Von Neumann–Morgenstern Axioms of Rationality and Inequalities in Analysis

The varying wealth of an agent betting on the results of tossing a fair coin is a classic example of a random process with martingale property. In such a game, there is no strategy that yields a positive expectation of profit. However, the problem of whether one can rationally choose a strategy other than inaction remains meaningful and nontrivial: it turns out that there is some “gap” between complete abandonment of the game and completely irrational economic behavior that violates the basic von Neumann–Morgenstern axioms of rationality. By solving the problem of finding this “gap” together with optimal strategies within it, we will arrive at Bellman functions that previously arose in completely abstract problems of finding sharp constants for inequalities in analysis. Thus, we will obtain a natural economic interpretation for these inequalities and for the corresponding Bellman functions.