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Joint work with Noam Berger, Nevin Kapur and Vijay V. Vazirani We study the decision theory of a maximally risk-averse investor --- one whose objective, in the face of stochastic uncertainties, is to minimize the probability of ever going broke. We examine a very simple model and obtain a characterization of best play by investors. Among other things, the model shows how poor and rich players, facing the same choices, may have different best strategies. It also makes some perhaps counterintuitive predictions about best play.