Brill Ryan S, Wyner Abraham J, Barnett Ian J
Graduate Group in Applied Mathematics and Computational Science, University of Pennsylvania, Philadelphia, PA 19104, USA.
Department of Statistics and Data Science, The Wharton School, University of Pennsylvania, Philadelphia, PA 19104, USA.
Entropy (Basel). 2024 Jul 23;26(8):615. doi: 10.3390/e26080615.
Much work in the parimutuel betting literature has discussed estimating event outcome probabilities or developing optimal wagering strategies, particularly for horse race betting. Some betting pools, however, involve betting not just on a single event, but on a tuple of events. For example, pick six betting in horse racing, March Madness bracket challenges, and predicting a randomly drawn bitstring each involve making a series of individual forecasts. Although traditional optimal wagering strategies work well when the size of the tuple is very small (e.g., betting on the winner of a horse race), they are intractable for more general betting pools in higher dimensions (e.g., March Madness bracket challenges). Hence we pose the multi-brackets problem: supposing we wish to predict a tuple of events and that we know the true probabilities of each potential outcome of each event, what is the best way to tractably generate a set of predicted tuples? The most general version of this problem is extremely difficult, so we begin with a simpler setting. In particular, we generate independent predicted tuples according to a distribution having optimal entropy. This entropy-based approach is tractable, scalable, and performs well.
在赛马投注文献中的许多工作都讨论了估计赛事结果概率或制定最优投注策略，尤其是针对赛马投注。然而，一些投注池不仅涉及对单个赛事进行投注，还涉及对一组赛事进行投注。例如，赛马中的六连胜投注、疯狂三月赛程挑战赛以及预测随机抽取的比特串，都涉及做出一系列单独的预测。尽管当事件组的规模非常小时（例如，对一场赛马的获胜者进行投注），传统的最优投注策略效果良好，但对于更高维度的更一般投注池（例如，疯狂三月赛程挑战赛），它们却难以处理。因此，我们提出了多赛程问题：假设我们希望预测一组赛事，并且我们知道每个赛事每个潜在结果的真实概率，那么以可处理的方式生成一组预测组合的最佳方法是什么？这个问题的最一般形式极其困难，所以我们从一个更简单的设置开始。具体来说，我们根据具有最优熵的分布生成独立的预测组合。这种基于熵的方法是可处理的、可扩展的，并且表现良好。
