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Online Learning in Betting Markets: Profit versus Prediction

Haiqing Zhu, Alexander Soen, Yun Kuen Cheung, Lexing Xie

TL;DR

This work develops a theoretical and algorithmic framework for online price-setting in binary betting markets, contrasting profit maximization with information elicitation. By modeling Kelly bettors and bookmaker beliefs under both unfair and fair odds, it reveals a fundamental incompatibility between profit and prediction, and shows that heavier-tailed bettor-belief distributions can enlarge bookmaker profit. The authors introduce two online algorithms—stochastic approximation (SA) for unfair odds and Follow The Leader (FTL) for fair odds—proving sublinear regret bounds and, in the fair-odds case, convergence to the global optimum. Empirical results corroborate the theory, demonstrating robust performance of the online methods across diverse belief distributions and highlighting the practical relevance for understanding large-scale, multi-agent betting and information markets. The findings offer a principled bridge between betting-market profitability and prediction-market information aggregation, suggesting broader implications for online multi-agent systems.

Abstract

We examine two types of binary betting markets, whose primary goal is for profit (such as sports gambling) or to gain information (such as prediction markets). We articulate the interplay between belief and price-setting to analyse both types of markets, and show that the goals of maximising bookmaker profit and eliciting information are fundamentally incompatible. A key insight is that profit hinges on the deviation between (the distribution of) bettor and true beliefs, and that heavier tails in bettor belief distribution imply higher profit. Our algorithmic contribution is to introduce online learning methods for price-setting. Traditionally bookmakers update their prices rather infrequently, we present two algorithms that guide price updates upon seeing each bet, assuming very little of bettor belief distributions. The online pricing algorithm achieves stochastic regret of $\mathcal{O}(\sqrt{T})$ against the worst local maximum, or $ \mathcal{O}(\sqrt{T \log T}) $ with high probability against the global maximum under fair odds. More broadly, the inherent trade-off between profit and information-seeking in binary betting may inspire new understandings of large-scale multi-agent behaviour.

Online Learning in Betting Markets: Profit versus Prediction

TL;DR

This work develops a theoretical and algorithmic framework for online price-setting in binary betting markets, contrasting profit maximization with information elicitation. By modeling Kelly bettors and bookmaker beliefs under both unfair and fair odds, it reveals a fundamental incompatibility between profit and prediction, and shows that heavier-tailed bettor-belief distributions can enlarge bookmaker profit. The authors introduce two online algorithms—stochastic approximation (SA) for unfair odds and Follow The Leader (FTL) for fair odds—proving sublinear regret bounds and, in the fair-odds case, convergence to the global optimum. Empirical results corroborate the theory, demonstrating robust performance of the online methods across diverse belief distributions and highlighting the practical relevance for understanding large-scale, multi-agent betting and information markets. The findings offer a principled bridge between betting-market profitability and prediction-market information aggregation, suggesting broader implications for online multi-agent systems.

Abstract

We examine two types of binary betting markets, whose primary goal is for profit (such as sports gambling) or to gain information (such as prediction markets). We articulate the interplay between belief and price-setting to analyse both types of markets, and show that the goals of maximising bookmaker profit and eliciting information are fundamentally incompatible. A key insight is that profit hinges on the deviation between (the distribution of) bettor and true beliefs, and that heavier tails in bettor belief distribution imply higher profit. Our algorithmic contribution is to introduce online learning methods for price-setting. Traditionally bookmakers update their prices rather infrequently, we present two algorithms that guide price updates upon seeing each bet, assuming very little of bettor belief distributions. The online pricing algorithm achieves stochastic regret of against the worst local maximum, or with high probability against the global maximum under fair odds. More broadly, the inherent trade-off between profit and information-seeking in binary betting may inspire new understandings of large-scale multi-agent behaviour.
Paper Structure (46 sections, 37 theorems, 136 equations, 8 figures, 2 tables, 3 algorithms)

This paper contains 46 sections, 37 theorems, 136 equations, 8 figures, 2 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Related work.
  3. Our contributions.
  4. Betting Markets
  5. Kelly bettors and their optimal strategies
  6. The bookmaker: maximising expected profit
  7. The bookmaker's belief versus event probabilities.
  8. Profit under fair odds and prediction market.
  9. Equilibria in Betting Markets
  10. Example: Profit maximisation and prediction aggregation are incompatible.
  11. Online Learning under Unfair Odds
  12. Setting.
  13. Stochastic Approximation (SA).
  14. Convergence.
  15. Regret.
  16. ...and 31 more sections

Key Result

Lemma 3.1

For any fixed $T$, the profit function $u_{1:T}$ in eq:argmax-profit is upper-bounded, and it admits at least one maximiser $({a}\xspace^\star, {b}\xspace^\star) \in (0,1)^2$.

Figures (8)

  • Figure 1: Left: Probability density function $f_{0.75,0.25,0.1}(p)$. Right: Profit function, with profit-maximising prices $({a}\xspace^\star, {b}\xspace^\star) = (0.70710,0.63395)$ marked by the green point.
  • Figure 2: Simulation of \ref{['alg:SA']} with 100,000 bettors. Left: The distribution of bettors' beliefs, unknown to bookmaker. Middle: Regrets over 100,000 iterations, comparing \ref{['alg:SA']} under different initialisation to risk-balancing. Markers "$\times$" indicate number of iterations on a log-scale $\{10^1, \ldots,10^5\}$. Right: Contour plot of $\delta(a,b) \stackrel{\mathrm{.}}{=} u_{1:T}({a}\xspace^\star,{b}\xspace^\star) - u_{1:T}({a}\xspace,{b}\xspace)$ where $T = 10^5$, darker colours means closer to maximum profit.
  • Figure 3: Simulating FTL and LMSR over 100,000 iterations. Left: Regret. Right: Price trajectory ${a}\xspace_t$.
  • Figure I: Illustrations of the roots of \ref{['app-eq: uniqueness']}. The dashed red line represents the value of RHS and all others represent the LHS with expectations taken w.r.t. different distributions. (left plot) Distributions used in \ref{['sec:Empirical']} and \ref{['app: Empirical']}. (right plot) Truncated exponential distributions with different parameters $\lambda \in \{1,2,5\}$.
  • Figure II: Illustrations of the roots of \ref{['app-eq: uniqueness']}. The dashed red line represents the value of RHS and all others represent the LHS with expectations taken w.r.t. different belief distributions. (left plot) Truncated Gaussian distributions with different means $\{0.5,0.6, 0.7, 0.8, 0.9\}$ and the same variance $0.2$. (right plot) Truncated Gaussian distributions with the same mean $0.2$ but different variances $\{0.2,0.1,0.05\}$.
  • ...and 3 more figures

Theorems & Definitions (60)

  • Definition 3.1
  • Lemma 3.1
  • Lemma 3.1
  • Proposition 3.1
  • Definition 3.2: dentcheva2003optimization
  • Proposition 3.2
  • Theorem 4.1
  • Definition 4.2: RM1951StochasticApproximation
  • Theorem 4.3
  • Theorem 4.4
  • ...and 50 more