Much work in the March Madness literature has discussed how to estimate the probability that any one team beats any other team. There has been strikingly little work, however, on what to do with these win probabilities. Hence we pose the multi-brackets problem: given these probabilities, what is the best way to submit a set of $n$ brackets to a March Madness bracket challenge? This is an extremely difficult question, so we begin with a simpler situation. In particular, we compare various sets of $n$ randomly sampled brackets, subject to different entropy ranges or levels of chalkiness (rougly, chalkier brackets feature fewer upsets). We learn three lessons. First, the observed NCAA tournament is a "typical" bracket with a certain "right" amount of entropy (roughly, a "right" amount of upsets), not a chalky bracket. Second, to maximize the expected score of a set of $n$ randomly sampled brackets, we should be successively less chalky as the number of submitted brackets increases. Third, to maximize the probability of winning a bracket challenge against a field of opposing brackets, we should tailor the chalkiness of our brackets to the chalkiness of our opponents' brackets.

翻译：关于疯狂三月（March Madness）的文献中，大量研究探讨了如何估计任意一支球队击败另一支球队的概率。然而，令人惊讶的是，关于如何处理这些胜率的研究却鲜有涉及。因此，我们提出了多支签表问题：给定这些概率，在疯狂三月签表挑战赛中提交一组$n$个签表的最佳方式是什么？这是一个极其困难的问题，因此我们从更简单的情况入手。具体而言，我们比较了多组随机抽样的$n$个签表，这些签表受限于不同的熵值范围或“爆冷程度”（简而言之，爆冷程度越低的签表包含的冷门越少）。我们得出三点结论。第一，观察到的NCAA锦标赛是一个具有特定“合适”熵值（大致对应于“合适”的冷门数量）的“典型”签表，而非低爆冷签表。第二，为了最大化一组随机抽样的$n$个签表的期望得分，随着提交签表数量的增加，我们应逐步降低爆冷程度。第三，为了在面临对手签表群体时最大化赢得签表挑战赛的概率，我们应根据对手签表的爆冷程度来调整自身签表的爆冷程度。