Can we say what these two operatic characters will do if all we assume is that they are rational?
Can we say what these two operatic characters will do if all we assume is that they are rational? Well, con- sider Tosca. If Scarpia chooses real, then Tosca’s payoff from choosing stab is 2 and from choosing consent is 1. (See FIGURE 3.3.) Thus, she clearly prefers to stab Scarpia if she expects him to have ordered the firing squad to use real cartridges. What about if she expects him to have chosen blank cartridges? Her payoff is 4 from stab and 3 from consent, so, once again, she prefers stab. Thus, regardless of what Tosca believes Scarpia will do, stab gives Tosca a strictly higher payoff than consent. Therefore, Tosca should most definitely not choose consent. Since she has to do something, she has nothing left to do but to stab Scarpia. To see this another way, stab is superior to consent regardless of what Scarpia will do, so a rational Tosca should most surely choose stab.
By a similar argument, the rationality of Scarpia implies that he will choose real. If Tosca chooses stab, he earns a payoff of 2 from killing Cavaradossi and only 1 from not doing so. If Tosca chooses consent, then the payoff from real is 4, which once again exceeds that from blanks (which is 3). Hence, regard- less of what he thinks Tosca will do, Scarpia should have the firing squad use real bullets. In conclusion, game theory makes a very clear (and bloody) pre- diction that Tosca will stab Scarpia and Scarpia will see that Cavaradossi dies at the hands of the firing squad.
The strategy consent is said to be strictly dominated by the strategy stab for Tosca, which just means that stab delivers a higher payoff than consent for any strategy of Scarpia.
✚ DEFINITION 3.1 A strategy strictly dominates a strategy if the payoff from is strictly higher than that from for any strategies chosen by the other players.*
A strategy that strictly dominates every other strategy for a player is said to be a dominant strategy. Obviously, with only two strategies, if consent is strictly dominated for Tosca, then stab must be the dominant strategy.
✚ DEFINITION 3.2 A strategy is the dominant strategy if it strictly domi- nates every other strategy.
s–s¿ s–s¿
FIGURE 3.2 The Tosca Game
2,2 4,1
1,4 3,3 Tosca
Scarpia
Stab
Consent
Real Blanks
FIGURE 3.3 Tosca’s Payoffs in the Event That Scarpia Chooses Real Bullets
2 ,2 4,1
1 ,4 3,3 Tosca
Scarpia
Stab
Consent
Real Blanks
*A more formal mathematical presentation of Definitions 3.1 and 3.2 is provided in Section 3.5, which is an appendix to this chapter.
If a strategy is strictly dominated, then it is not optimal for any beliefs re- garding what other players will do; thus, a rational player will avoid using such a strategy. Furthermore, if a player has a dominant strategy, then, if he is rational, he will use it. When each player has a dominant strategy, the unique reasonable solution is that each player uses his or her dominant strategy.
A rational player never uses a strictly dominated strategy. A rational player always uses a dominant strategy.
Before we apply these new tools to a few other games, take note of an in- teresting property of the outcome of Tosca. When Tosca stabs Scarpia and Scarpia has Cavaradossi killed, they each receive a payoff of 2. Now consider the alternative strategy pair in which Tosca consents and Scarpia has the fir- ing squad use blanks, so that Cavaradossi survives. Now Tosca and Scarpia each earn a payoff of 3; they are both better off!
We see in the Tosca situation an important distinction between individual rationality and collective rationality: It is individually rational for Tosca to stab Scarpia (because it is her dominant strategy), and it is individually ra- tional for Scarpia to use real bullets (because it is his dominant strategy); however, it is collectively rational—in the sense that everyone would be bet- ter off—if Tosca and Scarpia were to commit, respectively, to consenting and using blanks. Thus, what may be in an individual’s best interests need not be in the best interests of the group. (We’ll have more to say on this matter later in the book.)