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TRANSCRIPT
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".
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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.
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CRIMINAL 2
Cooperate Defect
CRIMINAL 1 Cooperate2 year sentence for Criminal 1,
2 year sentence for Criminal 2
5 year sentence for Criminal 1,
1 year sentence for Criminal 2
Defect1 year sentence for Criminal 1,
5 year sentence for Criminal 2
4 year sentence for Criminal 1,
4 year sentence for Criminal 2
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.
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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".
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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.
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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.
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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.
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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.
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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,
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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.
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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.
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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.
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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.
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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.
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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
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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
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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.
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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
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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.
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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,
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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
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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.
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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.
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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.
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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.
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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
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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.
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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".
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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..
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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;
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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.
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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
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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.
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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.
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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
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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.
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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
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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.
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C. Avery Notes on Microeconomic Theory ver: January 2010
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):
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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.
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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
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C. Avery Notes on Microeconomic Theory ver: January 2010
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
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C. Avery Notes on Microeconomic Theory ver: January 2010
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").
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C. Avery Notes on Microeconomic Theory ver: January 2010
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.
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