496 PART • 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 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 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