Additive Comparisons of Stop Rule and Supremum Expectations of Uniformly Bounded Independent Random Variables

Let XI, X2, . . . be independent random variables taking values in [a, b], and let T denote the stop rules for X1, X2, Then E(sup,, X,,) sup{EX,: t E T) < (1/4)(b a), and this bound is best possible. Probabilistically, this says that if a prophet (player with complete foresight) makes a side payment of (b a)/8 to a gambler (player using nonanticipating stop rules), the game becomes at least fair for the gambler.