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Appendix to Chapter 10 Mixed Strategies 441

payoff is (x)(2) (1 x)(7). If, instead, it uses C3, its expected payoff is

(x)(4) (1 x)(5). Firm 2 is indifferent between C1 and C3 when these

expected payoffs are equal:

[10A.1]

or 12 18x. Thus, x 2/3. In equilibrium, firm 1 uses R1 and R3 with prob-

abilities 2/3 and 1/3, respectively. Turning to firm 2, let y denote the proba-

bility it plays C1 and 1 y the probability it plays C3. If firm 1 uses R1, its

expected payoff is (y)(2) (1 y)(4). If, instead, it uses R3, its expected

payoff is (y)(7) (1 y)(5). Equating these expected payoffs implies:

[10A.2]

or 9 18y. Thus, y 1/2. In equilibrium, firm 2 uses C1 and C3, each with

probability 1/2. In Table 10A.1b, we display these mixed strategies.^1 Finally,

what is each firm’s expected payoff when it uses its mixed strategy? If we sub-

stitute x 2/3 into either side of Equation 10A.1, we find that firm 2’s

expected payoff is 1 from either of its pure strategies. Thus, the expected pay-

off for its mixed strategy is also 1. Similarly, firm 1’s expected payoff is 1 from

either of its strategies. These expected payoffs also are shown in Table 10A.1b.

In short, when both sides use their optimal mixed strategies, firm 1’s expected

gain in market share (and firm 2’s expected loss) is 1 percent.

REMARK In this equilibrium, neither side can improve its expected payoff by

switching from its mixed strategy. In fact, a player actually does not lose by

switching to some other strategy proportion. For instance, as long as firm 2

uses its 50–50 mixed strategy, firm 1 earns the same expected payoff from any

mixture of R1 and R3 (one-third/two-thirds, 50–50, etc.). The penalty for

switching from equilibrium proportions comes in a different form: A smart

opponent can take advantage of such a switch. Using its equilibrium strategy,

firm 1 guaranteesitself an expected payoff of 1 against the equilibrium play of

2y4(1y)7y5(1y),

2x7(1x)4x5(1x),

(^1) There is a simple rule for finding the mixed strategies in a 2-by-2 payoff table like the accompa-
nying one. Firm 1’s mixed-strategy proportions are x (d c)/[(d c) (a b)] and 1 x 
(a b)/[(d c) (a b)]. Firm 2’s proportions are y (d b)/[(d b) (a c)] and 1 
y (a c)/[(d b) (a c)]. To find the R1 chance, take the difference between the entries
in the opposite row(d c) and then divide by the sum of the row differences, (d c) (a b).
The same opposite-row rule works for the R2 chance, and an opposite-columnrule works for com-
puting firm 2’s mixed strategy proportions.
y1 y
C1 C2
xR1 ab
1 xR2 cd
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