Optimal Online Bookmaking for Any Number of Outcomes

2025-06-19

AI Summary

This paper investigates the dynamic odds-setting problem for bookmakers in online gambling: how to design an odds strategy that maximizes expected revenue while strictly bounding losses against adversarial bettors—i.e., under worst-case betting behavior, arbitrary numbers of outcomes, and rounds. We propose the Bellman–Pareto frontier framework, which for the first time exactly characterizes the optimal loss as the largest root of a Hermite-type polynomial. Leveraging a vectorized repeated-gaming model, we integrate dynamic programming with polynomial root theory to derive tight theoretical bounds. We further design an efficient adaptive algorithm that achieves the loss lower bound against optimal bettors and attains opportunistic optimality against suboptimal ones, while guaranteeing zero bankruptcy risk and fairness. Key innovations include an explicit analytical characterization of loss, a deep connection between the Pareto frontier and special orthogonal polynomials, and a robust-yet-adaptive odds-adjustment mechanism.

Technology Category

• Game Theory and Economic Paradigms: Adversarial Learning
• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Online Learning & Bandits

Application Category

• Economics, Online Markets and Human Computation: Uses of LLMs and GenAI for marketplace design, bidding, and strategic interactions
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: Attacks and countermeasures in recommendation systems

Abstract

We study the Online Bookmaking problem, where a bookmaker dynamically updates betting odds on the possible outcomes of an event. In each betting round, the bookmaker can adjust the odds based on the cumulative betting behavior of gamblers, aiming to maximize profit while mitigating potential loss. We show that for any event and any number of betting rounds, in a worst-case setting over all possible gamblers and outcome realizations, the bookmaker's optimal loss is the largest root of a simple polynomial. Our solution shows that bookmakers can be as fair as desired while avoiding financial risk, and the explicit characterization reveals an intriguing relation between the bookmaker's regret and Hermite polynomials. We develop an efficient algorithm that computes the optimal bookmaking strategy: when facing an optimal gambler, the algorithm achieves the optimal loss, and in rounds where the gambler is suboptimal, it reduces the achieved loss to the optimal opportunistic loss, a notion that is related to subgame perfect Nash equilibrium. The key technical contribution to achieve these results is an explicit characterization of the Bellman-Pareto frontier, which unifies the dynamic programming updates for Bellman's value function with the multi-criteria optimization framework of the Pareto frontier in the context of vector repeated games.

Problem

• Dynamic betting odds adjustment for profit maximization
• Optimal loss characterization via polynomial roots
• Efficient algorithm for subgame perfect Nash equilibrium

Innovation

• Dynamic odds adjustment based on gamblers' behavior
• Optimal loss characterized by polynomial roots
• Efficient algorithm for Bellman-Pareto frontier optimization