Entropy-Based Strategies for Multi-Bracket Pools

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.