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Optimal Online Bookmaking for Binary Games

Alankrita Bhatt, Or Ordentlich, Oron Sabag

TL;DR

The paper studies optimal online bookmaking for binary outcomes, formulating a repeated vector-valued game where a bookmaker updates odds $r_t$ and gamblers place bets $q_t$ over $T$ rounds. It shows decisiveness of gamblers is optimal and introduces bi-balancing trees to characterize the set of attainable losses, proving the exact optimal house loss $L_T^*=T+\sqrt{T}$ and giving an efficient forward algorithm to achieve it for decisive gamblers. For non-decisive bets, a Monte Carlo-based method approximates the optimal strategy with provable guarantees, and a Blackwell-inspired approach provides a suboptimal algorithm with tractable, horizon-free performance. The work links online bookmaking to dynamic programming, Blackwell approachability, and universal compression, and outlines clear directions for extending to multi-outcome settings and horizon-independent strategies.

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.

Optimal Online Bookmaking for Binary Games

TL;DR

The paper studies optimal online bookmaking for binary outcomes, formulating a repeated vector-valued game where a bookmaker updates odds and gamblers place bets over rounds. It shows decisiveness of gamblers is optimal and introduces bi-balancing trees to characterize the set of attainable losses, proving the exact optimal house loss and giving an efficient forward algorithm to achieve it for decisive gamblers. For non-decisive bets, a Monte Carlo-based method approximates the optimal strategy with provable guarantees, and a Blackwell-inspired approach provides a suboptimal algorithm with tractable, horizon-free performance. The work links online bookmaking to dynamic programming, Blackwell approachability, and universal compression, and outlines clear directions for extending to multi-outcome settings and horizon-independent strategies.

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.
Paper Structure (15 sections, 7 theorems, 76 equations, 3 figures, 2 algorithms)

This paper contains 15 sections, 7 theorems, 76 equations, 3 figures, 2 algorithms.

Table of Contents

  1. Introduction and Main results
  2. Mathematical formulation of the online bookmaking game
  3. Related Work and Connections to Other Problems
  4. Paper Structure
  5. Algorithms for Optimal Bookmaking
  6. An Optimal Algorithm for Decisive Gamblers
  7. Algorithm for Non-Decisive Gamblers (Continuous Bets)
  8. Proof of Main Results
  9. Decisive Gamblers are Optimal
  10. Optimal Solution via Balancing Trees
  11. The optimal strategy
  12. Proof of Theorem \ref{['th:main']} and Claim \ref{['claim:equalizing']}
  13. Monte-Carlo-Based Approximation of the Optimal Algorithm
  14. Sub-optimal Blackwell-Inspired Algorithm
  15. Conclusions and Open Problems

Key Result

Theorem 1

The optimal house loss for the binary online bookmaking game is Moreover, if the gambler is decisive, Algorithm alg:bets_for_bin_2 (Section sec:alg) defines a house strategy $\{r_t^{\texttt{ALG}}\}=\{\phi_t^{\texttt{ALG}}\}$ that can be computed with $T$ simple operations, and achieves the optimal house loss for all $q^T\in\{0,1\}^T$. If the gambler is non-decisive, i.e., $q^T\in[0,1]^T$, the ho

Figures (3)

  • Figure 1: A binary tree that describes the online bookmaking setup with $T=2$. First, the house chooses $r_1$, followed by the gambler's bet $q_1\in\{0,1\}$. The house then chooses the odd $r_2(q_1)$ based on the bet $q_1$, and lastly, the bet $q_2$ is placed. As shown in Claim \ref{['claim:winning_horse']}, the last bet $q_2$ is equal to the winning team.
  • Figure 2: The set of achievable value functions in Theorem \ref{['th:bi-balanced']} as a function of the remaining depth, normalized by depth. The curves are symmetric around the line $V^1=V^0$ due to symmetry between the teams. The optimal house loss for an horizon $d$ can be obtained by intersecting its curve with the dashed line $V^0 = V^1$. As $d$ grows, the intersection approaches the lower bound $(V^0,V^1) = (1,1)$ with a convergence rate of $\frac{1}{\sqrt{d}}$.
  • Figure 4: Illustration of the regions defined in \ref{['eq:BlackwellRegions']} and in \ref{['eq:BlackwellRegionsRefinement']}. Here $\mathcal{A}_1=\mathcal{A}_1^{-}\cup\mathcal{A}_1^{+}$ and $\mathcal{A}_2=\mathcal{A}_2^{-}\cup\mathcal{A}_2^{+}$.

Theorems & Definitions (15)

  • Remark 1
  • Theorem 1
  • Theorem 2
  • Claim 1
  • Example 1
  • Claim 2
  • Definition 1: Individual value function
  • Definition 2: Bi-Balanced Trees
  • Theorem 3
  • Claim 3
  • ...and 5 more