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.

摘要

1956 年，约翰·凯利制定了一种最优策略，即所谓的“凯利准则”，用于在已知获胜几率和概率的情况下在有利游戏的每一步出价。凯利准则用于理论上最大化长期回报。但是，在实际情况下，不可能无限时间地玩游戏。在这项研究中，我们分析了当玩家必须在有限的时间步数后出价时游戏的回报。我们证明了在对有限时间步长的最优分数（对应于玩家对事件发生的信念概率与实际结果的比例之间的差异）出价时游戏回报的对数。而且，当上述两者相等时，回报的对数达到最大值。最后，出于说明目的，进行了一些模拟实验。我们还提供了一个股票交易案例来证明本研究的实用性。