Abstract
A prophet is a player with complete foresight; a gambler knows only the past and the present, but not the future. If each of them bets on differences of consecutive nonnegative random variables $X_i$ such that $E(X_i|X_{i - 1}) = EX_i$, the players multiplying their stakes by uniformly bounded variables, then the expected gain of the prophet is at most three times that of the gambler. The constant 3 is optimal.
Citation
Ulrich Krengel. Louis Sucheston. "Prophet Compared to Gambler: An Inequality for Transforms of Processes." Ann. Probab. 15 (4) 1593 - 1599, October, 1987. https://doi.org/10.1214/aop/1176991996
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