Analyzing Payoff Tables 407from its equilibrium play, it reduces its own payoff and increases the competi-
tor’s payoff. Indeed, there should be no real uncertainty about how the game
will be played. Each side should anticipate equilibrium behavior from the other.
The resulting equilibrium outcome is called the valueof the game.^4(^4) A zero-sum game always possesses an equilibrium. The value of the game is unique; there cannot
be two equilibria having different values. However, equilibrium behavior may require the use of
randomized actions (so-called mixed strategies) by the players. We discuss the use of mixed strate-
gies in the appendix to this chapter.
CHECK
STATION 2
The following payoff table lists the respective market shares of the two firms in the
advertising competition. Indeed, it is derived directly from Table 10.2 under the assump-
tion that the firms’ initial shares are 45 percent and 55 percent, respectively. For instance,
according to Table 10.2, the play of R1 versus C1 results in a 2-percentage-point loss for
firm 1; this translates into 43 percent and 57 percent market shares in the table. Given
the form of the payoff table, explain why this competition can be referred to as a con-
stant-sum game. Determine the equilibrium. Is there any strategic difference between a
zero-sum game and a constant-sum game?
Firm 2
C1 C2 C3
R1 43, 57 44, 56 49, 51
Firm 1 R2 50, 50 47, 53 48, 52
R3 52, 48 42, 58 40, 60
A REMINDER It is important to distinguish clearly between a Nash equilib-
rium that involves dominant strategies and one that does not. Here is the dif-
ference:
In a dominant-strategy equilibrium, each player chooses an action that is a best
response against any action the other might take.
In a Nash equilibrium, each player takes an action that is a best response to the
action the other takes.
Both kinds of equilibrium share the essential feature of stability. In equilib-
rium, there is no second guessing; it is impossible for either side to increase its
payoff by unilaterally deviating from its chosen strategy.
The concepts differ in one important respect. When a player has a domi-
nant strategy, there is no circumstancein which doing anything else ever makes
sense. The player always should use this strategy. Of course, in many, if not
most, competitive situations, players will not have available a single strategy that
is dominant. However, as in the market-share competition, there still will be a
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