Analysis of Kelly betting on finite repeated games

摘要

The Kelly criterion can be used to maximize returns in a game with win rate p and odds b; however, optimization theoretically requires wagering over an infinite number of time steps. Despite the fact that Kelly's theory has been extended to most of the trading strategies used in financial markets, there is still a large gap between the theoretical determination of optimal bidding fractions and practical application of these methods. In this paper, we illustrate the difference between the theoretical and simulation results obtained from a gambling situation involving a finite number of bidding steps T( = W + L), where W and L respectively denote the numbers of wins and losses. The optimal bidding fraction based on the Kelly criterion should employ the win\loss proportion W/T rather than the win rate p; however, it is not possible to obtain the value of a priori. W. Thus, profits under the Kelly formula are calculated by applying win rate p and the win\lose proportion W/T. In this paper, we denote pt as the current win\lose proportion before time step t as an alternative to win rate p. The proposed approach does away with the need to apply win rate p and produces profits that are nearly optimal under Kelly betting.