1. The President P has been accused of colluding with Russian intelligence to tilt the election in his favor. Let g denote "how guilty" the President is of these charges. Assume g is either 0, 1, or 2, where 2 means "guilty as sin" and 0 means "not at all guilty." Assume that g is chosen by nature and a voter V believes each value is equally likely (voters are pretty skeptical of politicians these days). A special counsel S has been appointed to investigate the accusation. If S is allowed to complete her work, she will learn the value of g and report it to the voter V . However, P might fire S before she is allowed to complete her work, in which case S cannot report anything to V . Assume that nature draws g, shows it to P, and then P decides whether to fire S or not. If so, the "game" ends; if not, S reports g to V , and the game ends. Assume P's utility is V's expectation of g at the end of the game (i.e., P is worse off, the more guilty the voter thinks he is).

(a)What are the possible types of P? What is the prior distribution over these types?

(b)What are the possible pure strategies for P?

(c)Suppose P's strategy is "Fire S for all g." When V observes that S was fired, what is V 's expected value of g?

(d)Continuing the previous part, is it a best response for type g = 0 to fire S, given that types g = 1 and 2 do so?

(e)Is it a perfect Bayesian equilibrium for all types g to fire S?

(f)Is it a perfect Bayesian equilibrium for no types g to fire S?