . . . × TI. 

F(t) is common knowledge among all players. 

Note that the types of different players may be correlated. 

Each player learns his own type and updates his beliefs about the 

probabilities of the types of the other players. The updated cumulative 

distribution function of player i of type ti over the possible types t−i of his 

opponents is denoted by F (t−i|ti). 

Then the game is played. 


Beliefs 

The cumulative distribution function F(t−i|ti) is called the “belief” of 

player i. 

The assumption that all players start out with the same prior probability 

distribution (the same “prior”) F (t) is very important. Players may have 

very different information assigned to them by nature, but they all agree 

which events are possible and which are not. It also implies that it is 

common knowledge that the beliefs of player i with type ti are pi(t−i | ti). 

If the types of the players are correlated, a player learns something about 

the probabilities of the types of his opponent when he learns his own 

type. In this case each player has to use Bayes’ rule to update his beliefs: 

prob(A | B) = 

Example: Auction for drilling rights. 

prob(A ∩ B) 

prob(B) 


If the types of the players are stochastically independent we have 

pi(t−i|ti) = p(t−i) · p(ti) 

p(ti) 

= p(t−i) ∀ti ∈ Ti 

where pi(t−i | ti) is the probability player i of type ti assigns to the type 

profiles t−i of his opponents. In this case a player’s beliefs are 

independent of his own type. 

Because of the importance of Bayes’ Rule games with incomplete 

information are called Bayesian Games. 


Strategies 

Different types of player i may choose different actions. Therefore, we have to 

refine the notion of a strategy. 

Definition 3.18 (Strategies in Games with Incomplete 

Information) 

A pure strategy of player i assigns to each possible type of player i an action 

ai, i.e. si : Ti → Ai. 

A mixed strategy of player i assigns to each possible type of player i a 

probability distribution over Ai, i.e. σi : Ti → ∆(Ai). 

Note that each player has to specify a strategy not just for his realized type but 

also for all other potential types that he could have had. Why? 


Expected Payoffs 

Suppose that player i is of type ti and expects his opponents to play the 

strategy profile 

s−i(t−i) = (s1(t1), . . . , si−1(ti−1), si+1(ti+1), . . . , sN(tN)) 

His updated beliefs about the types of his opponents are p(t−i|ti). 

If player i with type ti chooses action ai, he gets with probability p(t−i|ti) the 

payoff 

ui(ai, s−i(t−i), ti). 

Thus, his expected payoff is given by 

E[ui(ai, s−i, ti)] = 

t−i ∈T−i 

pi(t−i|ti) ui(ai, s−i(t−i), ti). 


Definition 3.19 (Normal Form of a Bayesian Game) 

The normal form of a game G = [I; {Si}; {Ti}; F (·); {ui}] consists of 

(1) the set of players, I = {1, . . . , n}, 

(2) the strategy sets of all players {Si} = {S1, . . . , Sn}, 

(3) the type spaces T1, . . . , Tn of the players, 

(4) the common prior F(t) 

(5) the payoff functions of all players {ui} = {u1, . . . , un}. 


Bayesian Nash Equilibrium 

Definition 3.20 

A strategy profile s ∗ = (s ∗ 1 , . . . , s∗ n) is a Bayesian Nash Equilibrium of a 

game with incomplete information, if for all i = 1, . . . , n and all ti ∈ Ti the 

action ai = s ∗ i (ti) maximizes E ui(ai, s ∗ −i , ti) . 

Remarks: 

1) The idea of a Bayesian Nash Equilibrium is exactly the same as the idea 

of a NE. Each player must play a best response given the strategies 

played by his opponents. But now this condition has to hold for all 

possible types of each player. 

2) The beliefs of each player about the types of his opponents have to be 

formed using Bayes’ rule. 


3) If players choose mixed strategies they take the expectation over all 

possible actions that are chosen with positive probability by each possible 

type: 

Let σk(ak|tk) denote the probability with which player k of type tk chooses 

action ak. Thus, if the type profile is t−i, a−i is played with probability 

σ−i(a−i|t−i) = 

k=i σk(ak|tk). 

Player i’s expected payoff if he is of type ti and chooses ai is therefore 

E[ui(ai, σ−i, ti)] = 

t−i ∈T−i 

pi(t−i|ti) 

a−i ∈A−i 

σ−i(a−i|t−i) ui(ai, a−i, ti). 

4) σi is a best response to σ−i if for all ti ∈ Ti each action ai with σi(ai|ti) > 0 

maximizes E[ui(ai, σ−i, ti)]. 

5) In finite games with incomplete information there always exists a 

Bayesian Nash Equilibrium, possibly in mixed strategies. The proof 

follows the same lines as the existence proof for NE. 


Purification of Mixed Strategies 

In Chapter 3.1 we offered one justification for mixed strategy equilibria that 

goes back to Harsanyi. Harsanyi argued that a mixed strategy equilibrium can 

be interpreted as the limit of a sequence of pure strategy equilibria of a 

perturbed game with incomplete information as the perturbation goes to zero. 

To illustrate this argument consider again the “battle of the sexes” game. 

However, we now assume that each player is imperfectly informed about the 

payoff function of his opponent: 

Player 2 does not know exactly what player 1’s payoff is if they both go to 

“Boxing”. She believes that his payoff is 2 + t1 in this case, where t1 

uniformly distributed on the interval [0, x]. 

Player 1 does not know exactly what player 2’s payoff is if they both go to 

“Ballet”. He believes that her payoff is 2 + t2 in this case, where t2 is 

uniformly distributed on the interval [0, x]. 

Note that the types of two players are uncorrelated. Thus, by learning his or 

her own type no player learns anything about the type of his or her opponent. 


2 

❅ 

❅ Boxing Ballet 

1 

❅ 

Boxing 

Ballet 

2 + t1, 1 0, 0 

0, 0 1, 2 + t2 

FIG. 3.29: Battle of the Sexes with Incomplete Information 

We are going to construct a Bayesian Nash Equilibrium in pure strategies in 

which 

player 1 chooses “Boxing” iff t1 ≥ c1; 

player 2 chooses “Ballet” iff t2 ≥ c2. 


In this equilibrium 

the probability assigned by player 2 to the event that player 1 chooses 

“Boxing” is equal to x−c1 

x ; 

the probability that player 1 assigns to the event that player 2 goes to 

“Ballet” is equal to x−c2 

x . 

For which values of c1 and c2 do these strategies form a Bayesian Nash 

Equilibrium? 


Player 1 prefers “Boxing” iff: 

(1 − 

Player 2 prefers “Ballet” iff: 

(1 − 

E(u1|Bo) ≥ E(u1|Ba) 

x − c2 

) · (2 + t1) ≥ 

x 

t1 ≥ x 

x − c2 

x 

c2 

· 1 

− 3 ≡ c1 

E(u2|Ba) ≥ E(u2|Bo) 

x − c1 

) · (2 + t2) ≥ 

x 

t2 ≥ x 

x − c1 

x 

c1 

· 1 

− 3 ≡ c2 


This implies for c1 and c2: 

Solving for c: 

c1 = c2 = c and c 2 + 3c − x = 0 

c = − 3 

2 + 

 

9 + 4x 

4 

Hence, the probability with which player 1 chooses “Boxing” (player 2 chooses 

“Ballet”) is: 

√ 

x − c 9 + 4x − 3 

= 1 − 

x 2x 


What happens if the incomplete information becomes small, i.e. if x → 0? 

lim 

x→0 

√ 9 + 4x − 3 

2x 

( 

= lim 

x→0 

√ 9 + 4x − 3)( √ 9 + 4x + 3) 

2x( √ 9 + 4x + 3) 

= lim 

x→0 

9 + 4x − 9 

2x( √ 9 + 4x + 3) 

2 

= lim √ = 

x→0 9 + 4x + 3 1 

3 

Conclusion: The mixed strategy equilibrium 

2 1 

3 , 3 of the “battle of the sexes” 

game with complete information can be interpreted as the mixed Bayesian 

Nash Equilibrium of a perturbed game with incomplete information when the 

perturbation goes to zero. 

Harsanyi (1973) has shown that this “purification” of mixed strategy equilibria 

works out in almost all games. 


3.4 Dynamic Games with Incomplete Information 

In dynamic games with incomplete information we want to refine the Bayesian 

Nash Equilibrium concept in order to rule out equilibria that are not credible. 

Problem: Subgame perfection has no bite in these games. 

If a player does not know the type of his opponent, there are no proper 

subgames starting from a singleton information set after nature has drawn the 

types of the players. The only subgame is the entire game! 

How to generalize the idea of sequential rationality to games with incomplete 

information? 


Generalization of the idea of subgame perfection to “continuation 

games” 

A “continuation game” may start at any information set 

But: a player must have “beliefs” about where he is in this information set, 

i.e. he must have a probability distribution over all possible decision 

nodes in this information set. 

Furthermore: These beliefs must be “consistent”. 

This generalization is useful not only in games of incomplete but also in 

games of complete information. 

It will lead us to the notion of Perfect Bayesian Equilibrium. 


Continuation Games 

Consider the following game that is due to Selten (1975): 

1 

. .. . 

ℓ 

✎ 

. 

. 

✍ 

L 

r 

2 

M 

ℓ 

☞ 

. . 

✌ 

r 

2 

1 

0 

0 

0 

2 

What should player 2 do in this game? 

R 

 

0 

1 

FIG. 3.30: Selten’s Horse 

 

1 

3 


If player 2 is called to move, it is a dominant strategy to play ℓ. 

Anticipating this, player 1 should play L. 

Does this game have other equilibria? 

2 

❅ 

❅ 

1 

❅ 

L 

M 

R 

ℓ r 

2, 1 0, 0 

0, 2 0, 1 

1, 3 1, 3 

FIG. 3.31: Normal Form of Selten’s Horse 


(R, r) is also a Nash Equilibrium of this game! 

Is this equilibrium subgame perfect? 

To impose sequential rationality we need the following definition: 

Definition 3.21 (Continuation Game) 

A continuation game is a subset of an extensive form game with the 

following properties: 

a) It begins at any information set, contains all the decision nodes that are 

successors (both immediate and later) of this node, and contains only 

these nodes. 

b) In every information set the player who is called to move must have a 

belief that specifies the probability he assigns to the event that he is at 

any decision node of this information set. 

Given the definition of continuation games we can now define “sequential 

rationality” as follows: 


Definition 3.22 (Sequential Rationality) 

A player must choose a strategy that is a best response to the strategies of 

his opponents given his beliefs in any continuation game of the game. 

Applying this definition to Selten’s Horse, we see that playing r is not an 

optimal response for player 2 no matter what he believes at which decision 

node he is: For any probability µ, 0 ≤ µ ≤ 1, that he is at the left decision 

node his optimal strategy is ℓ. 

This example is very simple because player 2 has a dominant strategy in the 

information set where he is called to move. If he does not have a dominant 

strategy, his optimal strategy depends in general on his beliefs. 


Consider the following modification of Selten’s Horse: 

1 

. .. . 

✎ 

✍ 

ℓ 

[µ] . 

. 

L 

r 

2 

M 

ℓ 

. . [1 − µ] 

r 

2 

1 

0 

0 

0 

0 

R 

 

0 

1 

 

1 

3 

☞ 

✌ 

FIG. 3.32: No Dominant Strategy for Player 2 


In this game, player 2 should choose ℓ only if µ ≥ 0.5. There are two 

sequentially rational equilibria of this game: 

(L, ℓ, µ ≥ 0.5) 

(R, r, µ ≤ 0.5) 

So far, we did not ask where the beliefs come from. 

Consider the first equilibrium (L, ℓ, µ ≥ 0.5): 

Given that player 2 chooses ℓ it is optimal for player 1 to choose L. 

Given the belief µ ≥ 0.5 it is optimal for player 2 to choose ℓ. 

Are all beliefs µ ≥ 0.5 consistent with player 1’s strategy to choose L with 

probability 1. No. If player 2 uses Bayes’ rule to derive the probability of 

being in the left decision node, he should conclude that µ = 1. 

Therefore, the only equilibrium with consistent beliefs is (L, ℓ, µ = 1). 


What about the equilibrium (R, r, µ ≤ 0.5)? 

Given that player 2 chooses r it is optimal for player 1 to choose R. 

Given the belief µ ≤ 0.5 it is optimal for player 2 to choose r. 

Now Bayes’ Rule cannot be applied. Given that player 1 chooses R it is a 

zero probability event that player 2 is called to move. Thus, any belief µ is 

consistent with player 1’s strategy. 

Therefore, for any µ ≤ 0.5 the equilibrium (R, r, µ ≤ 0.5) is consistent. 


This discussion motivates the following definition: 

Definition 3.23 (Perfect Bayesian Equlibrium) 

A (weak) Perfect Bayesian Equilibrium is a profile of strategies σ ∗ and a 

system of beliefs µ ∗ such that 

the strategies σ ∗ of all players are sequentially rational given the system 

of beliefs µ ∗ 

the system of beliefs µ ∗ is consistent with the strategies s ∗ , i.e. they are 

derived from the equilibrium strategies using Bayes’ Rule whenever it 

applies. 


Remarks: 

1. If players use mixed strategies that assign positive probabilities to all 

possible actions in all information sets, then all decision nodes will be 

reached with positive probability. In this case Bayes’ rule can always be 

used to update beliefs and it pins down beliefs uniquely. 

2. If a strategy chooses an action with probability zero then it is possible that 

some information sets are reached with probability zero. In these 

information sets Bayes’ rule cannot be applied. So far we allow for 

arbitrary beliefs in these information sets. 

3. The same equilibrium outcome may be supported by different out off 

equilibrium beliefs. Thus, (R, r, µ ≤ 0.5) is not one Perfect Bayesian 

Equilibrium but a continuum of Perfect Bayesian Equilibria. 


Applying Bayes’ Rule: 

Consider again Selten’s game with the payoffs left unspecified and suppose 

that player 1 chooses the mixed strategy (p, q, 1 − p − q). 

. .. . 

[µ] 

. [1 − µ] 

. 

1 (1 − p − q) 

p q R 

✎ 

✍ 

ℓ 

L 

. 

r 

2 

ℓ 

M ☞ 

✌ 

r 

FIG. 3.33: A Fully Mixed Strategy of Player 1 


Suppose that player 2 is called to move. What is the probability that he is at 

the left decision node? 

There are three possible events: L, M and R. 

When player 2 is called to move, he knows that either L or M has happened. 

Define the following events: 

A = {L} 

B = {L, M} 

Now we can derive the following probabilities: 

prob(A ∩ B) = p 

prob(B) = p + q 

prob(A | B) = prob(A∩B) 

prob(B) 

= p 

p+q 


Examples: 

1 

. 

D 

... 

2 . 

. 

✎ 

✍ 

ℓ 

[µ] . 

 

L 

r 

3 

R 

ℓ 

. . [1 − µ] 

r 

⎛ 

⎝ 1 

2 

1 

⎞ 

⎠ 

⎛ 

⎝ 3 

3 

3 

⎞ 

⎠ 

⎛ 

⎝ 0 

1 

2 

⎞ 

⎠ 

A 

⎛ 

⎝ 0 

1 

1 

⎛ 

⎝ 2 

0 

0 

⎞ 

⎠ 

☞ 

✌ 

FIG. 3.34: Example 3.1: A PBE that is not a SPNE 


⎞ 

⎠

Analysis of Example 3.1: 


1 

. 

... 

2 . 

.. 

A 

A 

. 

. 

L 

3 

R 

. . 

′ 

D 

⎛ 

⎝ 

ℓ r ℓ r 

2 

0 

0 

⎛ 

⎝ 

⎞ 

⎠ 

0 

2 

⎞ 

⎠ 

✎ 

✍ 

[µ] [1 − µ] 

0 

☞ 

✌ 

⎛ 

⎝ 1 

2 

1 

⎞ 

⎠ 

⎛ 

⎝ 3 

3 

3 

⎞ 

⎠ 

⎛ 

⎝ 0 

1 

2 

⎞ 

⎠ 

⎛ 

⎝ 0 

1 

1 

FIG. 3.35: Example 3.2: Bayes’ Rule Cannot Be Applied 


⎞ 

⎠



1 . 

.. 

✎ 

✍ 

[µ] . 

. 

L 

2 

M 

. . [1 − µ] 

ℓ r ℓ r 

1 

1 

5 

0 

3 

1 

R 

 

4 

3 

 

0 

2 

☞ 

✌ 

FIG. 3.36: Example 3.3: A Mixed Strategy PBE 




Restricting Out-off Equilibrium Beliefs 

Our definition of a (weak) Perfect Bayesian Equilibrium is the definition that is 

used in most of the applied literature. However, it is still very weak, because it 

imposes no constraints whatsoever on beliefs that are formed out of 

equilibrium. 

To see that this may be too weak consider the following two examples: 


Nature 

. 

 

1 

1 

2 

✎ 

[0.5] . . 

✍ 

Player 1 

2 

☞ 

. . [0.5] 

✌ 

X Y 

✎ 

[0.9] . . 

✍ 

Player 2 

Y 

☞ 

. . [0.1] 

✌ 

X 

L R L R 

 

2 

10 

0 

5 

5 

2 

 

0 

 

5 

5 10 

 

2 

10 

FIG. 3.37: Example 3.4: Signal What You Don’t Know 

PBE: (X, L, µ1 = 0.5, µ2 = 0.9). 

What is fishy here? 


0 

2 

E 

. 

. 

Out In 

. . 

E 

✎ 

[1.0] 

✍ 

F 

. . M 

A 

☞ 

. . [0.0] 

✌ 

f a f a 

 

−3 

 

1 

−1 −2 

−2 

−1 

 

3 

1 

FIG. 3.38: Example 5: A Weak PBE That Is Not Subgame Perfect 

PBE: ((Out, A), f , µ = 1.0). 

Why is this a PBE even though it is not subgame perfect? 


Sequential Equilibria 

Kreps and Wilson (1982) were the first to come up with the idea to explicitly 

specify the beliefs of the players at every information set of the game. They 

introduced the notion of Sequential Equilibrium. 

Definition 3.24 (Sequential Equilibrium) 

A strategy profile σ ∗ and a system of beliefs µ ∗ form a Sequential Equilibrium 

if they have the following properties: 

(i) The strategy profiles σ∗ is sequentially rational given the belief system µ ∗ . 

(ii) k ∞ 

There exists a sequence of completely mixed strategies {σ k=1} with 

limk→∞ σk = σ, such that µ = limk→∞ µ k , where µ k denotes the beliefs 

derived from strategy profile σk using Bayes’ rule. 


Remarks: 

1. All Sequential Equilibria are Perfect Bayesian Equilibria, but not all PBE 

are SE. 

2. Sequential Equilibria rule out certain out of equilibrium beliefs: 

◮ In Example 4 Sequential Equilibrium requires that µ = 0.5. Why? 

◮ In Example 5 Sequential Equilibrium requires that µ = 0.0 and M chooses a. 

Why? 

◮ In games with more than two players Sequential Equilibrium requires that all 

players update their beliefs off the equilibrium path in the same way. Why? 

3. Even though these are very desirable properties, it is very difficult to 

show that an equilibrium candidate is a Sequential Equilibrium in 

practice. This is why most applied papers construct Perfect Bayesian 

Equilibria, call them “Sequential Equilibria”, but do not check whether 

they do indeed satisfy condition (ii). 

4. Kreps and Wilson (1982) have shown that Sequential Equilibria exist 

under rather weak conditions. 


5. Sequential Equilibrium is closely related to (Trembling Hand) Perfect 

Equilibrium, but slightly weaker: Every Perfect Equilibrium is a Sequential 

Equilibrium but not the other way round. (Trembling Hand) Perfect 

Equlibria require that trembles are uncorrelated and that we find a 

sequence of ɛ− constrained equilibria, while Kreps and Wilson allow for 

correlated trembles and require only a sequence of ɛ−constrained 

strategies. 

6. For many games the set of Sequential Equilibria and the set of Perfect 

Equilibria coincide. See Kreps and Wilson (1982) and Fudenberg and 

Tirole (1991) for a discussion. 


Signaling Games 

An important class of games with incomplete information are signaling 

games: 

Two players: a sender and a receiver 

Nature chooses the type of the sender out of T = {t1, . . . , tI} according to 

the probability distribution µ(t). 

The sender learns his type and chooses a message 

m ∈ M = {m1, . . . , mJ}. 

The receiver observes the message (but not the type) of the sender and 

chooses an action a ∈ A = {a1, . . . , aK }. 

Payoffs U S (t, m, a) and U R (t, m, a) are realized. 


Examples of Signaling Games: 

Job market signaling 

Initial public offering 

Limit pricing 

... 

We start out with an abstract signaling game with two types, two messages 

and two actions. 

This material is covered by MGW, Chapter 13C, 450-460. 


a 1 

a 2 

a 1 

a 2 

✎☞ 

m1 . .. . 

S 

t1 

m2 a1 ✎☞ 

. 

R Nature R 

. 

.. . 

✍✌ 

...... . 

. 

✍✌ 

m 1 

t 2 

S 

µ 0 

1 − µ 0 

FIG. 3.39: Structure of a 2 × 2 × 2 Signaling Game 


m 2 

a 2 

a 1 

a 2

In this game each player has four possible strategies. [Recall: A strategy 

specifies for every information set what the player who is called to move in this 

information set is going to do.] 

A possible strategy of the sender is (m2, m1), i.e. “Choose m2 if your type 

is t1 and choose m1 if your type is t2.” 

A possible strategy of the receiver is (a2, a1), i.e. “Choose a2 íf the 

sender has chosen m1 and choose a1 if the sender has chosen m2.” 

Furthermore, the receiver has to specify his beliefs about the type of the 

sender after receiving the senders message. Because there are two different 

information sets for the receiver, he must have two beliefs (µ1, µ2), where µi is 

the probability the receiver attaches to the event that the sender is of type t1 if 

he observes message mi. 


There are two types of pure strategy equilibria: 

Separating equilibria: Different types of the sender choose different 

messages. Thus, the receiver perfectly learns the type of the sender from 

observing his message. 

Pooling equilibria: All types of the sender choose the same message. 

Thus, the receiver learns nothing about the type of the sender from 

observing his message. 

If there are more than two types we can also have partially separating 

equilibria: Some messages are used by some types and not by others, but 

some messages are used by different types. Thus, there is some separation 

but it is not perfect. 

With mixed strategies there may also be hybrid equilibria: One type sends 

one message with probability one, the other type randomizes between the two 

messages. 


Example: 

 

1 

3 

 

4 

0 

 

2 

4 

 

0 

1 

✎☞ 

u 

µ l 

. .. . 

ℓ 

S 

t1 

r 

✎☞ 

u 

µr 

. 

d 

u 

R Nature 

........ 

1 

2 

R 

u 

. 

.. . . 

. 

1 − µ l 

d 

✍✌ 

ℓ t2 S 

r 1 − µr 

d 

✍✌ 

FIG. 3.40: Example of a Signaling Game 

1 

2 

d 

 

2 

1 

 

0 

0 

 

1 

0 

 

1 

2 


Let us go through the four possible pure strategy equilibrium candidates: 

1. Pooling on ℓ: If S chooses ℓ, R does not learn anything, so µl = 0.5. 

Thus, R will choose u. 

Is it indeed optimal for both types of S to choose ℓ? It is clearly optimal for 

type t2 who always gets 1 if he chooses r. For type t1 it is also optimal if R 

chooses d after observing r. This is optimal for the receiver if µr ≤ 2 

3 . 

Note that we cannot use Bayes’ rule after observing r to update beliefs. 

Thus, the following strategies and beliefs form a PBE: 

(ℓ, ℓ)(u, d), µl = 0.5, µr ≤ 2 

 

3 . 

2. Pooling on r: Cannot be a PBE. Why not? 

3. Separating (ℓ, r): Cannot be a PBE. Why not? 

4. Separating (r, ℓ): Can be sustained as a PBE. How? 


Multiple Equilibria and Equilibrium Refinements 

Signaling games have many equilibria. Thus, it is difficult to predict the 

outcome of these games. 

The problem of multiple equilibria is mainly due to the lack of restrictions on 

out of equilibrium beliefs. Several authors have further refined the equilibrium 

concept in order to rule out “implausible” out of equilibrium beliefs. 

All these refinements ask: 

Is there a type who may benefit from deviating? 

Are there any types who can never benefit from this deviation? 

The latter are less likely to deviate then the former. 


Consider the following variation of Selten’s Horse: 

1 

. .. . 

✎ 

✍ 

ℓ 

[µ] . 

. 

L 

r 

2 

M 

ℓ 

. . [1 − µ] 

r 

3 

1 

0 

0 

1 

0 

R 

 

0 

1 

 

2 

2 

☞ 

✌ 

FIG. 3.41: Example for the Dominance Argument 


Analysis 

The analysis of the normal form shows that there are two Nash Equilibria: 

(L, ℓ) and (R, r). 

There is no proper subgame, so both equilibria a subgame perfect. 

What are the corresponding PBE? 

◮ (L, ℓ, µ = 1) 

◮ (R, r, µ ≤ 0.5). Note: In this equilibrium player 2’s information set is reached 

with probability 0 on the equilibrium path. Thus, we cannot use Bayes’ rule 

to update µ. 

Is it really plausible to assume that µ ≤ 0.5. This assumes that a 

deviating player 1 is more likely to choose M than to choose choose L. 

But M is a strictly dominated strategy! Player 1 should never choose M, 

and player 2 should not believe that he did so. Thus, we should have 

µ = 1 which destroys the second equilibrium. 


This motivates the following condition: 

Criterion 3.1 (Dominance Criterion) 

If there are decision nodes in an information set off the equilibrium path that 

can be reached only if one of the players has chosen a strictly dominated 

strategy then these decision nodes have to be assigned probability 0, 

provided that there is at least one other decision node that can be reached 

without using a strictly dominated strategy. 

Remarks: 

1. The equilibrium (R, r, µ ≤ 0.5) does not satisfy Criterion 3.1. Thus, the 

Dominance Criterion rules out this equilibrium. 

2. If the payoff for player 1 after (L, ℓ) is changed to 1.5 rather than 3, then 

both L and M are strictly dominated. In this case Criterion 3.1 has no bite. 


The Dominance Criterion in a Signaling Game 

 

3 

2 

 

2 

0 

 

1 

0 

 

1 

1 

✎☞ 

✎☞ 

u 

[µ l ] 

. .. . 

L 

S 

t2 

R 

[µr ] 

. 

u 

d 

R Nature R 

u 

........ 

1 

2 

u 

. 

.. . . 

. 

[1 − µ l ] 

d 

✍✌ 

L t1 S 

R [1 − µr ] 

d 

✍✌ 

FIG. 3.42: Dominance in a Signaling Game 

1 

2 

d 

 

1 

0 

 

0 

1 

 

2 

1 

 

0 

0 


Analysis 

The following strategies and beliefs form a pooling equilibrium: 

[(L, L), (u, d), µl = 1 

2 , µr ≥ 1 

2 ] 

Because R is chosen with 0 probability in equilibrium, the belief µr can be 

chosen arbitrarily. 

But: For type t2 strategy R is strictly dominated, while it is not strictly 

dominated for type t1. 

Hence, Criterion 3.1 requires that µr = 0. The pooling equilibria described 

above do not satisfy Criterion 3.1. 


The Intuitive Criterion 

The Dominance Criterion rules out some equilibria but still not enough. The 

“Intuitive Criterion” that is due to Cho and Kreps (1987) goes much further. 

Consider the following signaling game: 

There are two types of senders: 

◮ a wimpy type: µ0 = 0.1 

◮ a surly type: 1 − µ0 = 0.9 

The receiver wants to fight the wimpy type but not the surly type. 

The sender can signal his type by the kind of breakfast he orders: Beer or 

Quiche. 

Payoffs are as follows: 

◮ Receiver gets 0 if he does not fight, 1 if he fights the wimpy type and -1 if he 

fights the surly type. 

◮ Sender: Both types prefer not to fight. The surly type prefers Beer for 

breakfast, the wimpy type prefers Quiche. The preferred breakfast gives an 

additional payoff of 1, not having to fight increases the payoff by 2. 


1 

1 

 

3 

0 

 

0 

−1 

 

2 

0 

✎☞ 

✎☞ 

f 

[µq] 

. .. . 

wimpy S 

Quiche t1 

Beer 

[µ b] 

. 

f 

n 

f 

0, 1 

R Nature R 

0, 9 

. 

.. . . 

. 

[1 − µq] Quiche 

n 

✍✌ 

t2 surly S 

Beer [1 − µ b] 

n 

✍✌ 

FIG. 3.43: The “Beer-Quiche” Game 

Consider the following pooling equilibria: 

 

(Quiche, Quiche), (n, f ), µq = 0.1, µb ≥ 1 

 

2 

n 

f 

 

0 

1 

 

2 

0 

 

1 

−1 

 

3 

0 


Analysis 

Along the equilibrium path both types eat Quiche and there is no fight. If 

the receiver observes “Beer”, however, he believes that the probability of 

the surly type is lower than 0.5 and fights. 

These equilibria satisfy the Dominance Criterion 3.1, because “Beer” is 

not strictly dominated for either type. 

Suppose that the sender after drinking his beer gives the following 

speech to the receiver: 

“Dear Receiver, the fact that I had Beer for breakfast should convince you 

that I am the surly type: 

◮ If I was the wimpy type, I could not improve my situation by drinking beer. 

Instead of my equilibrium payoff of 3 I would get either 0 or 2. 

◮ If I was the surly type, however, I could benefit. My payoff would increase 

from 2 to 3 if I manage to convince you that I am the surly type. 

Hence my deviation makes sense only if I am the surly type.” 

Would you be convinced by this speech? If so, you will also believe in the 

“Intuitive Criterion”. 


Criterion 3.2 (Intuitive Criterion) 

Consider a PBE (m ∗ , a ∗ ) of a signaling game. If the information set that is 

reached after message ˜m has been sent is off the equilibrium path, and if 

message ˜m is equilibrium dominated for type i, i.e., if 

U S (ti, m ∗ (ti), a ∗ (m ∗ (ti))) ≥ maxa∈AU S (ti, ˜m, a) , 

then the receiver should assign probability 0 to type i of the sender, provided 

that there exists at least one other type tj for whom ˜m is not equilibrium 

dominated. 


Remarks: 

1. The Intuitive Criterion compares the highest possible payoff when ˜m is 

sent not with the lowest possible payoff that type ti could have received 

by sending another message, but rather with the payoff that type ti gets in 

equilibrium. This is much more demanding than Criterion 3.1. 

2. Criterion 3.2 implies Criterion 3.1. Why? 

3. The Intuitive Criterion assumes some sort of hyperrationality: If there is a 

deviation from the equilibrium path, then the player should try to 

“rationally” explain this deviation. This is in stark contrast to the trembling 

hand story. Cho and Kreps show that every signaling game has at least 

one PBE satisfying the Intuitive Criterion. 

In the next chapter we will consider an application of signaling games to labor 

markets.

3. Game Theory - Economic Theory (Prof. Schmidt) - LMU

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