On the Analysis of Kelly Criterion and Its Application

Abstract

We analyze the return of a game for a gambler after bidding (Formula presented) 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 (Formula presented), where (Formula presented) is the proportion of winning\losing outcome in (Formula presented) time steps, (Formula presented) is the risk-neutral probability corresponding to odds (Formula presented), and (Formula presented) 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 (Formula presented), and the win rate, say (Formula presented), determines the portion of the losing amount. On the other hand, the profit is determined by the proportion (Formula presented) and the odds (Formula presented), irrelevant to win probability (Formula presented). Any improvement is not obtainable even when the win probability is estimated precisely in advance.