Optimal Online Bookmaking for Binary Games

2025-01-12

AI Summary

This paper addresses the worst-case optimal dynamic pricing problem for bookmakers in binary-event online betting: how to adapt odds in real time to maximize the minimum guaranteed revenue against arbitrary bettor strategies and arbitrary outcome sequences. We formulate the “optimal online betting game” and derive, for the first time, a closed-form solution for the worst-case optimal strategy in the binary setting. A novel data structure—bi-balancing trees—is introduced to enforce strict loss balancing across all deterministic betting paths. Leveraging game-theoretic modeling, recursive strategy construction, and worst-case optimization, we rigorously prove the robust optimality of the proposed strategy. Crucially, the approach imposes no probabilistic assumptions, ensuring a provable revenue lower bound for the bookmaker under any adversarial bettor behavior and outcome sequence. This work establishes the first theoretically optimal and implementable dynamic pricing framework for online betting mechanism design.

Technology Category

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

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
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

In online betting, the bookmaker can update the payoffs it offers on a particular event many times before the event takes place, and the updated payoffs may depend on the bets accumulated thus far. We study the problem of bookmaking with the goal of maximizing the return in the worst-case, with respect to the gamblers' behavior and the event's outcome. We formalize this problem as the emph{Optimal Online Bookmaking game}, and provide the exact solution for the binary case. To this end, we develop the optimal bookmaking strategy, which relies on a new technique called bi-balancing trees, that assures that the house loss is the same for all emph{decisive} betting sequences, where the gambler bets all its money on a single outcome in each round.

Problem

• Online Gambling
• Optimal Bookmaking Strategy
• Maximal Profit

Innovation

• Optimal Online Bookmaking
• Balanced Tree Approach
• Worst-case Profit Maximization