On the Analysis of Kelly Criterion and Its Application

Abstract

We analyze the return of a game for a gambler after bidding $T$ time steps. Consider a gamble with known odds and win rate, the optimal solution is to use Kelly criterion which determines the optimal fraction in each bidding step. In this paper we show that the logarithm of return when bidding optimal fraction is ${\text{KL}}\left( {{\text{R}}||{\text{P}}(b)} \right) - {\text{KL}}\left( {{\text{R}}||{\text{P}}} \right)$, where ${\text{R}}$ is the proportion of winning\losing outcome in $T$ time steps, ${\text{P}}\left( b \right)$ is the risk-neutral probability corresponding to odds $b$, and ${\text{P}}$ is the gambler’s individual belief about the win probability of the game. This argument shows that, in a gamble with fixed odds, the KL divergence of the win\lose proportion, say ${\text{R}}$, and the win rate, say ${\text{P}}$, determines the portion of the losing amount. On the other hand, the profit is determined by the proportion ${\text{R}}$ and the odds $b$, irrelevant to win probability ${\text{P}}$. Any improvement is not obtainable even when the win probability is estimated precisely in advance.