introduction to game theory - semantic scholar · 2016-01-22 · introduction to game theory 11.1...

Chapter 11

Introduction to Game Theory

11.1 Overview

All of our results in general equilibrium were based on two critical assumptions that consumers

and �rms take market conditions for granted when they decide what to buy and what to produce.

First, we assumed that all actors take market conditions as �xed. Second, we assumed that the

consumption and production decisions of each actor only a¤ect the utilities of other actors if they

change what those other actors consume. We relaxed these results by allowing for �rms to exert

market power in our analysis of monopoly �thus, the �rm considers the e¤ect of its actions on the

market conditions (e.g. the price). We also relaxed these results by allowing for externalities.

In the context of economics, game theory is the study of situations that do not meet these

two conditions and a game is any interaction where each person�s (or organization�s) actions a¤ect

the outcomes of others. Our �rst example was in our study of oligopoly. In both Cournot

and Bertrand competition, there are externalities becase the action by one �rm a¤ects the price-

quantity relationship (and ultimately the pro�ts) for the other �rm. Further, each �rm recognizes

that its own action in�uences the market price. These elements require a more new notion of

equilibrium that incorporates a description of how �rms would react to each other�s actions. In

this chapter, we develop a broader de�nition of games that go beyond the market framework of

oligopoly. Then in succeeding chapters, we consider speci�c applications of game theory under the

heading of "Information Economics".

273

C. Avery Notes on Microeconomic Theory ver: January 2010

11.2 Examples of Games

We begin with three well-known examples to provide a sense of the considerations that pervade

game theory. The �rst and best-known of these games is the Prisoners�Dilemma. In its original

form, the Prisoners�Dilemma is based on the negotiations between the police and two partners

who have been caught committing a crime together. The police have enough evidence to convict

both of them for a minor crime, but would like to get at least one criminal to inform on the other

so that they can convict at least one of them for a major crime. The police split up the criminals

and put them in di¤erent rooms where they cannot communicate. They then o¤er each the chance

for a lesser sentence if he will confess and testify against the other person.

The criminals, of course, have made a prior agreement with each other than neither will ever

testify against the other, but now that agreement is in jeopardy. Thus, the criminals must make

separate and simultaneous choices between two actions: 1) maintain the agreement and refuse the

o¤er from the police; 2) break the agreement and accept the o¤er from the police. We refer to

these two actions as the strategies "Cooperate" (C) and "Defect" (D), where cooperation refers to

the original agreement between the criminals (and does not meet cooperating with the police).

We de�ne a strategy to be a complete description of how a participant will act at di¤erent

points during the game. In the Prisoners�Dilemma, each person makes a single choice of action

with no knowledge of the other criminal�s choice. Therefore, each person�s choice of "Cooperate" or

"Defect" represents that person�s strategy for the game, since this single choice describes a person�s

actions for the entire game. We de�ne a simultaneous move game to be any game where all

participants choose a single action at the same time, without knowledge of the actions chosen by

the others. In a simultaneous move game, an action is equivalent to a strategy. All three games

discussed in this section are simultaneous move games, so we use the terms action and strategy

interchangeably in discussion of these examples.1

Table 1 shows the prison sentences the result for both criminals as a function of their strategies.

The fact that each person�s outcome depends on both her only strategy and on the other person�s

stragey underscores the interactive nature of the Prisoners�Dilemma as a game.

1An important caveat is that a strategy can involve a deliberate randomization - such as choosing one action 60%

of the time and another action 40% of the time. Such a strategy is known as a mixed strategy and is discussed

below.

274


CRIMINAL 2

Cooperate Defect

CRIMINAL 1 Cooperate2 year sentence for Criminal 1,

2 year sentence for Criminal 2

5 year sentence for Criminal 1,


Defect1 year sentence for Criminal 1,


4 year sentence for Criminal 1,


Table 1: Sentences in the Prisoners�Dilemma

Suppose that the Bernoulli utilities for these outcomes are the same for both criminals: u(1

year sentence) = 5, u(2 year sentence) = 4, u(4 year sentence) = 1, u(5 year sentence) = 0. Then

we can convert this table into a payo¤ matrix for the game, as shown in Table 2.

Cooperate Defect

Cooperate 4, 4 0, 5

Defect 5, 0 1, 1

Table 2: Payo¤s in the Prisoners�Dilemma

Table 2 is known as the Normal Form for the Prisoners�Dilemma that we have described

above.2 Each cell of the matrix lists a pair of utility outcomes corresponding to a particular set of

actions chosen by the two criminals. In the context of game theory, we describe each participant

as a "player". By convention, the rows correspond to player 1�s strategies and the columns

correspond to player 2�s strategies. To distinguish between the players, we assume that player 1 is

female and player 2 is male.

In each pair of payo¤s, player 1�s utility is listed �rst and player 2�s utility is listed second.

For example, if player 1 choose "Defect" and player 2 chooses "Cooperate", the outcome (5, 0)

indicates that player 1 receives utility of 5 and player 2 receives utility of 0.

In consumer theory, we assumed that each person made choices to maximize her own utility, as

represented by the solution to the Consumer Problem. In game theory, we make the same assumtion

but the utility maximization problem for each player is not as well de�ned as the Consumer Problem

was in consumer theory. For example, player 1�s utility maximizing strategy may depend on the

strategy of player 2 and further, player 1 may not be certain about what player 2 will do.

2The Normal Form is also known as the Matrix Form or the Strategic Form representation of a game.

275


The Prisoners�Dilemma is particularly straightforward to analyze because these complications

do not a¤ect the result of player 1�s utility maximization problem. If player 2 chooses "Cooper-

ate", then player 1 gets utility 5 by choosing "Defect" (1 year sentence) and utility 4 by choosing

"Cooperate" (2 year sentence). Therefore, player 1 gets a lesser sentence and higher utility from

"Defect" if player 2 chooses "Cooperate". In technical language, we say that player 1�s best

response to "Cooperate" is "Defect". Similarly, if player 2 chooses "Defect", then player 1

gets utility 1 by choosing "Defect" (4 year sentence) and utility 0 by choosing "Cooperate" (5 year

sentence). Therefore, player 1 gets a lesser sentence and higher utility from "Defect" if player 2

chooses "Defect". Once again, player 1�s best response to "Defect" is also "Defect".

Thus, for either action by player 2, player 1 has the same best response, "Defect", meaning

that "Defect" gives higher utility for player 1 than for player 2, regardless of what strategy player 2

chooses.3 When one strategy for a particular player is a strict best response to all combinations of

strategies for other players we say that it is a strictly dominant strategy for that player. (See

Section 11.4.1 for detailed analysis of dominant strategies.) In the Prisoners�Dilemma, "Defect" is

a strictly dominant strategy for each player, essentially eliminating the complexity of interaction in

the Prisoners�Dilemma. Although each player�s action a¤ects the other�s utility, it does not a¤ect

the other player�s maximizing decision. To maximize personal utility, each player should choose

"Defect" regardless of what the other player does.

We feel con�dent in predicting that the outcome of the Prisoners�Dilemma will be the dominant

strategy outcome, ("Defect", "Defect"), meaning that each player receives a four-year sentence and

utility of 1. Note that the choice of speci�c utility values for the four possible combinations of

strategies is not critical to the prediction that ("Defect", "Defect") will be the result of the game.

The choice to "Defect" reduces one�s own sentence from two years to one if the other player chooses

"Cooperate" or from �ve years to four if the other player chooses "Defect". Given this comparision,

"Defect" will dominate "Cooperate" so long as a longer sentence gives each player less utility than

a shorter sentence.

The perplexing element of the Prisoners�Dilemma is that the dominant strategy outcome is not

3Player 1�s strategy "Cooperate" also provides higher utility than "Defect" even if we allow player 2 to play a

"mixed strategy", choosing "Cooperate" with some probability p and "Defect" with probability 1 � p. Since all

utilities are Bernoulli values, each player will act to maximize expected utility when at least one player is playing a

mixed strategy. In this case, player 1�s expected utility from "Cooperate" is equal to 4p+0(1� p) or 4p, while player

1�s expected utility from "Defect" is 5p+ 1(1� p) = 4p+ 1. Since 4p+ 1 > 4p for any p, player 1�s best response to

any mixed strategy by player 2 is "Defect".

276


a Pareto optimum. If both players cooperate, they would receive shorter sentences than if they

both defect. This outcome emphasizes the importance of externalities in games. The choice to

defect reduces one�s own sentence by one year, but increases the sentence to the other player by

three years. Thus, each choice to defect produces an aggregate increase in two years in jail time

for the two players. If both players follow their dominant strategies and defect, the net result is

that they each get a sentence that is two years longer than if both cooperate. By chasing the small

gain of a one-year reduction in sentence, the players ensure themselves the maximum combined

sentence in total years served.

Prisoners�Dilemma in Oligopoly

The Prisoners�Dilemma can arise in a variety of more traditional economic situations. Consider

the numerical example of Cournot competition from the last chapter. With demand given by

p = 20�Q and constant marginal costs of 8 for each �rm, the Cournot equilibrium is for each �rm

to produce qi = 4, for a total quantity of 8. A monopolist would limit quantity to 6, so if the Cournot

�rms agreed to collude, they would each produce half the monopoly quantity or 3. Consider a

simultaneous move game where the players are the two �rms and they make simultaneous choices

to produce individual quantities of 3 ("Low") or 4 ("High"), as shown in Table 3. We assume that

each player�s utility is simply equal to her pro�t.

Low, q2 = 3 High, q2 = 4

Low, q1 = 3 18, 18 15, 20

High, q1 = 4 20, 15 16, 16

Table 3: Prisoners�Dilemma Version of Cournot Competition

If player 2 chooses "Low", then player 1 gets utility 20 from "High" and utility 18 from "Low".

If player 2 chooses "High", then player 1 gets utility 16 from "High" and utility 15 from "Low".

In either case, player 1 gets greater pro�t from "High" than from "Low", so "High" is a dominant

strategy for player 1. Similarly, "High" is a dominant strategy for player 2, so we predict that

each player will choose "High" and that each player will receive pro�t of 16.4

In this game, however, total pro�ts decline in the total quantity produced (assuming that total

quantity is greater than the monopoly quantity of 6). Here each �rm increases its own pro�ts by

4With a greater choice of strategies, Firm 1�s best response to "Low" by Firm 2 is q1 = 3.5, not q1 = 4. For

illustrative purposes, we restrict each �rm�s quantity to integer levels of production for this example.

277


deviating from the choice of "Low" to "High", but reduces the pro�ts to the other �rm by a larger

amount. So when both deviate from "Low" to "High", each ends up with a lower payo¤.5 This

game is just another version of the Prisoners�Dilemma, since the dominant strategy outcome from

("High" "High") is Pareto dominated by the outcome from ("Low", "Low").

Note that this particular Prisoners�Dilemma produces a bad outcome from the perspective of

the �rms, but a good outcome in terms of overall welfare. With a constant marginal cost of 8

and demand function p = 20�Q, a total quantity Q = 12 is socially e¢ cient and would result in

general equilibrium with many identical �rms. However, with only a small number of �rms, those

�rms exercise market power and limit production below the socially e¢ cient level. Comparing the

monopoly and duopoly outcomes, an increase in production from the monopoly level Qm = 6 to

the Cournot level Qc = 8 increases societal welfare (the sum of Consumer Surplus and Producer

Surplus) though it reduces the pro�ts to the �rms (i.e. Producer Surplus).

11.2.1 The Stag Hunt Game

A second well-known game, "Stag Hunt", is based loosely on a section from Rousseau�s "A Discourse

on Inequality". In this game, two hunters make simultaneous choices about what to hunt: "Stag"

or "Hare". Stag is too di¢ cult for a single hunter to catch, but hare is less valuable. Suppose

that if both hunters choose to hunt stag, they work together and catch one stag for total pro�ts of

$400 each. If only one of them chooses to hunt stag, she goes home empty-handed. Either can

choose to hunt hare on her own for a total pro�t of $100. Table 4 shows the Normal form for this

game, once again assuming that the utility for a hunter is equal to her total pro�t.

Stag Hare

Stag 400, 400 0, 100

Hare 100, 0 100, 100

Table 4: Stag Hunt

Neither player has a dominant strategy in this game. If player 2 chooses "Stag", then player 1

maximizes her utility by choosing "Stag", but if player 2 chooses "Hare", then player 1 maximizes

5Unlike the original Prisoners�Dilemma, the net e¤ect of switching from "Low" to "High" depends on the other

player�s strategy in this example. If player 2 chooses "Low", then a switch by player 1 from "Low" to "High" reduces

aggregate pro�ts by 1 unit. If player 2 chooses "High", then a switch by player 1 from "Low" to "High" reduces

aggregate pro�ts by 3 units.

278


utility by choosing "Hare". This is known as a coordination game because the players receive

higher payo¤s if they manage to coordinate their actions with both choosing "Hare" or with both

choosing "Stag" rather than having one choose "Hare" and the other choose "Stag". Thus analysis

of this game requires more sophistication than did analysis of the Prisoners�Dilemma.

If the players cannot communicate and have no way of coordinating their actions,6 then each

might assign probabilities to the other player�s choice. This is the same approach that we took in

our study of uncertainty, with the di¤erence that the players are assigning probabilities to human

choices rather than to actions that no one controls (often called "actions by nature"). If player 1

assigns probability p2 that player 2 will choose "Stag", then player 1�s expected utility from "Stag"

is 400 p2, while player 1�s expected utility from "Hare" is 100. So player 1 should choose "Stag"

if 400 p2 � 100 or p2 � 1=4. (Note that if p2 = 1=4 then player 1 gets expected utility 100

from "Stag" and expected utility 100 from "Hare". In this case, both of these strategies are best

responses for player 1.) Similarly, player 2 should choose "Stag" if 400 p1 � 100 or p1 � 1=4,

where p1 is the probability that player 1 will choose "Stag".

Two combinations of probabilities are consistent with the solutions to these two decision prob-

lems: 1) both choose "Stag", in which case p1 = p2 = 1 , or 2) both choose "Hare", in which case

p1 = p2 = 0.7 Each of these beliefs is self-con�rming. If each hunter is nearly certain that the

other will not show up to help hunt stag, then both will end up hunting hare. But if the hunters

have con�dence in each other, then both will show up to work together and they will receive the

larger payo¤ of 400 from catching stag rather than hare.

This game highlights an important richness and also a de�ciency of game theory. Although we

have yet to give a formal de�nition of equilibrium for games, it seems natural that ("Stag", "Stag")

and ("Hare", "Hare") will qualify as equilibria for any reasonable de�nition that we could create.

So abstract analysis cannot pinpoint a particular result of this game �even if we take a leap of

faith to believe that the world is in equilibrium. Instead, it is necessary to consider speci�c history

and institutional detail to understand which of two (or more) plausible outcomes is most likely in

a particular application.

6One way that they could coordinate their actions would be to hunt stag on warm days and hunt hare on cold days.

In that case, they would be able to coordinate their actions on each day even if they were not able to communicate

on that particular day. The use of an external factor to coordinate play is called a "Correlated Equilibrium" and

was �rst described by Robert Aumann in 1974.7 In fact, there is a third plausible outcome to this game where the players each choose a randomized strategy.

This outcome is shown in Figure 8.c and discussed in the context of mixed strategy equilibrium later in this chapter.

279


11.2.2 Rock, Paper, Scissors

A third well-known game is the children�s game known as "Rock, Paper, Scissors". In this game,

two players each choose simultaneously between these three di¤erent actions. If the players choose

di¤erent actions, then one wins and the other loses according to the following rules: "Rock" beats

"Scissors", "Scissors" beats "Paper", and "Paper" beats "Rock". If both players choose the same

action, then the game is a tie. Suppose that if the outcome is decisive, the loser pays a dollar to

the winner and that each player�s utility is simply his net payo¤ for the game.

Rock Paper Scissors

Rock 0, 0 -1, 1 1, -1

Paper 1, -1 0, 0 -1, 1

Scissors -1, 1 1, -1 0, 0

Table 5: Rock, Paper, Scissors

This game is known as a zero-sum game because if one player gains, then the other player

loses an equal amount. Much of the early analysis in game theory focused on applications to battle

strategies in war with the view that war was a zero-sum game.8 If neither side has a dominant

strategy in a zero-sum game, then there is no obvious outcome to the game. If either player is

too predictable, then the other player can take advantage. For example, in one episode of the

American television series, "The Simpsons", the Simpson children Bart and Lisa decide to write

a television script together. To settle the thorny question of �rst authorship, they agree to play

"Rock-Paper-Scissors". Bart�s immediate thought is "Good ol�rock. Nothing beats that." Lisa

is younger, but smarter, and her �rst thought is "Poor predictable Bart. He always takes Rock".

Naturally, Lisa wins the game.

Rock-Paper-Scissors illustrates another challenge for game theory. Even if we assume some

possibility of coordination between the players, there is no obvious outcome to the game. Thus,

any de�nition of equilibrium must also allow for the possibility of probabilities or randomized play.

11.3 Formal De�nition of Games

We now build a formal framework for game theory that incorporates all of these examples.8Thomas Schelling was one of the �rst to argue that wars are not always zero-sum games. For example, both

sides could gain from a well-constructed peace treaty, or both sides could lose in battles that kill civilians.

280


The basic components of a game are:

� A set of I players

� A description of the rules of the game. This consists of the sequence of possible decisions by

players, where each decision is a choice among a set of actions the player can take. The players

may also choose some or all of their actions simultaneously.

� The set of payo¤s to the players corresponding to each possible combination of strategies.

These payo¤s are in units of Bernoulli utilities so that the players maximize "Expected Utility" in

terms of known probabilities of di¤erent outcomes.

11.3.1 Strategies in Extensive and Normal Form Representations

The examples above are all simultaneous move games and can be easily represented in the Normal

Form, as described above. But games may also be sequential, with one player moving �rst and the

other player moving second. More complex games may also involve a series of moves in some order

as well as moves by nature. These games can be represented in diagrammatic form known as a

game tree or as the Extensive Form representation. Figure 1 is the extensive form representation

for a game where player 1 moves �rst and chooses "Up" or "Down", then player 2 observes player

1�s choice and responds by choosing "Left" or "Right".

The extensive form is often called a game tree because it represents each possible sequence of

play as one of many branches in a tree. Each point in a game tree where one player is called upon

to act is called a decision node. In Figure 1, there are three decision nodes: node A where player

1�s starts the game by choosing an action and nodes B and C, where player 2 may be called upon

to move. Even though player 2 only acts once, the game tree includes two separate decision nodes

for player 2 because player 2�s action follows player 1�s action and it is not known in advance what

player 1 will do.

A strategy for a player is a complete contingent plan of actions for the entire game for that

player. For the game in Figure 11.1, player 1�s strategy must specify her action at node A, while

player 2�s strategy must specify his action at node B and his action on node C. In player 2�s

strategy, each action that is speci�ed for a given node is contingent upon that node being reached

in the game.9

9The de�nition of a strategy as a contingent plan of action includes a philosophical assumption that each player

can anticipate every contingency in the game and identify how he would respond in that contingency. In practice,

some people may say that they cannot form a plan of action for some contingencies prior to the start of the game,

281


1

1:pdf

Figure 11.1: An Extensive Form Game

A pure strategy for a player requires that player to chose an action with probability 1 at each

decision node (i.e. "for certain"). Thus, there are two pure strategies for player 1 in Figure 1:

"Up" and "Down". By contrast, a pure strategy for player 2 consists of one action chosen with

probability 1 at node B and a separate action chosen with probability 1 at node C. There are two

possible options at each decision node for player 2, so there are 2 x 2 = 4 possible pure strategies

for player 2. We can represent player 2�s possible strategies in an ordered pair, where the �rst

entry in the pair represents the action that player 2 will take at Node B if player 1 moves "Up"

and the second entry represents the action that player 2 will take at Node C if player 1 moves

"Down". The four possible pure strategies for player 2 for the game in Figure 1 are "Left, Left",

"Left, Right", "Right, Left", "Right, Right".

Table 6a shows the actions taken by the players in the extensive form game in Figure 1 as

a function of their pure strategies. Table 6a would be a normal form representation of this

game, except that it lists actions rather than payo¤s in each cell. Table 6b is the normal form

representation for this game.

particularly shocking ones. (e.g. "I can�t tell if I�ll want to take revenge if you betray me, because it�s unthinkable

to me that you would betray me.") A separate practical di¢ culty is that in complicated games (e.g. chess), it may

be impossible for each player to contemplate all possible contingencies.

282


Left, Left Left, Right Right, Left Right, Right

Up Up, Left Up, Left Up, Right Up, Right

Down Down, Left Down, Right Down, Left Down, Right

Table 6a: Actions Chosen by Players in Figure 1 as a Function of Their Strategies

Left, Left Left, Right Right, Left Right, Right

Up 1, 4 1, 4 4, 1 4, 1

Down 3, 2 2, 3 3, 2 2, 3

Table 6b: Normal Form Representation for the Extensive Form Game in Figure 1

In some cases, a player may be called upon to act, but may not know precisely which node in

the tree has been reached. For example, if player 2 does not observe player 1�s action at node A

in Figure 1, then he would not know whether he was making a choice at node B or at node C. We

describe this possibility with the use of an information set, which is a set of decision nodes for a

particular player who cannot distinguish among these nodes at the time she must choose an action.

Note that the set of possible actions must be the same at each decision node in an information set,

for otherwise, the player could use the set of possible actions to distinguish among at least some of

the nodes in the information set.

Figure 2 shows the extensive form representation for a simultaneous move game where player 1

chooses either "Up" or "Down" and player 2 chooses either "Left" or "Right". Figure 2 is identical

to Figure 1 except for the addition of the dotted line connecting nodes B and C. This dotted line

indicates that these two nodes are in the same information set, meaning that player 2 cannot tell

if he is at node B or at node C when he is called upon to move.

From a strategic standpoint, information is more important that the exact timing of decisions.

If player 1 chooses a strategy at 11:30 and player 2 chooses a strategy at 11:45, then the game is

literally sequential. But if player 2 does not observe player 1�s choice prior to 11:45, then from

player 2�s perspective, the game might as well be simultaneous since player 2 has exactly the same

information (none) about player 1�s choice at 11:45 as when the game started. For this reason, we

use the same extensive form representation for a simultaneous move game and a sequential move

game where player 1 moves before player 2 but player 2 does not observe player 1�s move.

283


2

2:pdf

Figure 11.2: Extensive Form for a Simultaneous Move Game

11.3.2 Mixed and Behavioral Strategies

A mixed strategy is a randomization, where a player puts positive probability on at least two of

her strategies. In a game where the players have just two strategies each, such as Stag Hunt, any

mixed strategy can be described by a single probability: if player 1 plays "Stag" with probability

p, then she must play "Hare" with the remaining probability 1 � p. In more complicated games,

a mixed strategy is a vector of probabilities, where the probabilities sum to 1 and each element

of the vector represents the probability of playing a particular pure strategy in the game. For

example, in Table 6b, a mixed strategy for player 2 is a vector of four probabilities (p1;p2;p3;p4)

where p1 + p2 + p3 + p4 = 1 and p1 represents the probability of pure strategy ("Left", "Left"),

p2 represents the probability of pure strategy ("Left", "Right"), p3 represents the probability of

pure strategy ("Right", "Left"), and �nally p4 represents the probability of pure strategy ("Right",

"Right").10

10Since the sum of probabilities for player 2�s pure strategies must add to 1, the values of any three of the strategies

are su¢ cient to determine the fourth probability. For this reason, we say that there are three degrees of freedom in

determining player 2�s mixed strategy among four possible pure strategies.

284


285


Another way of representing randomization for player 2 is to allow player 2 to randomize

between strategies at each node. A node-by-node list of randomizations for a player is known as

a behavioral strategy. Returning to the game in Figure 11.1, a behavioral strategy for player

2 consists of two probabilities pB and pC , where pB is the probability that player 2 chooses "Left"

at node B and pC is the probability that player 2 chooses "Left" at node C, as shown in Figure

11.3. We do not emphasize the distinction between behavioral and mixed strategies, because it

is possible to convert a behavioral strategy into a mixed strategy with equivalent probablities for

player 2�s action at each node. To convert a behavioral strategy in Figure 11.3 to a mixed strategy,

assume that player 2 actually chooses an action at both node B and node C even though only one of

those nodes can be reached in the game. Further assume these choices are statistically independent

so that P(Left, Left) = pB � pC .11 Then the behavioral strategy (pB; pC) corresponds to the

mixed strategy (p1;p2;p3;p4) where p1 = pB � pC , p1 = pB � (1 � pC); p3 = (1-pB) � pC ; p4 =

(1-pB) � (1� pC). Note that P (Left j Up) = p1+ p2 = pB � pC+ pB � (1� pC) = pB and similarly

that P (Left j Down) = pC . Thus, this mixed strategy produces the same probabilities for player

2�s actions as the behavioral strategy, so the two are truly equivalent.

Since it is generally easier to analyze the best response to a mixed strategy than the best response

to a behavioral strategy, and there is an equivalence between mixed and behavioral strategies, we

will concentrate on mixed strategies in further discussion.

11.3.3 Equivalence of Extensive and Normal Form Representations

Our method for converting the extensive form game in Figure 1 to the normal form game in Table

6b can be generalized into an algorithm to convert any �nite extensive form game into a normal

form game. (Any game with a �nite number of players and a �nite number of actions at each

information set is a �nite game.). To identify the pure strategies for each player, �rst identify the

number of information sets at which that player could be called upon to move. Then create a set of

strategies where each strategy is a vector containing an entry with an action for each information

set. For example, if there are �ve information sets where player 1 could be called upon to move

11 If we assume that there is statistical correlation between the resolution of uncertainty for player 2�s actions

at nodes B and C, we would still be able to convert player 2�s behavioral strategy into a mixed strategy, but we

would identify a di¤erent mixed strategy than when these actions are statistically independent. However, this mixed

strategy would have the same probability of each action at each node and so player 1�s best response would be the

same in both cases.

286


and three possible actions at each information set, then each strategy would be a vector with �ve

entries and there would be a total of 35 pure strategies for player 1. Naturally, as the game becomes

more complex, the number of possible strategies for player 1 grows exponentially. It may not be

practical to represent a game with many information sets for each player in the normal form, but

it is at least theoretically possible to do so.

In some cases, our procedure for converting an extensive form game to a normal form game

may appear to include some redundant strategies. For example, Figure 4 shows a case where

player 1 may be called upon to move twice, but only if his �rst move is "Continue". There are

four possible strategies for player 1: ("Continue", "Up"), ("Continue", "Down"), ("Stop", "Up"),

("Stop", "Down"), but both ("Stop", "Up") and ("Stop", "Down") both end the game immediately.

Despite the fact that the strategies ("Stop", "Up") and ("Stop", "Down) may seem equivalent,

it is important to include both of them in the analysis of the game, as indicated by the following

logic.

287


Player 1 can end the game immediately by choosing "Stop" at node A for a payo¤ of (2, 0).

Player 1�s payo¤ from "Continue" at node A depends on player 2�s (anticipated) action at node B.

If player 2 would choose "Left" at node B, then player 1�s would get a payo¤ of 5 from ("Continue",

"Up") and "3 from ("Continue", "Down"). Either of these outcomes gives player 1 a higher payo¤

than if player 1 chooses "Stop" at node A. But if player 2 would choose "Right" at node B, then

player 1 does better by choosing "Stop" than "Continue" at node A.

To this point, we have not distinguished in this analysis between ("Stop", "Up") and ("Stop",

"Right"). This distinction is paramount for player 2�s choice of action at node B. If player 2 is

called upon to move at node B, he gets a higher payo¤ from "Left" if player 1 would choose "Down"

at node C and a higher payo¤ from "Right" if player 1 would choose "Up" at node C. That is,

the di¤erence between ("Stop", "Up") and ("Stop", "Down") determines whether player 2 should

choose "Continue" or "Stop" at node B. If we merge player 1�s strategies ("Stop", "Up") and

("Stop", "Right") into the single strategy "Stop", then it would be impossible to identify player 2�s

utility maximizing strategy at node B in response to player 1�s strategy, "Stop". Then in turn, it

would be impossible for player 1 to determine if "Stop" is her utility maximizing strategy at node

A if player 2�s action at node B is not speci�ed. For this reason, we include both ("Stop", "Up")

and ("Stop", "Down") as distinct pure strategies for player 1 in the analysis of the extensive form

for this game.

However, a di¤erent convention applies to the representation of this game in the normal form.

Both strategies ("Stop", "Up") and ("Stop", "Down") yield the same outcome (2, 0), regardless of

player 2�s strategy, as shown in Table 6c.

Left Right

Continue, Up 5, 1 1, 2

Continue, Down 3, 3 1, 2

Stop, Up 2, 0 2, 0

Stop, Down 2, 0 2, 0Table 6c: Full Normal Form for the Extensive Form Game in Figure 4

288


In the normal form, there is no reason or way to distinguish between ("Stop, Up") and ("Stop,

Down"). Therefore, it is appropriate to combine these two strategies in the reduced normal

form of the game, as shown in Table 6d. But this distinction in the conventional representation of

the game �four pure strategies for player 1 in the extensive form of the game, but only three pure

strategies for player 1 in the reduced normal form of the game - suggests that a subtle di¤erence

between the extensive and normal form representations of games where at least one player acts more

than once. This distinction in�uences the analysis of dynamic games - as discussed later in these

notes under the headings of Subgame Perfect equilibrium and Perfect Bayesian equilibrium.

Left Right

Continue, Up 5, 1 1, 2

Continue, Down 3, 3 1, 2

Stop 2, 0 2, 0Table 6d: Reduced Normal Form for the Extensive Form Game in Figure 4

Converting a Normal Form Game into an Extensive Form Game

There are multiple ways to convert a normal form representation into an extensive form game with

the same set of strategies and payo¤s. For example, the extensive form games in both Figure 11.1

and Figure11 5 correspond to the same normal form game in Table 6b.

The simplest method for converting a normal form to an extensive form game is simply to

condense all of the actions for each player into a single choice of moves - including all possible

pure strategies as separate actions at a single information set for that player. This is an awkward

choice that meets the literal de�nition of the extensive form, but takes the spirit of a normal

form (simultaneous move) game. In addition, brute force conversion from the normal form to the

extensive form may suppress important strategic considerations. For example, the representation

in Figure 1 highlights player 2�s ability to observe player 1�s move and to respond optimally to it,

but the representation in Figure 5 obscures player 2�s strategic advantage in the game.

The equivalence of the extensive form games in Figures 1 and 5 is based on the assumption

that the players make complete contingent plans prior to the start of the game. Even with this

assumption, it is natural to prefer the representation in Figure 1, which accurately depicts the

series of moves in the game, to the representation in Figure 5 which suppresses this information.

For our purpose of exposition here, it is only important that it is possible to represent a normal

form game in the extensive form and vice versa. Now that we have established this possibility, we

289


4

5:pdf

Figure 11.3: Second Extensive Form for Simultaneous Move Game

have the freedom to represent any game in either form in further discussion.

11.3.4 Moves by Nature

A game may include elements of uncertainty that are common to all players, and some or all of these

uncertainties may be resolved during the game. For example, in a game that involves negotiation

between a venture capitalist and an entrepreneur, there might be a preliminary report about the

entrepreneur�s pro�tability during the course of the negotiation. In game theory, we describe the

resolution of uncertainty as a move by nature, where nature is modeled as a player that acts

probabilistically rather than to achieve a particular objective.

In the simplest case of uncertainty caused by a move by nature, two players do not know which

of two simultaneous move games that they are playing. For example, players 1 and 2 may not

know whether they are playing the game in Table 7a or the game in Table 7b.

290


Left, Right

Up 4, 0 1, 2

Down 3, 4 2, 1

Table 7a: One of Two Possible GamesLeft, Right

Up 2, 1 3, 4

Down 1, 2 4, 0

Table 7b: The Second of Two Possible Games

Suppose that the players choose their strategies simultaneously, and then only learn which of

the two games they were actually playing when they learn their payo¤s. This uncertainty can be

represented in the extensive form as a move by nature that takes place after the moves by each

player. It can also be represented in the extensive form as a move by nature that takes place prior

to the moves by the two players.

So long as we assume that the players know the probability, �, that they are playing game 1

(so that they are playing game 2 with probability 1 � �), then we can incorporate the move by

nature into their normal form payo¤s. If the players play ("Up", "Left") for example, they receive

payo¤s of (4,0) from Game 1 with probability � and they receive a payo¤ of (2; 1) from Game 2

with probability 1� �. Since their payo¤s in each game are assumed to be Bernoulli utilities, the

players act to maximize their expected utilities when uncertainties (either due to moves by nature

or mixed strategies by other players) are involved. Combining the two possible payo¤s for ("Up",

"Left"), the expected utilities for the two players are are (4� + 2(1� �); 1(1� �)) or (2+2�, 1-�).

Table 7c represents the normal form for the probability weighted combination of these two possible

games.

Left, Right

Up 2+2�, 1-� 3-2�; 4 + 2�

Down 1+2�, 2+2� 4-2�, �

Table 7c: Normal Form Incorporating a Move by Nature into Expected Utility

In cases where there are moves by nature, but none of the players observe those moves, then

uncertainty about nature�s actions can simply be incorporated into the (expected) payo¤s of the

game, as shown by example in Table 7c above. When some, but not all players observe one of

291


nature�s moves, then the game involves asymmetric information. We discuss these games separately

in the section on Bayes-Nash equilibrium below and subsequently in the chapters on Information

Economics.

11.4 Solution Concepts in Game Theory

Now that we have a method for representing games in the normal and extensive forms, we would

like to be able to make robust predictions about how economically rational actors might play those

games. We call each rule for predicting the outcome of a game a solution concept. Almost all

solution concepts are based on the idea of optimal action / best response by each player; solution

concepts di¤er by requiring more or less restrictive assumptions about what other players will do.

11.4.1 Iterative Solution Methods

The simplest concept is that of Dominance, which we explored in the context of the Prisoners�

Dilemma above. We say that strategy A strictly dominates strategy B if strategy A gives a strictly

higher payo¤ than does strategy B for each possible combinations of strategies by other players.

We say that strategy A weakly dominates strategy B if strategy A gives at least as high a payo¤

as strategy B for each possible combinations of strategies by other players and strategy A gives

a strictly higher payo¤ than strategy B for some possible combination(s) of strategies by other

players.12

Dominance generalizes to a procedure of elimination of strategies by iterated strict dom-

inance. Roughly, iterated strict dominance says that if A is preferred to B in all but a ridiculous

set of circumstances, then we should select A over B. The de�nition of the ridiculous set of circum-

stances are those in which other players select clearly faulty strategies. In Table 8a below, Middle

dominates Right for player 2, but there are no other dominated strategies.

Left Middle Right

Up 4,-4 1, 4 0, -3

Down 5, 3 2, 2 -1, -2

Table 8a: A Normal Form Game with One Dominant Strategy12 If strategies A and B give the same payo¤ for each possible combination of strategies for other players, then these

strategies are equivalent. We could say that A and B weakly dominate each other in this case.

292


As this example illustrates, Dominance emphasizes the strategy that is dominated. In this

case, we can eliminate "Right" since player 2 would always get higher utility with "Middle" than

with "Right". However, the fact that "Middle" rather than "Left" is the strategy that dominates

"Right" does not necessarily mean that there is any reason to prefer "Middle" to "Left" in the

choice of the two remaining strategies for player 2. Instead, we proceed by eliminating "Right"

and then examining the resulting 2x2 game shown in Table 8b for dominant strategies.

Left Middle

Up 4, -4 1, 4

Down 5, 3 2, 2

Table 8b: The Normal Form Game with Removal of One Dominated Strategy

After the elimination of Right, Down dominates Up. So we can eliminate Up. This leaves only

one strategy for player 1, Down, and two strategies for player 2, as shown in Table 8c.

Left Middle

Down 5, 3 2, 2

Table 8c: The Normal Form Game with Removal of Another Dominated Strategy

Comparing the payo¤s for player 2, Left gives the higher payo¤ and is the best response to

Down �meaning that Left dominates Middle after the elimination of Up for player 1. Thus, we can

eliminate Middle, leaving only one strategy for each player, Down for player 1 and Left for player

2, as shown in Table 8d.

Left

Down 5, 3

Table 8d The Normal Form Game with Removal of Another Dominated Strategy

With only one strategy left for each player, we predict the outcome of the game to be (Down,

Left). If iterated dominance yields an exact prediction for each player�s strategy, then we say that

the game is dominance solvable. (Note that any game with a dominant strategy for each player,

293


such as the Prisoners�Dilemma, is de�ned to be solvable by iterated dominance even though the

dominance procedure does not have to be iterated to produce this solution.)

Intuitively, when iterated dominance yields a speci�c prediction about the outcome of a game,

each player is playing a best response to all possible combinations of strategies by other players,

excluding implausible strategies � an implausible strategy means any strategy excluded by the

iterated dominance procedure. For example, examining Table 8a, player 1 should only consider

"Up" if player 2 is expected to play "Right", but "Right" is a dominated strategy. This logic,

which corresponds to the �rst two steps in iterated elimination of dominated strategies, provides a

strong case for the prediction of ("Down", "Left").

Dominance is the most stringent criterion in common use in game theory for predicting the

outcome of a game. The downfall to this criterion is that many games, such as "Stag Hunt"

and "Rock-Paper-Scissors", are not dominance solvable. In general, if iterated (strict) dominance

identi�es a solution to a game, we tend to believe that sophisticated players will follow the dominant

strategy outcome �particularly if only one or two stages of elimination are required to identify

the solution. Even though the dominant strategy outcome in the Prisoners�Dilemma involves

regrettably little cooperation, we still believe that both sides will defect in any Prisoners�Dilemma.

However, there are some extreme cases where the a dominant strategy outcome is unlikely to be

played. Table 8c illustrates one such case.

Left Middle Right

Up 4,-4 1, 4 0, -3

Down 5, 3 2, 2 -1,000,000, -2

Table 8e: A Game where Players May Deviate from Dominance Predictions

Table 8e repeats the game from Table 8a with a single change in payo¤s: player 1�s payo¤

from ("Down", "Right") has been changed from -1 to -1,000,000. This change in payo¤s does not

a¤ect the results of iterated dominance. As before, "Middle" dominates "Right", then "Down"

dominates "Up" after the elimination of "Right", and �nally "Left" dominates "Middle" after the

elimination of "Right" and "Up". The iterated dominance solution to this game remains ("Down",

"Left"). But now the possibility of payo¤ -1,000,000 for player 1 weakens the prediction that the

players will play ("Down", "Left").

The argument in favor of "Down" is that with these payo¤s, player 2 would not play "Right".

Assuming that player 2 would not play "Right", player 1 increases her payo¤ by 1 with the choice

294


of "Down" instead of "Up". However, if there is even a small probability that player 2 would play

"Right", the possibility of a payo¤ of -1,000,000 from ("Down", "Right"), could in�uence player 1

to choose "Up". For example, if P(Right) = 1 / 1,000,000, then player 1 would get very slightly

higher expected utility from "Down" instead of "Up".

In practice, we suspect that very few people would choose "Down" as player 1 in this game.

There are at least three reasons that player 2 could play "Right": 1) this assessment of player 2�s

payo¤s is incorrect and in fact, player 2 would bene�t rather than lose by playing "Right"; 2)

player 2 could make a mistake and play "Right" even with the knowledge that this is a dominated

strategy; 3) player 2 might derive positive utility from imposing a disastrous result on player 1.

Considering any one of these reasons would likely be su¢ cient for player 1 to avoid playing "Down".

This example highlights several assumptions that are necessary to translate the reasoning from

iterated dominance into a prescription for play. Speci�cally, iterated dominance requires each

player not only to make sophisticated calculations, but also to assume that other players will make

those same sophisticated calculations. This requirement is known as common knowledge of

economic rationality. It is not a trivial assumption, but it is standard in game theory. In

addition, iterated dominance requires common knowledge of the payo¤s in the game - each

player must be certain of the payo¤s for other players in the game.

Each additional step to eliminate strategies requires one more level of certainty in terms of

common knowledge. For player 1 to choose "Up" in place of "Down", she must be certain of player

2�s payo¤s and also that player 2 is sophisticated enough to complete one round of dominance

reasoning to eliminate "Down". For player 2 then to eliminate "Middle", he must be certain of

player 1�s payo¤s, that player 1 knows player 2�s payo¤s and will conclude that player 2 to eliminate

"Right", and that player 1 will be sophisticated enough to choose "Down" after concluding that

player 2 will not play "Right". Common knowledge is often represented as a chain. Here from

player 2�s perspective, the choice to eliminate "Middle" is predicated on reasoning of the form, "I

know that you know my payo¤s and that I am economically rational." The assumption of common

knowledge in game theory allows for reasoning of this form of any length, thus allowing any number

of steps necessary to identify a solution using the iterated dominance procedure. However, it is

clear that as we add more stages of reasoning, the common knowledge assumption becomes more

burdensome. At least for complicated applications of iterated dominance, it may be desirable to

ask oneself if the prediction relies too heavily on common knowledge to be believable in practice.

295


11.4.2 Nash Equilibrium

The greatest defect of the dominant strategy criterion for solving games is that many games cannot

be solved by iterated dominance. The more general concept of Nash equilibrium applies to games

that do not have dominant strategy outcomes. A Nash equilibrium is a list of strategies for the

players in a game such that each player�s strategy is a best response to the strategies of the other

players.

If a game can be solved by iterated dominance, then the solution is a Nash equilibrium.13 So,

Nash equilibrium is a strictly weaker requirement than iterated dominance. We consider two types

of Nash equilibrium outcomes in turn.

Pure Strategy Nash Equilibrium

In a pure strategy Nash equilibrium, both players select a strategy with probability 1 (i.e. no mixed

/ randomized strategies). It is usually easiest to analyze a game in the normal form to �nd pure

strategy Nash equilibria. We use the game in Table 9a to illustrate how to �nd a pure strategy

Nash equilibrium.

Left Middle Right

Up 2, 5 3, 4 7, 8

Down 1, 6 6, 7 4, 2

Table 9a: A Normal Form Game

One way to identify pure strategy Nash equilibria when neither player has a dominant strategy is

to "Guess and Verify". There are six possible combinations of pure strategies, so it is possible

to check combination individually to see if it produces a Nash equilibrium. For example, the

combination ("Up", "Left") is not a Nash equilibrium because player 2�s best response to "Up" is

"Right". That is, in the combination ("Up", "Left"), player 1 is playing a best response to player

2�s strategy, but player 2 is not playing a best response to player 1�s strategy. Exhaustive use of

13We don�t prove this formally here, but the reasoning is straightforward. If a game can be solved by iterated

dominance, then the solution is clearly a Nash equilibrium among all strategies that remain after the removal of

dominated strategies at early stages of reasoning. So the only way that the iterated dominance solution could fail to

be a Nash equilibrium would be if one player could improve her utility by switching to a strategy that was eliminated

in an earlier stage of reasoning. But if this is possible, that strategy should not have been removed at any earlier

stage - meaning that in fact, the iterated dominance solution must in fact be a Nash equilibrium.

296


the "guess and verify" method in this example will identify two pure strategy Nash equilibria: (Up,

Right) and (Down, Middle).

Fortunately, there is a simpler method than "Guess and Verify" for identifying pure strategy

Nash equilibria in a normal form game: highlight the best responses for each player to the pure

strategies of the other player(s). Table 9b identi�es each best response by putting a box around

the payo¤s that correspond to a each player�s best reponses to the opponent�s possible strategies.

For example, "Up" is player 1�s best response to "Left", so Table 8b highlights player 1�s payo¤ of

2 in the cell corresponding to ("Up", "Left").

Left Middle Right

Up 2 , 5 3, 4 7 , 8

Down 1, 6 6 , 7 4, 2

Table 9b: Identifying Pure Strategy Nash Equilibria

The highlighting of best response payo¤s reduces the work of "Guess and Verify" to a single

visual inspection of the normal form game. Any cell with both payo¤s highlighted is a pure strategy

Nash equilibrium. In Table 9b, both of the payo¤s in the cells corresponding to ("Down", "Middle")

and ("Up", "Right") are highlighted. These are Nash equilibria because the highlighting indicates

that each player�s pure strategy is a best response to the other�s pure strategy. Any other cell,

where at least one of the payo¤s is not highlighted, is not a Nash equilibrium because (at least) one

player�s pure strategy is not a best response to the other player�s pure strategy. For example, "Up"

is player 1�s best response to "Left" because player 1�s payo¤ is highlighted in this cell. However,

player 2�s best response to "Up" is not left �instead it is "Right" - so ("Up", "Left") cannot be a

Nash equilibrium.

Mixed Strategy Nash Equilibrium

Many games such as "Rock-Paper-Scissors" have no pure strategy Nash equilibrium. Table 10a

shows a simpler version of "Rock-Paper-Scissors" known as "Matching Pennies," where two players

each select a penny and choose one side of it, "Heads" or "Tails". If the players match their

choices, then player 1 wins both pennies, while if one player chooses "Heads" and the other player

chooses "Tails", then player 2 wins both pennies. Thus, for any combination of pure strategies,

one player is exploiting the other, meaning that the losing player is not playing a best response to

the other�s strategy. For this reason, there can be no pure strategy Nash equilibrium in this game.

297


Heads Tails

Heads 1, -1 -1, 1

Tails -1, 1 1, -1

Table 10a: Normal Form Representation for Matching Pennies

Table 10b shows the normal form representation for this game with the best responses for each

player highlighted. This veri�es the intution provided above that there cannot be a pure strategy

Nash equilibrium for this game. In each cell, one player, the winner, is playing a best response to

the other and one player, the loser, is not doing so.

Heads Tails

Heads 1 , -1 -1, 1

Tails -1, 1 1 , -1

Table 10b: Matching Pennies with Best Responses Highlighted

But there is a Nash equilibrium in mixed strategies for both players in Matching Pennies.

Suppose that player 1 plays a mixed strategy that selects "Heads" with probability p and "Tails"

with probability 1� p. Player 2�s expected utility for "Heads" as a function of p is p(�1) + (1�

p) (1) = 1�2p, while Player 2�s expected utility for "Tails" as a function of p is p(1)+ (1�p) (�1) =

2p� 1: Comparing these two expected payo¤s, we �nd that player 2 gets a higher expected payo¤

from "Heads" than from "Tails" if p < 1=2 and a higher expected payo¤ from "Tails" than from

"Heads" if p > 1=2.

Intuitively, if p > 1=2, then player 1 tends to play "Heads" more often than "Tails". Since player

2 wishes to choose the opposite strategy from player 1, player 2 can exploit player 1�s tendency by

selecting a pure strategy of "Tails" if p > 1=2, Similarly, if p < 1=2, then player 1 tends to play

"Tails" more often than "Heads" and so player 2 should select a pure strategy of "Heads". The

one instance where player 2 cannot exploit player 1 is when p = 1=2. In this case, each of player

2�s strategies yields an expected payo¤ of 0, so player 2 is indi¤erent between her two possible pure

strategies. In addition, when p = 1=2, any mixed strategy for player 2 also gives an expected payo¤

of 0 and is a best response to player 1�s strategy. A similar analysis from the perspective of player

1 indicates that player 1 has a strict best response of "Heads" if q > 1=2, a strict best response

of "Tails" if q < 1=2, and that player 1 is indi¤erent between "Heads" and "Tails" if q = 1=2.

298


Further, each pure or mixed strategy for player 1 gives an expected payo¤ of 0 in this game when

q = 1=2:

Combining these observations, there is a unique mixed strategy equilibrium of Matching Pennies

where player 1 randomizes between "Heads" and "Tails" with probability p = 1=2 and player

2 randomizes between "Heads" and "Tails" with probability q = 1=2. In this mixed strategy

equilibrium, each player is indi¤erent between the two possible pure strategies, and each player

received expected utility of 0.

A General Method for Finding Mixed Strategy Equilibrium in 2x2 Games Generalizing

from our analysis of Matching Pennies, we can identify mixed strategy equilibria of all 2x2 games

�i.e. games with two players and two pure strategies each. Table 11 lists payo¤s in a general 2x2

game as variables ai; bi; ci;and di.

Left Right

Up a1; a2 b1; b2

Down c1; c2 d1; d2Table 11: Finding Mixed Strategy Equilibria in a General 2x2 Game

If player 2 plays "Left" with probability q and "Right" with probability 1-q, then player 1�s

expected payo¤ from "Up" is qa1 + (1 � q) b1 and player 1�s expected payo¤ from "Down" is

qc1 + (1 � q) d1. Setting these two equal, we �nd that player 1 is indi¤erent between "Up" and

"Down" if qa1 + (1� q) b1 = qc1 + (1� q) d1, which is satis�ed for

q� = (d1 � b1)=(a1 � c1 + d1 � b1).

Similarly, player 2 is indi¤erent between "Left" and "Right" if player 1 plays "Up" with prob-

ability p� as given by the following equation:

p� = (d2 � c2)=(a2 � b2 + d2 � c2).

For example, in the "Stag Hunt" game, a1 = 400; b1 = 0; c1 = 100; d1 = 100; a2 = 400; b2 =

100; c2 = 0; d2 = 100. Substituting these values into the formulas above, there is a mixed strategy

equilibrium with p� = 1=4 and q� = 1=4.

Note that this mechanical analysis produces unique candidate probabilities p� and q� for a

mixed strategy as a function of parameters (a1; a2; b1; b2; c1; c2; d1; d2). Therefore, there is at most

299


one mixed strategy Nash equilibrium of any 2x2 game with the speci�c mixed strategy probabilities

p� for player 1 and q� for player 2.14 However, if the formula for either p� or q� produces a number

greater than 1 or less than 0, then the formula does not produce a valid probability, indicating that

there is no mixed strategy equilibrium for a given game.

Proposition 1 There is a unique mixed strategy equilibrium of any 2x2 game if and only if neither

player has a (weakly) dominated strategy.

If player 1 does not have a (weakly) dominated strategy then either

1) a1 > c1 and d1 > b1 so that "Up" is a strict best response to "Left" and "Down" is a strict

best response to "Right" or

2) a1 < c1 and d1 < b1 so that "Down" is a strict best response to "Left" and "Up" is a strict

best response to "Right".

In either case, 0 < q� < 1 based on the equation for q� above. Similarly, if player 2 does not

have a (weakly) dominated strategy, then 0 < p� < 1 based on the equation for q� above. That

is, the formulas for p� and q� produce probability values between 0 and 1 whenever neither player

has a (weakly) dominated strategy and produce invalid probability values whenever at least one

player has a (weakly) dominated strategy. So we conclude that there is a single mixed strategy

Nash equilibrium for any 2x2 game with no (weakly) dominated strategies

There is a natural intuition for the existence of a mixed strategy Nash equilibrium to match

the mechanical analysis above. If either player has a strictly dominant strategy, then the game

is solvable by iterated dominance and clearly there is no mixed strategy equilibrium. If neither

player has a (weakly) dominant strategy, then each player�s pure strategy is a best response to one

of the other player�s pure strategies.15 For expositional purposes, suppose that "Up" is player 1�s

best response to "Left" by player 2 and "Down" is player 1�s best response to "Right" by player

2. If player 2 plays "Left" with probability q and q is very close to 1, then player 1 should prefer

"Up", since player 2�s mixed strategy is very close to a pure strategy of "Left". Similarly, if player

2 plays "Left" with probability very close to 0, then player 1 should prefer "Down", since player 2�s14 In addition, if player 1 plays a pure strategy, then player 2 must play a pure strategy best response unless one

of player 2�s strategies weakly dominates the other. That is, there is only a Nash equilibrium of a 2x2 game where

one player plays a pure strategy and the other player plays a mixed strategy if one player has a weakly dominant

strategy.15See Figure 6 and the associated discussion below for analysis of mixed strategies when one player has a weakly

dominant strategy.

300


mixed strategy is very close to a pure strategy of "Right". In addition, if player 2 increases q from

0 to 1, then "Up" becomes more attractive relative to "Down" for player 1 since an increase in q

indicates that player 2 is more likely to play "Left" (and "Up" gives a higher payo¤ than "Down"

for player 1 in response to "Left"). That is, player 1 strictly prefers "Down" for q close to 0,

becomes strictly more inclined to "Up" as q increases, and strictly prefers "Up" for q close to 1.

By this logic, there must be an single intermediate value, q� where player 1 is indi¤erent between

"Up" and "Down". This con�rms the intution that if neither player has a dominant strategy, there

must be some intermediate mixed strategy probability for player 2 such that player 1 is indi¤erent

between "Up" and "Down", and similarly there is some intermediate mixed strategy probability

for player 1 such that player 2 is indi¤erent between "Left" and "Right".

Identifying Mixed Strategy Equilibria in More Complicated Games In games with more

strategies or more players, it is more di¢ cult to identify all mixed strategy equilibria. Table 13

gives the payo¤s for a game with just one more pure strategy for one player. There are two pure

strategy equilibria in this game, as indicated by the combination of best responses: (Up, Right),

(Middle, Down).

Left Middle Right

Up 2 , 4 3, 0 7 , 10

Down 1, 9 6 , 10 4, 0

Table 13: Finding Mixed Strategy Equilibria in a 2x3 Game

The procedure we have demonstrated for �nding a mixed strategy Nash equilibrium in a 2x2

game is essentially a "Guess and Verify" method. That is, we consider all possible mixed strategies

for each player (as indexed by p� and q�), identify the condition for each player to be indi¤erent

between the two possible pure strategies, and then check that these conditions yield valid mixed

strategies for a Nash equilibrium. We can use the same method here for each of three di¤erent

2x2 games that would result if we assume that player 2 does not play one of his two strategies, as

shown in Table 14a, which eliminates "Right", Table 14b which eliminates "Middle", and Table

14c, which eliminates "Left".

301


Left Middle

Up 2, 4 3, 0

Down 1, 9 6, 10

Table 14a: A 2x2 subportion of a 2x3 game

In the 2x2 game shown in Table 14a, there are two pure strategy equilibria, ("Up", "Left") and

("Down", "Middle") and a mixed strategy equilibrium where player 1 plays "Up" with probability

0.2 and "Down" with probability 0.8 while player 2 plays "Left" with probability 0.75 and "Middle"

with probability 0.25.

Left Right

Up 2, 4 7, 10

Down 1, 9 4, 0

Table 14b: A 2x2 subportion of a 2x3 game

In the 2x2 game shown in Table 14b, "Up" dominates "Down" for player 1. This game is

dominance solvable and the unique Nash equilibrium is ("Up", "Right").

Middle Right

Up 3, 0 7, 10

Down 6, 10 4, 0

Table 14c: A 2x2 subportion of a 2x3 game

In the 2x2 game shown in Table 14c, there are two pure strategy equilibria, ("Up", "Right") and

("Down", "Middle") and a mixed strategy equilibrium where player 1 plays "Up" with probability

0.5 and "Down" with probability 0.5 while player 2 plays "Middle" with probability 0.5 and "Right"

with probability 0.5.

Any outcome that is an equilibrium of one of these three games is a potential Nash equilibrium

of the full 2x3 game. The only question is whether each of the equilibria in a 2x2 game remains

an equilibrium when we consider the third possible strategy for player 2. Across all three games,

there are three potential pure strategy equilibria: ("Up", "Left"), ("Up", "Right"), and ("Down",

"Middle"), but we already know from our analysis above that both ("Up", "Right") and ("Down",

"Middle") are Nash equilibria and that ("Up", "Left") is not a Nash equilibrium of the full 2x3

game..

302


We identi�ed two potential mixed strategy Nash equilibria in the analysis of these three partial

games. We can describe each mixed strategy for player 1 as an ordered pair where the �rst entry is

the probability of "Up" and the second entry is the probability of "Down". We can describe each

mixed strategy for player 2 as an ordered triple where the �rst entry is the probability of "Left," the

second entry is the probability of "Down," and the third entry is the probability of "Right". Then

the two potential mixed strategy equilibria are [(0:2; 0:8); (0:75; 0:25; 0)] and [(0:5; 0:5); (0; 0:5; 0:5)].

To check if each of these is a mixed strategy equilibrium, we need to verify that player 2�s omitted

strategy is not a best response to player 1�s mixed strategy.

Case 1: In the �rst possible mixed strategy equilibrium, player 1 plays "Up" with probability

0.2 and "Down" with probability 0.8, and player 2 mixes between "Left" and "Middle". Given

player 1�s mixed strategy, player 2�s expected utility from either "Left" or "Middle" is 8 (note that

player 1�s mixing probabilities are chosen to equate player 2�s expected utility from "Left" and

"Middle"), while player 2�s expected utility from "Right" is equal to 0:2 � 10 + 0:8 � 0 = 2. So

"Right" is not a best response to player 1�s mixed strategy, [(0.2, 0.8)] indicating that the mixed

strategy equilibrium in Table 14a (with "Right" omitted) is a mixed strategy equilibrium of the

full 2x3 game. That is, [(0:2; 0:8); (0:75; 0:25; 0)] is a mixed strategy equilibrium.

Case 2: In the second possible mixed strategy equilibrium, player 1 plays "Up" with probability

0.5 and "Down" with probability 0.5. and player 2 mixes between "Middle" and "Right". Given

player 1�s mixed strategy, player 2�s expected utility from either "Middle" or "Right" is 5 (note

that player 1�s mixing probabilities are chosen to equate player 2�s expected utility from "Middle"

and "Right"), while player 2�s expected utility from "Left" is equal to 0:5 � 4 + 0:5 � 9 = 6:5. So

"Left" is not a best response to player 1�s mixed strategy [(0.5, 0.5)], indicating that the mixed

strategy equilibrium in Table 14c (with "Left" omitted) is not a mixed strategy equilibrium of the

full 2x3 game. That is, [(0:5; 0:5); (0; 0:5; 0:5)] is not a mixed strategy equilibrium.

We have now checked for all possible ways that player 2 could play a mixed strategy involving

just two of his three pure strategies. The last possibility is that player 2 could play a mixed

strategy that places positive probability on all three of his pure strategies. This is only possible

if player 1 plays a strategy that gives player 2 the same expected utility from all three of his pure

strategies. Suppose that player 1 plays "Up" with probablity p and "Down" with probability 1�p.

Then player 2�s expected utilities are as follows:

EU("Left") = 4p+ 9(1� p) = 9� 5p;

EU("Middle") = 10p+ 0(1� p) = 10p;

303


Figure 7: Using Expected Utilities toIdentify a Mixed Strategy Equilibrium

A

B

C

0

1

2

3

4

5

6

7

8

9

10

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Player 1 P("Up")

Play

er 2

Ex

pect

ed U

tility

EU(Left)EU(Middle)EU(Right)

EU("Right") = 0p+ 10(1� p) = 10� 10p.

Setting all three of these expected utilities equal gives two equations in one unknown: 9� 5p =

10p and 10p = 10�10p, which have di¤erent solutions p = 0:2 and p = 0:5:16 This shows that there

cannot be a mixed strategy equilibrium where player 2 puts weight on all three pure strategies. In

short, since player 1 only has two strategies, there is no way for her to adjust the weight between

the two of them to make player 2 indi¤erent between all three of his strategies.

This analysis of these three separate 2x2 subportions of the 2x3 game is cumbersome. One

way to illuminate these calculations is to compare the expected utilities for player 2 as a function

of player 1�s randomizing probability.

Figure 7 graphs the expected utilities for each of player 2�s possible strategies as a function of p,

player 1�s probability of "Up" in a mixed strategy. The expected utility for each pure strategy is

linear in p and there are three intersection points between the expected utility lines: A = (0:2; 0:8);

16 It could be said that there are actually three equations in one unknown: 9 � 5p = 10p; 10p = 10 � 10p;and

9� 5p = 10� 10p. But if any two of these equations are satis�ed, the third must be as well.

304


B = (0:5; 0:5); C = (0:6; 6). At any value for p other than 0.2, 0.5, and 0.6, player 2�s best

response to player 1�s strategy is a pure strategy. If p < 0:2, player 2�s best response is "Middle".

If 0:2 < p < 0:6, player 2�s best response is "Left". If p > 0:6, player 2�s best response is "Right".

Each of the points A;B;C represents a potential mixed strategy equilibrium. However, point

B does not even represent a best response for player 2. Even though "Middle" and "Right" give

the same expected utility at point B, "Left" gives a higher utility in response to player 1�s strategy

at point B, so this cannot be a mixed strategy equilibrium. Points A and C both represent mixed

strategy best responses by player 2 for the given strategy for player 1. The only question is

whether there are mixed strategies for player 2 between "Left" and "Middle" (point A) or "Left"

and "Right" (point C) that make player 1 indi¤erent between "Up" and "Down" to justify player

1�s proposed mixed strategies at these points. We have seen from the analysis above that if player

2 does not play "Middle", then "Up" strictly "Dominates" down for player 1. This eliminates the

possibility of a mixed strategy equilibrium at point C, leaving only the mixed strategy equilibrium

at point A that we identi�ed in the analysis of Table 14a: [(0:2; 0:8); (0:75; 0:25; 0)].

Figure 7 also demonstrates why there is no mixed strategy equilibrium where player 2 puts

positive weight on all three pure strategies. Player 2�s expected utility for each of his pure strategies

is a linear function of p, player 1�s probability of "Up". There is no single point where all three

lines cross in Figure 7, so there is no value of p that sets the expected utilities equal for all three

of player 2�s pure strategies. Thus, there is no mixed strategy equilibrium that includes all three

of player 2�s strategies because it is impossible for player 2 to be indi¤erent among all three pure

strategies simultaneously.17

In fact, it is a property of almost all 2x3 games that there is no mixed strategy equilibrium

where player 2 plays all three strategies with positive probability, precisely because three lines with

intercepts and slopes chosen at random (equivalent to a random choice of payo¤s) will intersect at

the same point with probability 0. It is possible to �nd 2x3 games where all three of player 2�s

expected utilities intersect at a single point, but these occur with probability 0 when the payo¤s

are chosen at random.18

17 In discussion of weakly dominant strategies below, we introduce the mathematical concept of genericity. It is a

generic property of two-player games that when if one player has m pure strategies and the other player has n > m

pure strategies that there are no Nash equilibria where the player with n strategies plays more than m of them with

positive probability.18More generally, in a game where player 1 has m pure strategies, player 2 has n pure strategies and n > m, it is

a generic property that there will be no mixed strategy where player 2 puts positive weight on more than m pure

305


Interpreting Mixed Strategy Nash Equilibria Our procedure for identifying mixed strategy

equilibria in 2x2 games highlights a key feature of all mixed strategy equilibria. A player who

plays a mixed strategy in a Nash equilibrium must be indi¤erent between all strategies that she

chooses with positive probability. (Otherwise, that player would have a strict preference for one

of those pure strategies over the others and could increase her expected payo¤ by switching to

a pure strategy.) But when one player is indi¤erent among a set of pure strategies, she has no

obvious incentives to select one relatively more often than the others. That is, in a mixed strategy

equilibrium, player i�s own payo¤s give no guidance for the mixed strategy probabilities that she

should adopt.

In fact, our formulas for the mixed strategy probabilities in a 2x2 Nash equilibrium yield a

value for player 2�s mixing probability as a function of player 1�s payo¤s and also yield a value

for player 1�s mixing probability as a function of player 2�s payo¤s That is, each player�s mixed

strategy probabilities serve to make the other player indi¤erent between two (or more) strategies in

a mixed strategy Nash equilibrium. In a zero sum game, such as "Matching Pennies", it is possible

to argue that it is in each player�s interest to follow her mixed strategy Nash equilibrium to limit

possibilities for exploitation by the other player. For instance, if player 1 plays "Heads" more often

than "Tails" (just as Bart Simpson plays "Rock" too frequently in "Rock-Paper-Scissors"), then

player 2 can make a positive expected payo¤ by playing "Tails" as a pure strategy.

In non-zero sum games, however, there is no obvious reason for either player to follow a mixed

strategy. For this reason, Harsanyi suggested a population-level interpretation of mixed strategy

equilibria.known as puri�cation of mixed strategy equilibrium. Consider the game of Chicken, as

shown in Table 15a. This game is featured in the James Dean movie "Rebel Without a Cause",

and is frequently used as a basic model of international relations (e.g. the Cuban missile crisis)..

In this game, two players conduct a contest of nerves, where they must choose among an aggressive

strategy, "Hawk" and a passive strategy "Dove". An aggressive player takes advantage of a passive

player, but there is a tremendous cost if both sides are aggressive. (In "Rebel Without a Cause",

the players were drivers in cars headed towards a cli¤, and the rules of the game speci�ed that the

�rst person to swerve away from the cli¤ would be a "chicken" who loses the game.)

Dove Hawk

Dove 8, 8 6 , 10

Hawk 10 , 6 -2, -2

strategies.

306


Table 15a: Payo¤s for "Chicken" Game

Table 15a shows that there are two pure strategy Nash equilibria of the game. In each of these

equilibria, one player is passive and the other is aggressive. Chicken is sometimes known as an

"anti-coordination" game because each player�s best response to a pure strategy by the opponent

is to play the other pure strategy. There is also a symmetric mixed strategy of the game where

each player plays "Dove" with probability 4/5 and "Hawk" with probability 1/5.

It is di¢ cult to explain how two randomly matched players would be able to play either of the

pure strategy equilibria. Even if they could communicate prior to choosing their strategies, each

would have an incentive to argue that she planned to play "Hawk". It is also di¢ cult to imagine how

two randomly matched players could implement the mixed strategy equilibrium. Harsanyi argued,

however, that the players might actually playing a perturbed game with very slightly di¤erent

payo¤s that would yield a natural outcome that is equivalent to the mixed strategy equilibrium in

terms of the frequency of choices of "Hawk" and "Dove".

Dove Hawk

Dove 8+"1, 8+"2 6 , 10

Hawk 10 , 6 -2, -2

Table 15b: Payo¤s for Harsanyi�s Perturbed "Chicken" Game

The perturbed game in Table 15b is equivalent to the original game in Table 15a except for the

addition of small incremental payo¤s "1and "2 for the two players in the case of ("Dove", "Dove).

We assume that "1and "2 are identically and independently distributed random variables with

P("j < 0) = 0:2;P(j"j j < 2) = 1.19 That is, the "j values are random components of the payo¤s

that are too small to a¤ect the pure strategy best responses as shown in Table 15a. However, these

values will a¤ect the strategies of the players for the mixed strategy Nash equilibrium.

Suppose that each player anticipates that the other player will follow the mixed strategy equi-

librium actions with P("Dove") = 0.8 and P("Hawk") = 0.2. Then each would be indi¤erent

between "Dove" and "Hawk" if "j = 0. But a player with "j > 0 will strictly prefer "Dove" and

a player with "j < 0 will strictly prefer "Hawk" in response to the mixed strategy (0.8, 0.2) by her

opponent. Since we assumed that P("j > 0) = 0.8, the empirical best response to (0.8, 0.2) is the

pure strategy "Dove" with probability 0.8 and the pure strategy "Hawk" with probability 0.2. In

19We assume that P("j = 0) = 0, so that P("j > 0) = 0:8.

307


formal language, the empirical distribution of actions induced by these pure strategies in tandem

with the "-values is (0.8, 0.2) �which matches the empirical distribution for the mixed strategy

equilibrium of the original game.

The point of Harsanyi�s construction and the perturbed game is that it explains how the players

can play pure strategies and yet produce a probability distribution of actions that corresponds to

the mixed strategy equilibrium. In essence, we assume that some people have a natural inclination

for "Dove" and others have a natural inclination for "Hawk", but that these inclinations are not

strong enough to change the underlying nature of the game. If the proportions inclined to each

action match the mixed strategy proportions for each equilibrium action, then if everyone follows

their inclinations, they will implement a version of the mixed strategy equilibrium.

Existence of Nash Equilibrium

The existence of a mixed strategy Nash equilibrium in Matching Pennies is not an accident. Instead,

it is a general phenomenon that makes Nash equilibrium the most ubiquitous solution concept in

game theory.

Theorem 2 Every game with a �nite number of players and a �nite number of pure strategies for

each player has at least one Nash equilibrium.

This result was proven by John Nash, a Nobel prize winner and the subject of the movie "A

Beautiful Mind" in 1951. A full proof of Nash�s theorem is beyond the scope of this text. However,

we can provide some of the intuition of the theorem by examining the 2x2 case - games with two

players and two pure strategies for each player - in some detail. Suppose that player 1 has two

strategies, "Up" and "Down", that player 2 has two strategies "Left" and "Right". Player 1

can play one of two pure strategies or a mixed strategy that randomizes between them. We can

describe the full set of strategies, including the mixed strategies, with a single parameter p, which

represents the probablity that player 1 plays "Up". (The values p = 0 and p = 1 denote player 1�s

pure strategies, while values of p in the interval between 0 and 1 represent mixed strategies.)

Figure 8a graphs the best responses for the players, q�(p) for player 2 and p�(q) for player 1,

in the Prisoners�Dilemma, where player 1�s probabilities are on the horizontal axis and player 2�s

probabilities are on the vertical axis.20 Here "Up" corresponds to "Cooperate" for player 1 and20The orientation of player 1�s best response function in this graph goes against ordinary conventions, which usually

graphs values on the y-axis as functions of values on the x-axis. To read player 1�s best response function in the

308


Figure 8a: Best Responses in the Prisoners' Dilemma

0.02, 0

A0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Player 1, P(Cooperate)

Play

er 2

, P(C

oope

rate

)

q*(p)p*(q)

"Left" corresponds to "Cooperate" for player 2. Since each player has a strictly dominant strategy,

the best response function is constant: q�(p) = 0 for player 2 (meaning that player 2�s best response

to any mixed strategy by player 1 is to "Defect" � i.e. probability 0 of "Cooperate" / "Right")

and p�(q) = 0 for player There1 (meaning that player 2�s best response to any mixed strategy by

player 1 is to "Defect" �i.e. probability 0 of "Cooperate" / "Up").

When player 1 has a strictly dominant strategy, then player 1�s best response function p�(q)

is a vertical line, while if player 2 has a strictly dominant strategy, then player 2�s best response

function q�(p) is a horizontal line.21 These two functions intersect at a single point (0, 0), labeled

A in Figure 8a, corresponding to the dominant strategy outcome of the game �("Defect", "Defect")

or p = 0; q = 0:

graph, however, it is necessary to reverse this relationship. Thus, the point (0, 0.8) in player 1�s best response

function in Figure 3 indicates that if player 2 plays Left ("Cooperate") with probability q = 0:8, then player 1�s best

reponse is to play Up ("Cooperate) with probability p = 0.21Most algebra texts denote functions in the form y(x), where the vertical dimension is shown as a function of the

horizontal dimension in an XY graph. The �gures in this section show player 2�s best responses in this form. But

the orientation of player 1�s best response function may seem unnatural at �rst, because this shows an x-value as a

function of the y-values.

309


We can use this same graphical technique to demonstrate the existence of a Nash equilibrium

in any 2x2 game. As above, we assume that player 1 has two strategies, "Up" and "Down", that

player 2 has two strategies, "Left" and "Right" and we use p to denote P(Left) for player 1 and q

to denote P(Up) for player 2. To simplify analysis, we assume that the four possible payo¤s for

any player are all di¤erent. Given this assumption, each player�s best response to a pure strategy

is always a unique pure strategy. Further, if either player has a dominant strategy, it is a strictly

dominant strategy. Then there are four possible cases, as listed in Table 16.

Case Description Set of Nash equilibria

1 Both have dominant strategies Unique Nash equilibrium in pure strategies

2 One has a dominant strategy Unique Nash equilibrium in pure strategies

3 Neither has a dominant strategy, co-

ordinated preferences.

Two pure strategy Nash equilibria and one

mixed strategy Nash equilibrium

4 Neither has a dominant strategy, un-

coordinated preferences.

No pure strategy Nash equilibrium, one mixed

strategy Nash equilibrium

Table 16: Classi�cation of 2x2 Games and Existence of Nash Equilibrium

If either player has a dominant strategy, then the game can be solved by iterated dominance

and that solution is a Nash equilibrium. Figure 8a illustrates the case where both players have

dominant strategies, where we know from Figure 8a that the best response functions intersect

at a single point - the corner point of the graph where both players are playing their dominant

strategies.

Figure 8b illustrates the best responses in the case where one player has a dominant strategy

and the other does not. Here, player 1 has a dominant strategy, "Down", and thus, player 1�s

best response function is the vertical line p�(q) = 0. However, player 2 does not have a dominant

strategy since q�(0) = 1 and q�(1) = 0 indicates that "Left" is player 2�s best response to "Up"

and Right" is player 2�s best response to "Down".

Since player 2 has di¤erent best responses to player 1�s pure strategies, there is some mixed

strategy for player 1 that makes player 2 indi¤erent between his two possible strategies. Intuitively,

if player 1 is almost certain to play "Up", then player 2 prefers to play "Left" and player 2�s best

response function falls somewhere on the horizontal line near point A. On the other hand, if

player 1 is almost certain to play "Down", then player 2 prefers to play "Right" and player 2�s

310


Figure 8b: Best Response Functions:One Player has a Dominant Strategy

DC

BA

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Player 1 P(Up)

Play

er 2

P(

Left)

q*(p)p*(q)

best response function falls somewhere on the horizontal line near point B: As we adjust the

probabilities for player 1, there must be some intermediate probability of "Up" so that player 2 is

indi¤erent between "Left" and "Right".22

In Figure 8b, player 2 is indi¤erent between "Left" and "Right" if P(Up) = 0.8 and P(Down) =

0.2. For this mixed strategy, indicated by p = 0:8, each mixed strategy for player 2 gives the same

payo¤, and so all possible mixed strategies for player 2 are best responses to p = 0:8 for player 1.

For this reason, player 2�s best response function includes a vertical line from point B; (0:8; 1), to

point C; (0:8; 0). This vertical line in player 2�s best response mapping includes all points of the

22We can state this observation more formally. The best response function in a 2x2 game always takes threshold

form for a player with no dominant strategy. If p is close to 1, then player 2�s best response is the pure strategy

that is the best response to "Up". If p is close to 0, then player 2�s best response is the pure strategy that is the

best response to "Down". Somewhere in the middle, there is a cuto¤ value p� so that when P(Left) = p�, player 2

is indi¤erent between the pure strategies "Left" and "Right". For all values of p between 0 and p�, player 2�s best

response is the pure strategy that is the best response to "Down". For all values of p between p� and 1, player 2�s

best response is the pure strategy that is the best response to "Up". This best response function takes a threshold

form because p� is the threshold that distinguishes between the two pure strategy best responses.

311


form (0:8; q), where 0 � q � 1.23

Despite the additional complexity for player 2�s best response mapping, the best responses again

intersect at a single corner point, A, where player 1 plays her dominant strategy and player 2 plays

his best response to that strategy. That is, there is a unique Nash equilibrium where player 1 plays

"Down", player 2 plays "Left", and this outcome is shown at point A where p = 0; q = 1:

When neither player has a dominant strategy in a 2x2 game, player 2 has a di¤erent best response

to player 1�s two possible pure strategies, "Up" and "Down", so that q�(0) 6= q�(1). Similarly,

player 1 has di¤erent best responses to "Left" and to "Right", so that q�(0) 6= q�(1). Then, there

are two possibilities corresponding to cases 3 and 4 in Table 16. First, as shown in Figure 8c, the

pure strategy best responses may be coordinated so that there are two pure strategy Nash equilibria

with either (1) p�(0) = q�(0) = 0 and p�(1) = q�(1) = 1 or (2) p�(0) = 1, q� (1) = 0 and p�(0) = 1,

q� (1) = 0: Second, as shown in Figure 8d, the pure strategy best responses may be uncoordinated

so that there are no pure strategy Nash equilibria with either with either (1) p�(0) = 0; q�(0) = 1

and p�(1) = 1; q�(1) = 0 or (2) p�(0) = 1, q� (0) = 0 and p�(1) = 0, q� (1) = 1:

Figure 8c depicts the best responses for the Stag Hunt game from Table 4, where the strategies

are labeled so that "Up" for player 1 and "Left" for player g2 represent the strategy "Stag". Player

2�s best response mapping follows the path A�C�D�E�G. Between point A, (0, 0), and point

C, (0.25, 0), player 1 is relatively unlikely to play "Stag", and so player 2�s best response is "Hare",

or q = 0. Between point E, (0.25, 1), and point G, (1, 1), player 1 is relatively likely to play "Stag"

and so player 2�s best response is "Stag" or q = 1. When player 1 chooses a mixed strategy with

P(Stag) = 0.25, player 2 is indi¤erent between "Stag", "Hare" and any mixed strategy between

these two pure strategies. So all points on the vertical line from C to E are included in player 2�s

best response mapping.

Similarly, Player 1�s best response mapping follows the path A � B � D � F � G. Between

point A, (0, 0), and point B, (0, 0.25), player 2 is relatively unlikely to play "Stag", and so player

1�s best response is "Hare", or p = 0. Between point D, (1, 0.25), and point G, (1, 1), player 2 is

relatively likely to play "Stag" and so player 1�s best response is "Stag" or p = 1. When player 2

chooses a mixed strategy with P(Stag) = 0.25, player 1 is indi¤erent between "Stag", "Hare" and

any mixed strategy between these two pure strategies. So all points on the horizontal line from B

23Technically, player 2�s best response mapping is a correspondence rather than a function. A function is a

one-to-one mapping, but player 2�s best response to player 1�s mixed strategy with p = 0:8 includes many values for

q�(p):

312


Figure 8c: Best Response Functions for Stag Hunt:Neither Player Has a Dominant Strategy, Coordinated Preferences

GE

D

C

FB

A0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Player 1: P(Stag)

Play

er 2

: P(S

tag)

q*(p)p*(q)

to F are including at player 1�s best response mapping.

Because the preferences of the players are coordinated in this game, there are two pure strategy

Nash equilibria, ("Stag", "Stag") and ("Hare", "Hare"). There is also a single mixed strategy

equilibrium at point D; (0:25; 0:25), the third point where the best response mappings intersect.

At this point, each player chooses "Stag" with probability 0:25 = 1=4, and each player is indi¤erent

between "Stag" and "Hare".

Figure 8d depicts the best responses for the case where neither player has a dominant strategy

and their preferences are uncoordinated, so that there are no pure strategy Nash equilibria. Player

2�s best response mapping follows the path C � F � E � D � G. Between point C, (0, 1), and

point F , (0.5, 1), player 1 is relatively unlikely to play "Up", and player 2�s best response is "Left",

or q = 1. Between point D, (0.5, 0), and point G, (1, 0), player 1 is relatively likely to play "Up"

and player 2�s best response is "Right" or q = 0. When player 1 chooses a mixed strategy with

P(Up) = 0.5, player 2 is indi¤erent between "Left", "Right" and any mixed strategy between these

two pure strategies. So all points on the vertical line from D to F are included in player 2�s best

response mapping.

313


Figure 8d: Best Response Functions:Neither Player Has a Dominant Strategy, Uncoordinated Preferences

GD

E

C F J

H

A

B

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Player 1 Probability(Up)

Play

er 2

Pr

obab

ility

(Lef

t)

q*(p)p*(q)

Similarly, Player 1�s best response mapping follows the path A�B�E�H�J . Between point

A, (0, 0), and point B, (0, 0.5), player 2 is relatively unlikely to play :"Left", and player 1�s best

response is "Down", or p = 0. Between point H, (1, 0.5), and point J , (1, 1), player 2 is relatively

likely to play "Left" and player 1�s best response is "Up" or p = 1. When player 2 chooses a mixed

strategy with P(Left) = 0.2, player 1 is indi¤erent between "Up", "Down" and any mixed strategy

between these two pure strategies. So all points on the horizontal line from B to H are including

at player 1�s best response mapping.

The best response mappings intersect at a single point, E, (0.5, 0.5). This is a mixed strategy

that produces the unique Nash equilibrium in the game, with player 1 choosing "Up" and "Down"

with equal probability and player 2 choosing "Left" and "Right" with equal probability.

Summary of Nash Equilibria in 2x2 Games Examining the results from Figures 8a

through 8d, there are several points to note. First, there exists a Nash equilibrium in every case.

In Figures 8a and 8b, the existence of a dominant strategy made it obvious that there would be a

pure strategy outcome, but this was less immediately obvious for Figures 8c and 8d. A separate

point of interest is that there are an odd number of Nash equilibria in all four graphs �a unique

314


Nash equilibrium in Figures 8a, 8b, and 8d and three Nash outcomes in Figure 8c.

The intuition for these results in the 2x2 case is straightforward. The best response mapping

for each player starts at one corner of the range of probabilities and ends at another corner of the

range, where each corner point represents an instance where player 1 and player 2 each choose a

pure strategy. Traversing the best response mapping from one corner of the graph to another,

a player either has a straight line best response mapping - indicating a dominant strategy - or a

stepwise path consisting of three perpendicular lines.

Figure 8d represents the only case where there is no pure strategy Nash equilibrium. In this

case, the players have pure strategy best responses at opposite corners of the graph. So player 1

must have a horizontal line in her best response mapping and player 2 must have a vertical line

in his best response mapping for these mappings to connect the opposite corners of the graph.

The intersection of these lines is a mixed strategy Nash equilibrium. This reasoning, which

emphasizes a mathematical understanding of graphical analysis rather than economic intuition,

can be generalized to demonstrate the existence of a Nash equilibrium with any �nite number of

players each with a �nite number of strategies.

An additional similarity between the results in Figures 8a and 8b is that each features an even

number of pure strategy Nash equilibria and an additional mixed strategy equilibrium, for an overall

total of an odd number of Nash equilibria. Thus, each of the four cases includes an odd number

of equilibria. This too is a general property that generalizes beyond the 2x2 case because of the

nature of the graphical interaction between best response mappings for player 1 and player 2. One

di¤erence between these two properties: 1) the existence of a Nash equilibrium; 2) the existence of

an odd number of Nash equilibria, is that the odd number of equilibria relies on the restriction that

there are no ties in pure strategy payo¤s for any one player. This assumption is necessary because

it rules out weakly dominated strategies - the sole case where there may not be an odd number of

Nash equilibria.

Left Right

Up 1, 4 3, 4

Down 5, 2 3, 1

Table 17: A Game with Weakly Dominated Strategies

Table 17 depicts a game where each player has a weakly dominant strategy. For player 1,

"Down" weakly dominates "Up", while for player 2, "Left" weakly dominates "Right". If player

315


Figure 9: Best Responses with Weakly Dominated Strategies

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Player 1 P(Up)

Play

er 2

P(

Left)

q*(p)p*(q)

2 plays the pure strategy "Right", then player 1 is indi¤erent among all mixed strategies between

"Up" and "Down", including the pure strategies "Up" and "Down". Thus, player 1�s best response

mapping includes the horizontal line between (0,0) and (1,0). But if player 2�s strategy has any

positive probability of "Left", then player 1 strictly prefers "Down" to any other strategy. Thus,

player 1�s best response mapping includes the vertical line between (0, 0) and (0, 1).

By similar reasoning, if player 1 plays the pure strategy "Up", then player 2 is indi¤erent among

all mixed strategies between "Left" and "Right", including the pure strategies "Left" and "Right".

Thus, player 2�s best response mapping includes the vertical line between (1,0) and (1,1). But if

player 1�s strategy has any positive probability of "Down", then player 2 strictly prefers "Left"

to any other strategy. Thus, player 2�s best response mapping also includes the horizontal line

between (0, 1) and (1, 1).

Figure 9 depicts the best responses for players 1 and 2 in the game in Table 17. There are two

intersection points between these best response mappings, and each of these points corresponds to

a pure strategy Nash equilibrium. The intersection (0, 1) corresponds to the Nash equilibrium

("Down", "Left") and the intection (1, 0) corresponds to the Nash equilibrium ("Up", "Right").

316


There can be no mixed strategy equilibrium because each player has a weakly dominated strategies,

so the best response for either player to a mixed strategy from the other player is the weakly

dominant pure strategy. Since the best response to a mixed strategy cannot be a mixed strategy,

there can be no mixed strategy equilibrium in this game.

Figure 9 indicates the the best response mappings for player 1 still includes a horizontal line and

the best response for player 2 still includes a vertical line. This property guaranteed the existence

of a mixed strategy equilibrium in Figures 8c and 8d when those lines did not correspond to pure

strategies. However, with weakly dominant strategies, these components of the best response

mappings lie on the edges of the graph and do not lead to mixed strategy outcomes. Apart from

unusual cases, such as the one depicted in Figure 9, every �nite player, �nite strategy game has an

odd number of Nash equilibria.

Mathematicians have developed a property known as genericity to formalize the concept of

"unusual cases". A property of games is deemed generic if it holds with probability 1 in games

where the payo¤s are chosen at random. While it would take some care to de�ne the process of

choosing the payo¤s, it should be clear that if the payo¤s are drawn from a range of real numbers

(i.e. numbers with many decimal places), the probability of picking the same payo¤ more than

once for any particular player is negligible. So, it is a generic property of games that a player will

not have any pure strategy that is weakly dominated but not strictly dominated by another pure

strategy, and as a result, it is also a generic property of games that there will be an odd number of

equilibria.24

24See Wilson (1971) for details.

317

introduction to game theory - semantic scholar · 2016-01-22 · introduction to game theory 11.1...

Documents