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 protestif at least m of the other players also protests, and it isif 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?