An Adaptive Kelly Betting Strategy for Finite Repeated Games

Abstract

Kelly criterion is the optimal bidding strategy when considering a series of gambles with the wining probability p and the odds $b$. One of the arguments is Kelly criterion is optimal in theory rather than in practice. In this paper we show the results of using Kelly criterion in a gamble of bidding T steps. At the end of T steps, there are $W$ times of winning and $L$ times of losing. i.e. $T = W + L$. Consequently, the best strategy for these bidding steps is using the probability $W/T$ instead of using $p$ in Kelly Criterion. However, we do not know the number of $W$, to put it better the information of $p$, before placing the bet. We first derive the relation of profits between using p and $W/T$ as the winning probability in the Kelly formula, respectively. Then we use the proportion of winning and bidding numbers before time step t, denoted as $p_{t}$, as the winning probability used in the Kelly criterion at time step $t$. Even we do not know the winning probability of $p$ in a gamble, we can use this method to achieve the profit near the optimal profit when using $p$ in the Kelly betting.