Weak Dominance
Not to be absolutely certain is, I think, one of the essential things in rationality. —BERTRAND RUSSELL
Returning to the world of Italian opera yet again, suppose we now assume that Scarpia, upon being stabbed by Tosca, does not care whether Cavaradossi is killed. The resulting payoff matrix is shown in FIGURE 3.6. Although stab continues to be the dominant strategy for Tosca (indeed, we haven’t changed her payoffs), using real bullets no longer strictly dominates using blanks for Scarpia. Nevertheless, real would seem the reasonable course of action for Scarpia. If Tosca consents, then Scarpia strictly prefers to have used real bullets; he receives a payoff of 4 as opposed to 3. If Tosca stabs him, then he doesn’t care, as his payoff is 2 re-
gardless of what he chooses. Thus, he can’t be any worse off by using real bul- lets, and he might just be better off. We say that the strategy real weakly dom- inates the strategy blanks.
✚ DEFINITION 3.3 A strategy weakly dominates a strategy if (1) the payoff from is at least great as that from for any strategies chosen by the other players; and (2) there are some strategies for the other players whereby the payoff from is strictly greater than that from .*
Because most people are cautious and lack absolute confidence as to what other players will do, it seems prudent to avoid weakly dominated strategies. Doing so means that you can never be any worse off and you just might end up being better off. There’ll be no regrets by avoiding weakly dominated strategies. And if there is a weakly dominant strategy—which means that it weakly dominates all other strategies—it would be wise to use that strategy.
A rational and cautious player never uses a weakly dominated strategy. A rational and cautious player always uses a weakly dominant strategy.