Question

Constantin Brancusi’s sculpture Bird in Space is on exhibit at MoMA, and four thieves contemplate a heist. It’s risky for them to communicate, and each thief has to decide independently whether to participate in the heist (strategy P) or not (strategy N). Due to MoMA’s elaborate security system, thieves need to work in pairs to succeed: the heist is successful only if exactly two or all four thieves participate. If the heist is successful, each of the four thieves gets a benefit of 2 (even those who don’t participate in the heist). Every thief who participates in the heist bears a cost of 1 (regardless of whether the heist succeeds or fails). A thief who doesn’t participate in the heist has cost 0. Therefore, the payoff of a thief who chooses P is 1 if the heist succeeds, and is -1 if the heist fails, while the payoff of a thief who chooses N is 2 if the heist succeeds, and is 0 if the heist fails.

(a) How many pure strategy profiles does this game have?

(b) What’s the best response of a thief to every possible combination of strategies for the other three thieves?

(c) Find all the pure-strategy Nash equilibria of this game. What’s the number of purestrategy Nash equilibria in the game?

(d) [Extra Credit] Compute a symmetric mixed-strategy Nash equilibrium in this game in which every thief participates in the heist with the same probability p ∈ (0, 1).

Answer 1