Relax, I’m not referring to actual Sicilians. I’m referring, of course, to Vizzini from the movie “The Princess Bride.” The hero, Westley, is trying to rescue his true love, Buttercup, from the clutches of Vizzini and his henchmen, Inigo Montoya and Fezzik. After outdueling Inigo and knocking out Fezzik, he overtakes Vizzini, who threatens to kill Buttercup if Westley comes any closer. This leads to an impasse: Vizzini cannot escape, but Westley cannot free Buttercup. So, Westley challenges Vizzini to a “battle of wits”:
The structure of the game is simple: there are two glasses of wine. Westley has placed poison (in the form of the odorless, tasteless, yet deadly iocaine powder) somewhere among the two cups, and allows Vizzini to choose which to take. Afterwards, they drink, and they see “who is right, and who is dead.”
Presumably, when Vizzini encounters the game, he is supposed to think that Westley has restricted himself to poisoning one of the glasses. In this case, we have a standard extensive form game of incomplete information, which is equivalent to a normal-form game:
|Vizzini\Westley||Poison Westley’s cup||Poison Vizzini’s cup|
|Drink Westley’s cup||(Dead, Right)||(Right, Dead)|
|Drink Vizzini’s cup||(Right, Dead)||(Dead, Right)|
|Fig. 1: Battle of Wits (outcomes)
Immediately we see that this game is symmetric (or, more precisely, anti-symmetric), in that whatever doesn’t happen to one player happens to the other. In this way, this game is strategically equivalent to the game of matching pennies. This lets us know right away that the equilibrium outcome is for Westley to randomize 50-50 between the choices: do anything else, and Vizzini has a better chance of winning if he plays optimally, as he could just choose the cup that is less likely to have the poison. Similarly, if Vizzini was a priori less likely to choose a given cup, then that is where Westley should have put the poison.
Yet Vizzini does not reason this way. Instead, he attempts to make vacuous arguments about the psyche of Westley, namely, where Westley would have put the poison. He may be reasoning as if Westley is a behavioral type, but clearly, that’s not the best thing to do in a “battle of wits,” where presumably everyone is rational. Instead of making the game-theoretic choice based on mixed strategies, he tries to find an optimal pure strategy.
In the end, Vizzini takes his own cup, which indeed contains the poison. As it turns out, both cups contained poison: Westley has built up tolerance to iocaine, and so it didn’t make any difference which was chosen. So in a way, Westley did make Vizzini indifferent between the two outcomes; it’s just that Vizzini was mistaken in which game was being played. In reality, no matter what, Vizzini would be dead, and Westley would win. This makes one wonder that perhaps Vizzini should have thought something was afoul when Westley proposed the game in the first place, and even more so when he falls for such an obvious trick of misdirection which tries to get Westley to look the other way (see 3:04 in the video). But no matter – while Vizzini may have been smarter than Plato, Aristotle, and Socrates, he could have used some of the 20th century wisdom of John Nash.