Question

Three friends saw a robbery happen (Y, X, and Z). Each friend can do one of the following: call the police or not, and they each decide simultaneously. If nobody calls they each get a payoff of 0. If anyone calls, the person who calls gets a payoff of 2/3 while the others get a payoff of 1.

a) Find the symmetric Nash equilibrium.

b) Assume N people saw the robbery. Find the symmetric Nash Equilibrium. Find the probability that no one calls. What happens to this probability as N -> infinity?

Answer 1

a) There is no symmetric nash equilibrium in this case. A symmetric nash equilibrium is the one in which each player chooses the same strategy. If no one calls then each would get a better payoff by calling i.e. from 0 to 2/3 and if everyone calls each would get a better pay off by choosing not to call i.e. from 2/3 to 1. Hence each player has a tendency to deviate provided other players dont change their strategies.

b) Again the argument for the non existence of symmetric nash equilibrium is same as part a).

The probability that no one calls is 1/2^N because out of the total of 2^N case of startegies (each of the N player has 2 strategies) there is only case where no one calls.

As N increases the denominator increases and hence the probability that no one calls also decreases