Consider a version of the "social unrest" game analyzed in Chapter 8 (including Exercise 8) with incomplete information. Two people (players 1 and 2) have to simultaneously choose whether to protest (P) or stay home (H). A player who stays home gets a payoff of 0. Player i's payoff of protesting is determined by this player's protest value xi and whether the other player also protests. Specifically, if player i decides to protest, then her payoff is xi - 1 3 if the other player also protests, whereas her payoff is xi - 2 3 if the other player stays home. Each player knows her own protest value but does not observe that of the other player. Thus, xi is player i's type. Assume that x1 and x2 are independently drawn from the uniform distribution on [0, 1].
(a) Calculate the Bayesian Nash equilibrium of this game. (Hint: Note that each player's strategy is a function from her type to {P, H}. Consider cutoff strategies, where player i will protest if and only if xi ≥ yi , for some constant yi .) Document your analysis and report the equilibrium cutoffs y* 1 and y* 2 .
(b) What is the set of rationalizable strategies in this game? Without providing mathematical details, try to give reasoning based on considering (i) a lower bound on the types that protest regardless of their beliefs and (ii) an upper bound on the types that stay home regardless of their beliefs
(c) Consider an n-player version of the game in which the payoff of protest
if at least m of the other players also protests, and it is
if fewer than m of the other players also protests. Can you find values of n and m for which this game has multiple Nash equilibria?