Three people are arrested after committing a crime. The police lacks sufficient resources to investigate the crime thoroughly. The chief investigator therefore presents the suspects the following proposal. If at least one of them confesses, every suspect who has confessed will serve a 1-year sentence, and the rest will be released. If nobody confesses, then the police will continue the investigation, at the end of which all of the suspects will receive a 9-year sentence.

Consider this situation as a simultaneous-move game, where the payoff of each suspect is 9 minus the number of years spent in prison.

(a) Find all the pure-strategy Nash equilibria. What is the intuitive meaning of such an equilibrium, and under what conditions is it reasonable for such an equilibrium to be played in the context of this game?

(b) Find a symmetric Nash equilibrium in mixed strategies. What is the probability that at this equilibrium nobody volunteers to confess?

(c) Suppose that the number of suspects is not 3, but n. How would the Nash equilibrium in mixed strategies change as n increases? What can you conclude from this analysis for the topic of volunteering in large groups?