Question

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 1, 2, or 3, where 3 means “guilty as sin” and 1 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 Ior not. If so, the “game” ends; if not, S reports g to V , and the game ends. Assume P ’s utility is ?1 times V ’s expectation of g at the end of the game (i.e., P is better off, the lower is the voter’s assessment of his guilt).

(c) Suppose all types except g = 1 fire S. What does V think is the expected value of g, given that S is fired? Is type g = 2 happy with that?

(d) Is it a perfect Bayesian equilibrium for all types to fire S? For type g = 1 not to fire S, and for the other types to fire S?

(e) Assuming P is rational, what should V conclude about g, if S is fired?

Answer 1

(c) S is fired if and only if g=2 or 3.

If S is fired then V knows that g is 2 or 3.

P(g is 2| S is fired)= P(g is 2 and S is fired)/P(S is fired)=P(g is 2)/ ( P(g is 2)+P( g is 3) ) = (1/3)/(1/3+1/3)=1/2

Hence P(g is 3| S is fired)=1/2 as well.

Hence E(g|S is fired)=2*1/2+3*1/2=2.5.

However, had g been 2, had P not fired S, V would have known g to be 2 and hence P would have gained a utility of -2, which is higher than -2.5, which is the utility in the case when S is fired. Hence g=2 would not be happy with that.

(d) Suppose the prior belief is that all types fire S.

Then if P is of type g=1, it gains a utility of -1 if S is not fired and -(1+2+3)*1/3=-2 if S is fired.

If P is of type g=2, it gains a utility of -2 if or if not S is fired.

If P is of type g=3, it gains a utility of -3 if S is not fired but -2 if S is fired.

Hence the equilibirum strategy for type g=1 is not to fire S, but for g=3 is to fire S and for g=2, it is just indifferent between firing or not firing S given this belief.

(e) However, now, suppose if S is fired if when g is 2, given that only types 2 or 3 fire S, then the expected utility of P is -2.5, which is less had P not fired S. Hence P would refrain from firing S if it is of type 2.

Hence if P is rational and S is fired, then V should conclude that P is of type g=3.