Abstract
This note considers the average-optimal expected return of two players observing independent random variables X 1, … , X n , whose distributions are generated at random. One player, the pseudo prophet, knows the distributions prior to observing the random variables. The other player, the gambler, has no such foresight. Sharp difference and ratio comparisons of the two players' optimal expected returns are given. The key step in the proof is a reduction to a classical prophet inequality for i.i.d. random variables proved by Hill and Kertz (Hill, T.P.; Kertz, R.P. Comparisons of stop rule and supremum expectations of i.i.d. random variables. Ann. Probab. 1982, 10 (2), 336–345).