论文标题
经典的凯利博彩公式对时间相关的情况
A Generalization of the Classical Kelly Betting Formula to the Case of Temporal Correlation
论文作者
论文摘要
对于连续博彩游戏，凯利的理论旨在最大化一个人的帐户价值的对数增长，涉及对所谓的投注分数$ k $的优化。在这封信中，我们扩展了经典的配方，以允许赌注之间的时间相关性。为了证明这种新范式的潜力，为了简单，我们主要解决了以偶数收益的货币投资游戏的案例。为此，我们解决了内存深度$ m $的问题。这样，我们的意思是，不再认为硬币翻转的结果是i.i.d.random变量。取而代之的是，翻转$ k $的头部可能性取决于以前的翻转$ k-1，k-2，...，k-m $。对于最简单的$ n $翻转情况，$ m = 1 $，我们获得了一个封闭式解决方案$ k_n $，用于最佳投注分数。这概括了无内存情况的经典结果。也就是说，我们的新分数$ k_n $而不是分数$ k^* = 2p-1 $，它遍布具有头部概率$ p \ geq 1/2 $的硬币的文献，取决于$ n $和与时间相关相关的参数。这些结果的概括为$ m> 1 $，还包括数值模拟。最后，我们指出该理论如何扩展到时变的反馈和替代性收益分布。
For sequential betting games, Kelly's theory, aimed at maximization of the logarithmic growth of one's account value, involves optimization of the so-called betting fraction $K$. In this Letter, we extend the classical formulation to allow for temporal correlation among bets. To demonstrate the potential of this new paradigm, for simplicity of exposition, we mainly address the case of a coin-flipping game with even-money payoff. To this end, we solve a problem with memory depth $m$. By this, we mean that the outcomes of coin flips are no longer assumed to be i.i.d.random variables. Instead, the probability of heads on flip $k$ depends on previous flips $k-1,k-2,...,k-m$. For the simplest case of $n$ flips, with $m = 1$, we obtain a closed form solution $K_n$ for the optimal betting fraction. This generalizes the classical result for the memoryless case. That is, instead of fraction $K^* = 2p-1$ which pervades the literature for a coin with probability of heads $p\geq 1/2$, our new fraction $K_n$ depends on both $n$ and the parameters associated with the temporal correlation. Generalizations of these results for $m > 1$ and numerical simulations are also included. Finally, we indicate how the theory extends to time-varying feedback and alternative payoff distributions.
