Consider a decision-maker whose subjective probability distribution over the set of possible states Ω is p = (p(s)) We ask him to tell us his subjective probability distribution, but he can lie and report any distribution in Δ(SZ) that he wants. To guide his reporting decision, we plan to give him some reward Y(q,!) that will be a function of the probability distribution q that he reports and the true state of nature .f that will be subsequently observed.
a. Suppose that his utility for our reward is u(Y(q,$),$) = q(s), for every q in 0(1)) and every s in SZ. Will his report q be his true subjective probability distribution p? If not, what will he report?
b. Suppose that his utility for our reward is u(Y(q,$),$) = loge(q(s)), for every q and s. Will his report q be his true subjective probability distribution p? If not, what will he report?