Optimal Strategies For Repeated Games

We extend the optimal strategy results of Kelly and Breiman and extend the class of random variables to which they apply from discrete to arbitrary random variables with expectations. Let Fn be the fortune obtained at the nth time period by using any given strategy and let $F_{n}^{*}$ be the fortune obtained by using the Kelly–Breiman strategy. We show (Theorem 1(i)) that $F_{n} / F_{n}^{*}$ is a supermartingale with $E(F_{n} / F_{n}^{*}) \leqq 1$ and, consequently, $E(\lim {F_{n} / F_{n}^{*}) \leqq 1$. This establishes one sense in which the Kelly–Breiman strategy is optimal. However, this criterion for ‘optimality’ is blunted by our result (Theorem 1(ii)) that $E(F_{n} / F_{n}^{*}) = 1$ for many strategies differing from the Kelly–Breiman strategy. This ambiguity is resolved, to some extent, by our result (Theorem 2) that $F_{n}^{*} / F_{n}$ is a submartingale with $E(F_{n}^{*} / F_{n}) \geqq 1$ and $E(\lim {F_{n}^{*}} / F_{n}) \geqq 1$; and $E(F_{n}^{*} / F_{n}) = 1$ if and only if at each time period j, 1≦j≦n, the strategies leading to Fn and $F_{n}^{*}$ are ‘the same’.