EXAMPLE 4 Matching Pennies

In Matching Pennies, each of two players simultaneously shows either a head or a tail . if the two coins match—either two heads or two tails—then the first player (Player I) receives both coins (a win of 1 for Player I). If the coins do not match, that is, if one is an and the other is a , then the second player (Player II) receives the two coins (a loss of 1 for Player I). These wins and losses for Player I are shown in the zero-sum payoff matrix in Table 15.5.

It is fruitless for one player to attempt to outguess the other in this game. Both should instead resort to mixed strategies and use expected values to estimate their likely gains or losses.

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To illustrate, assume that Player I randomly selects half the time and half the time. This mixed strategy can be expressed as

Note that the probabilities of choosing and () of choosing do indeed sum to 1, as required; in particular, when .

Player II gains nothing by knowing that Player I is using the optimal mixed strategy . However, Player I must not reveal to Player II whether or will be displayed in any given play of the game before Player II makes his or her own choice of or . Even without this information, if Player II knew that Player I was using a particular nonoptimal mixed strategy , where (i.e., not choosing a 50-50 mixture between and ), then Player II could take advantage of this knowledge and increase his or her average winnings over time to something greater than the value of 0 (see Exercise 18 on page 661).

This mixture can be realized in practice by the flip of a coin. Whenever Player II plays (first column of Table 15.5), Player I’s resulting expected value is

Similarly, whenever Player II plays (second column), Player I’s resulting expected value is

We will develop the tools later in this section to see that neither player can guarantee a better payoff than to choose and with equal likelihood (i.e., p = ), making this strategy optimal (see Exercise 18).