When every player has a dominant strategy, we call the outcome of the
game an equilibrium in dominant strategies. Such games are straightforward
to analyze because each player’s optimal strategy can be determined without
worrying about the actions of the other players.
Unfortunately, not every game has a dominant strategy for each player. To see
this, let’s change our advertising example slightly. The payoff matrix in Table 13.2
is the same as in Table 13.1 except for the bottom right-hand corner—if neither
firm advertises, Firm B will again earn a profit of 2, but Firm A will earn a profit
of 20. (Perhaps Firm A’s ads are expensive and largely designed to refute Firm
B’s claims, so by not advertising, Firm A can reduce its expenses considerably.)
Now Firm A has no dominant strategy. Its optimal decision depends on what
Firm B does. If Firm B advertises, Firm A does best by advertising; but if Firm B
does not advertise, Firm A also does best by not advertising. Now suppose both
firms must make their decisions at the same time. What should Firm A do?
To answer this, Firm A must put itself in Firm B’s shoes. What decision is best
from Firm B’s point of view, and what is Firm B likely to do? The answer is clear:
Firm B has a dominant strategy—advertise, no matter what Firm A does. (If
Firm A advertises, B earns 5 by advertising and 0 by not advertising; if A doesn’t
advertise, B earns 8 if it advertises and 2 if it doesn’t.) Therefore, Firm A can conclude
that Firm B will advertise. This means that Firm A should advertise (and
thereby earn 10 instead of 6). The logical outcome of the game is that both firms
will advertise because Firm A is doing the best it can given Firm B’s decision;
and Firm B is doing the best it can given Firm A’s decision.