(8th edition) (the pearson series in economics) robert pindyck, daniel rubinfeld microecon 520
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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 2
should happen to make a mistake and fail to invest, it would be extremely costly
to Firm 1. (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 2 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 1 does not invest and Firm 2 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 1 knew for certain
that Firm 2 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 1 is unsure about what Firm 2
will do 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
1 thinks that there is only a 10-percent chance that Firm 2 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 1 should invest.
On the other hand, suppose Firm 1 thinks that the probability that Firm 2
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 1 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 2. 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.
