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 (X1, X2, …,Xm) ∼ F, where Xi ⩾ 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 $\underline{B} \in \mathbb{R}^{m} , \underline{B} \geqslant \underline{0}, E \sum\nolimits_{i = 1}^{m} \underline{B}_{i} = 1$. The payoff to player 1 for policy $\underline{B}_{1} \; \textrm{vs.} \; \underline{B}_{2}$ is $ P \{ \underline{B}^{t}_{1}\underline{X} \geqslant \underline{B}^{t}_{2}\underline{X} \}$. The optimal policy is shown to be $\underline{B}^{*} = U\underline{b}^{*}$, where U is a random variable uniformly distributed on [0, 2], and $\underline{b}^{*}$ maximizes E In $\underline{b}^{\prime} \underline{X}$ over $\underline{b} \geqslant \underline{0}, \sum {b_{i} = 1}$.Curiously, this competitively optimal investment policy $\underline{b}^{*}$ is the same policy that achieves the maximum possible growth rate of capital in repeated independent investments (Breiman (1961) and Kelly (1956)). Thus the immediate goal of outperforming another investor is perfectly compatible with maximizing the asymptotic rate of return.