Abstract. We study the decision theory of a maximally risk-averse investor | one whose objec-
tive, in the face of stochastic uncertainties, is to minimize the probability of ever going broke. With
a view to developing the mathematical basics of such a theory, we start with a very simple model
and obtain the following results: a characterization of best play by investors; an explanation of
why poor and rich players may have di®erent best strategies; an explanation of why expectation-
maximization is not necessarily the best strategy even for rich players. For computation of optimal
play, we show how to apply the Value Iteration method, and prove a bound on its convergence rate.