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584 CHAPTER 14 GAME THEORY AND STRATEGIC BEHAVIOR
The fact that games can have Nash equilibria in the form of mixed strategies
illustrates that unpredictability can have strategic value. When your opponent can
predict what you will do, you can leave yourself vulnerable to being exploited by your
opponent. Athletes in sports such as baseball, soccer, and tennis have long understood
this point, and the World Cup game illustrates it nicely. If the kicker knew which way
the goalie was going to dive, the kicker could simply aim the other way and score the
goal. There is value in being unpredictable, and mixed strategies illustrate how this
value is present in game theory.

SUMMARY: HOW TO FIND ALL THE NASH
EQUILIBRIA IN A SIMULTANEOUS-MOVE GAME
WITH TWO PLAYERS

We can summarize the lessons of this section by outlining a five-step approach to
identifying the Nash equilibria in simultaneous-move games involving two players.

1. If both players have a dominant strategy, these constitute their Nash equilibrium
strategies.
2. If one player, say Player 1, has a dominant strategy, this is the player’s Nash
equilibrium strategy. We then find Player 2’s best response to Player 1’s dominant
strategy to identify Player 2’s Nash equilibrium strategy.
3. If neither player has a dominant strategy, we successively eliminate each player’s
dominated strategies in order to simplify the game, and then search for Nash
equilibrium strategies.
4. If neither player has dominated strategies, we identify Player 1’s best response to
each of Player 2’s strategies and then identify Player 2’s best response to each of
Player 1’s strategies. In a table representing the game, the Nash equilibria are the
cells where a Player 1 best response occurs together with a Player 2 best response.
(This approach, which is guaranteed to identify all the pure-strategy Nash equi-
libria in a game, was demonstrated in Learning-By-Doing Exercise 14.2.)
5. If the approach in Step 4 does not uncover any pure-strategy Nash equilibria—
that is, if the game does not have a Nash equilibrium in pure strategies, as in the
Womens’ World Cup game—we look for an equilibrium in mixed strategies.

14.2 A key lesson of the prisoners’ dilemma is that the individual pursuit of profit max-
THE imization does not necessarily result in the maximization of the collective profit of a
group of players. But the prisoners’ dilemma is a one-shot game, and you might
REPEATED wonder if the game would turn out differently if it was played over and over again by
PRISONERS’ the same players. When we allow the players to interact repeatedly, we open the pos-
DILEMMA sibility that each player can tie its current decisions to what its opponent has done in
previous stages of the game. This expands the array of strategies that the players can
follow and, as we will see, can dramatically alter the game’s outcome.
To illustrate the impact of repeated play, consider the prisoners’ dilemma game in
Table 14.12. For each player, “cheat” is a dominant strategy, but the players’ collec-
tive profit is maximized when both play “cooperate.” In a one-shot game, the Nash
equilibrium would be for both players to choose “cheat.”