(8th edition) (the pearson series in economics) robert pindyck, daniel rubinfeld microecon 521
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496 PART 3 • Market Structure and Competitive Strategy
Therefore, the outcome in which both prisoners confess is both a Nash equilibrium and a maximin solution. Thus, in a very strong sense, it is rational for each
prisoner to confess.
*Mixed Strategies
• pure strategy Strategy in
which a player makes a specific
choice or takes a specific action.
• mixed strategy Strategy in
which a player makes a random
choice among two or more
possible actions, based on a set
of chosen probabilities.
In all of the games that we have examined so far, we have considered strategies
in which players make a specific choice or take a specific action: advertise or
don’t advertise, set a price of $4 or a price of $6, and so on. Strategies of this kind
are called pure strategies. There are games, however, in which a pure strategy is
not the best way to play.
MATCHING PENNIES An example is the game of “Matching Pennies.” In
this game, each player chooses heads or tails and the two players reveal their
coins at the same time. If the coins match (i.e., both are heads or both are tails),
Player A wins and receives a dollar from Player B. If the coins do not match,
Player B wins and receives a dollar from Player A. The payoff matrix is shown
in Table 13.6.
Note that there is no Nash equilibrium in pure strategies for this game.
Suppose, for example, that Player A chose the strategy of playing heads. Then
Player B would want to play tails. But if Player B plays tails, Player A would
also want to play tails. No combination of heads or tails leaves both players satisfied—one player or the other will always want to change strategies.
Although there is no Nash equilibrium in pure strategies, there is a Nash
equilibrium in mixed strategies: strategies in which players make random choices
among two or more possible actions, based on sets of chosen probabilities. In this game,
for example, Player A might simply flip the coin, thereby playing heads with
probability 1/2 and playing tails with probability 1/2. In fact, if Player A follows this strategy and Player B does the same, we will have a Nash equilibrium: Both players will be doing the best they can given what the opponent is
doing. Note that although the outcome is random, the expected payoff is 0 for
each player.
It may seem strange to play a game by choosing actions randomly. But put
yourself in the position of Player A and think what would happen if you followed
a strategy other than just flipping the coin. Suppose you decided to play heads. If
Player B knows this, she would play tails and you would lose. Even if Player B
didn’t know your strategy, if the game were played repeatedly, she could eventually discern your pattern of play and choose a strategy that countered it. Of
course, you would then want to change your strategy—which is why this would
not be a Nash equilibrium. Only if you and your opponent both choose heads or
tails randomly with probability 1/2 would neither of you have any incentive to
change strategies. (You can check that the use of different probabilities, say 3/4
for heads and 1/4 for tails, does not generate a Nash equilibrium.)
TABLE 13.6
MATCHING PENNIES
Player B
Player A
Heads
Tails
Heads
Tails
1, ؊1
؊1, 1
؊1, 1
1, ؊1
