Competitive Optimality of Logarithmic Investment

Consider the two-person zero-sum game in which two investors are each allowed to invest in a market with stocks ( X 1 , X 2 , ..., X m ) (sim) F , where X i (ge) 0. Each investor has one unit of capital. The goal is to achieve more money than one’s opponent. Allowable portfolio strategies are random investment policies B (in) R m , B (ge) 0 , E (sum) m i = 1 B i = 1. The payoff to player 1 for policy B 1 vs. B 2 is P { B t 1 X (ge) B t 2 X }. The optimal policy is shown to be B * = U b *, where U is a random variable uniformly distributed on [0, 2], and b * maximizes E ln b t X over b (ge) 0 , (sum) b i = 1 .Curiously, this competitively optimal investment policy b * is the same policy that achieves the maximum possible growth rate of capital in repeated independent investments (Breiman [Breiman, L. 1961. Optimal gambling systems for favorable games. Fourth Berkeley Symposium. 1 65--78.] and Kelly [Kelly, J. 1956. A new interpretation of information rate. Bell System Tech. J. 917--926.]). Thus the immediate goal of outperforming another investor is perfectly compatible with maximizing the asymptotic rate of return.