The Volunteer's Dilemma Ten people are arrested after committing a crime. The police lack sufficient resources to investigate the crime thoroughly. The chief investigator therefore presents the suspects with the following proposal: if at least one of them confesses, every suspect who has confessed will serve a one-year jail sentence, and all the rest will be released. If no one confesses to the crime, the police will continue their investigation, at the end of which each one of them will receive a ten-year jail sentence.
(a) Write down this situation as a strategic-form game, where the set of players is the set of people arrested, and the utility of each player (suspect) is 10 minus the number of years he spends in jail.
(b) Find all the equilibrium points in pure strategies. What is the intuitive meaning of such an equilibrium, and under what conditions is it reasonable for such an equilibrium to be attained?
(c) Find a symmetric equilibrium in mixed strategies. What is the probability that at this equilibrium no one volunteers to confess?
(d) Suppose the number of suspects is not 10, but n. Find a symmetric equilibrium in mixed strategies. What is the limit, as n goes to infinity, of the probability that in a symmetric equilibrium no one volunteers ? What can we conclude from this analysis for the topic of volunteering in large groups?