CHAPTER 13 • Game Theory and Competitive Strategy 495 $20 million) than by not investing (and losing $10 million) Clearly the outcome (invest, invest) is a Nash equilibrium for this game, and you can verify that it is the only Nash equilibrium But note that Firm 1’s managers had better be sure that Firm 2’s managers understand the game and are rational If Firm should happen to make a mistake and fail to invest, it would be extremely costly to Firm (Consumer confusion over incompatible standards would arise, and Firm 1, with its dominant market share, would lose $100 million.) If you were Firm 1, what would you do? If you tend to be cautious—and if you are concerned that the managers of Firm might not be fully informed or rational—you might choose to play “don’t invest.” In that case, the worst that can happen is that you will lose $10 million; you no longer have a chance of losing $100 million This strategy is called a maximin strategy because it maximizes the minimum gain that can be earned If both firms used maximin strategies, the outcome would be that Firm does not invest and Firm does A maximin strategy is conservative, but it is not profit-maximizing (Firm 1, for example, loses $10 million rather than earning $20 million.) Note that if Firm knew for certain that Firm was using a maximin strategy, it would prefer to invest (and earn $20 million) instead of following its own maximin strategy of not investing MAXIMIZING THE EXPECTED PAYOFF If Firm is unsure about what Firm will but can assign probabilities to each feasible action for Firm 2, it could instead use a strategy that maximizes its expected payoff Suppose, for example, that Firm thinks that there is only a 10-percent chance that Firm will not invest In that case, Firm 1’s expected payoff from investing is (.1)(Ϫ100) + (.9)(20) = $8 million Its expected payoff if it doesn’t invest is (.1)(0) + (.9)(Ϫ10) = Ϫ$9 million In this case, Firm should invest On the other hand, suppose Firm thinks that the probability that Firm will not invest is 30 percent Then Firm 1’s expected payoff from investing is (.3) (Ϫ100) + (.7)(20) = Ϫ$16 million, while its expected payoff from not investing is (.3)(0) + (.7)(Ϫ10) = Ϫ$7 million Thus Firm will choose not to invest You can see that Firm 1’s strategy depends critically on its assessment of the probabilities of different actions by Firm Determining these probabilities may seem like a tall order However, firms often face uncertainty (over market conditions, future costs, and the behavior of competitors), and must make the best decisions they can based on probability assessments and expected values THE PRISONERS’ DILEMMA What is the Nash equilibrium for the prisoners’ dilemma discussed in Chapter 12? Table 13.5 shows the payoff matrix for the prisoners’ dilemma Recall that the ideal outcome is one in which neither prisoner confesses, so that both get two years in prison Confessing, however, is a dominant strategy for each prisoner—it yields a higher payoff regardless of the strategy of the other prisoner Dominant strategies are also maximin strategies TABLE 13.5 PRISONERS’ DILEMMA Prisoner B Prisoner A Confess Don’t confess Confess Don’t confess ؊5, ؊5 ؊1, ؊10 ؊10, ؊1 ؊2, ؊2 • maximin strategy Strategy that maximizes the minimum gain that can be earned For a review of expected value, see §5.1, where it is defined as a weighted average of the payoffs associated with all possible outcomes, with the probabilities of each outcome used as weights