 
Summary: SOLVENCY GAMES
NOAM BERGER, NEVIN KAPUR, LEONARD J. SCHULMAN, AND VIJAY V. VAZIRANI
Abstract. We study the decision theory of a maximally riskaverse 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 different 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.
1. Introduction
A key concern in computer science and operations research is decisionmaking under uncertainty.
We define a very simple game that helps us study the issue of solvency, or indefinite survival, in
the presence of stochastic uncertainties. In Section 1.1 below we provide some motivating reasons
for studying this issue.
We start by defining the model. A state of the game is an integer, which we call the wealth of
the player. An action (representing, say, an investment choice) is a finitely supported probability
distribution on the integers; this distribution specifies the probabilities with which various payoffs
are received, if this action is chosen. Let w be the wealth of the player at time t. Let A be a set
of actions. Suppose that after choosing a particular action from A, the random variable sampled
from that action is a. Then at time t + 1 the wealth of the player is w + a. The game terminates
