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 ) ∼ F, where X i ≥ 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 ∈ R m , B ≥ 0, E ∑ m i = 1 B i = 1. The payoff to player 1 for policy B 1 vs. B 2 is P {B t 1 X ≥ B t 2 X}. The optimal policy is shown to be B* = Ub*, where U is a random variable uniformly distributed on [0, 2], and b* maximizes E ln b t X over b ≥ 0, ∑ 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.