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Determination of Profit Quantification on Kelly Betting Based on Relative Entropy

ABSTRACT In 1956, John Kelly formulated an optimal strategy, the so-called ‘Kelly criterion’, for bidding at each step of a favorable game when the odds and probability of winning are known. The Kelly criterion is used to theoretically maximize long-run return. However, in practical situations, it is impossible to play a game for an unlimited time. In this study, we analyze the return of a game when a player must bid after a finite number of time steps. We demonstrate that the logarithm of the return of a game when bidding the optimal fraction (corresponding to the difference between the player’s belief probability of the event occurrence and the proportion of real outcomes) for finite-time steps. Moreover, the maximum of the logarithm of the return is achieved when the above two are equal. Finally, a few simulation experiments are conducted for illustrative purposes. We also present an example of a case in stock trading to demonstrate the practicality of this study.