Entropy-Based Strategies for Multi-Bracket Pools

TL;DR

The paper addresses the challenge of generating multiple bracket predictions in high-dimensional parimutuel pools. It proposes an entropy-based, information-theoretically grounded approach that samples $n$ i.i.d. brackets from a parametric distribution, with performance improving as bracket entropy increases with $n$ and with opponent entropy. Through a canonical bitstring guessing example and real-world cases (pick six and March Madness), the authors show that optimizing bracket distribution entropy yields tractable solutions and practical gains, especially under favorable carryover or against diverse opponent strategies. The framework connects to the Asymptotic Equipartition Property and provides actionable guidance for designing multi-bracket strategies with scalable performance. The work lays a foundation for further refinement under unknown probabilities and more nuanced, round-wise entropy tuning.

Abstract

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 $n$ predicted tuples? The most general version of this problem is extremely difficult, so we begin with a simpler setting. In particular, we generate $n$ independent predicted tuples according to a distribution having optimal entropy. This entropy-based approach is tractable, scalable, and performs well.

Figures (11)

• Figure 1: The expected maximum Hamming score ($y$-axis) of $n$ submitted Bernoulli($q$) bitstrings relative to a reference Bernoulli($p$) bitstring as a function of $p$ ($x$-axis), $q$ (color), and $n$ (facet) in the "guessing a randomly drawn bitstring" contest with $p \equiv p_{\mathsf{rd} }$, $q \equiv q_{\mathsf{rd} }$, $r \equiv r_{\mathsf{rd} }$, and $R=6$ rounds. As $n$ increases, we want to increase the entropy of our submitted brackets.
• Figure 2: The probability (color) that the maximum Hamming score of $n$ submitted Bernoulli($q$) brackets relative to a reference Bernoulli($p$) bracket exceeds that of $k$ opposing Bernoulli($r$) brackets as a function of $q$ ($y$-axis), $r$ ($x$-axis), and $n$ (facet) for $p=0.75$ and $k=100$ in the "guessing a randomly drawn bitstring" contest with $p \equiv p_{\mathsf{rd} }$, $q \equiv q_{\mathsf{rd} }$, $r \equiv r_{\mathsf{rd} }$, and $R=6$ rounds. We should increase entropy as $n$ increases and as our opponents' entropy increases.
• Figure 3: The expected maximum ESPN score ($y$-axis) of $n$ submitted bitstrings, with Bernoulli($q_E$) bits in early rounds and Bernoulli($q_L$) bits in later rounds, relative to a reference Bernoulli($p$) bitstring as a function of $q_E$ ($x$-axis), $q_L$ (color), $n$ (columns), and the partition $(q_E, q_L)$ (rows) in the "guessing a randomly drawn bitstring" contest with $R=6$ rounds and $p=0.75$. The circles indicates the best strategy in each setting. As $n$ increases, we want to increase the entropy of our bracket predictions in both early and late rounds.
• Figure 4: The probability (color) that the maximum ESPN score of $n$ bitstrings, with Bernoulli($q_E$) bits in early rounds and Bernoulli($q_L$) bits in later rounds, relative to a reference Bernoulli($p$) bitstring exceeds that of $k$ opposing bitstrings, with Bernoulli($r_E$) bits in early rounds and Bernoulli($r_L$) bits in later rounds, as a function of $q_E$ ($y$-axis), $r_E$ ($x$-axis), $q_L$ (rows), and $r_L$ (columns) for $p=0.75$, $k=100$. and $n=100$ in the "guessing a randomly drawn bitstring" contest with $R=6$ rounds. Figure (a) uses the partition where the first three rounds are the early rounds (e.g., $q_E = q_1 = q_2 = q_3$ and $r_E = r_1 = r_2 = r_3$) and Figure (b) uses the partition where just the first round is an early round (e.g., $q_E = q_1$ and $r_E = r_1$). We should still increase the entropy of our bracket predictions as our opponents increase entropy.
• Figure 5: Note that these figures are not drawn to scale. First line: the probability mass of an individual low entropy (chalky) bracket is much larger than the probability mass of an individual typical bracket, which is much larger than the probability mass of an individual high entropy (rare) bracket. Second line: there are exponentially more rare brackets than typical brackets, and there are exponentially more typical brackets than chalky brackets. Third line: the typical brackets occupy most of the probability mass on aggregate.

Theorems & Definitions (2)

• Theorem 1: Asymptotic Equipartition Property
• proof : Proof (Theorem \ref{['thm:equipartition_thm']})