Game Theory in The Princess Bride
In one famous scene from the 1987 cult classic film, The Princess Bride, two characters challenge each other to a battle of the wits, a battle which draws upon the principles of game theory and Nash equilibrium.
The scene in question can be seen in this link (https://www.youtube.com/watch?v=3EkBuKQEkio)
In this scene, one character known only as the Masked Man poisons one of two cups of wine, with a fictional odorless tasteless poison called Iocane Powder. He places one cup in front of himself and another in front of his rival. The other character, Vizzini, must then decide whether to drink the cup before himself or to drink the cup in front of the Masked Man.
First let’s operate under the assumption that the poisoned cups game works as the masked man explains. the Masked man (M) chooses to poison his cup Sp (self poison) or Vizzini’s cup Op (other poison), and Vizzini (V) must then then decide whether to drink from his cup Sd (self drink) or drink from M‘s cup Od (other drink). In this case there are two binary payoffs 1 (live) or 0 (die).
The Payoff Matrix for this game can be seen below
| Masked Man | |||
| Self Poison | Other Poison | ||
| Self Drink | (1,0) | (0,1) | |
| Vizzini | Other Drink | (0,1) | (1,0) |
It’s obvious that by any pure strategy, there is no Nash equilibrium, regardless of what each character chooses, one of them would have benefited from making the opposite choice.
Therefore, to create an equilibrium each player needs to employ a mixed strategy where M must take a number p to represent the probability that M takes the strategy Sp (and 1-p for Op) and V must take a number q to represent the probability of Sd (and 1-q for Od)
Now, this randomization strategy only works when p and q are both .5, essentially Sd and Od must have the same expected payoff for V to avoid missing an unknown best dominant strategy (vice versa for M). Sd and Od only have the same expected payoff when p = .5 (and by extension 1-p = .5, q=.5, and 1-q=.5). Therefore, through randomization the situation may reach Nash equilibrium. However, V believes that by analyzing his opponent’s mind, he can deduce which strategy M was more likely to take. He employs his wits to determine a more realistic value for p. If p is not equal to .5 then V can take a better bet than M, the system would be out of equilibrium.
In the end, however, it is revealed to the audience that both cups were poisoned. Poor Vizzini never had a chance.
Another interesting example of game theory in the princess bride is when a weakened Wesley challenges Humperdink “to the pain” and Humperdink has to decide to fight or surrender. Depending on if Wesley has actually recovered or is simply bluffing, Humperdink faces a variety of payoffs. https://www.youtube.com/watch?v=I_keWS1i3RA
Afterward: I acknowledge that the topic of the poison cup scene in Princess Bride has been adressed in an older blog post https://blogs.cornell.edu/info2040/2011/09/24/game-theory-in-movies-the-princess-bride/ however I disagree with the author’s assertion of there being two separate payoff matrices, and their analysis fails to acknowledge the possibility of achieving Nash equilibrium through a mixed, randomized, strategy