Exciting games and Monge-Ampère equations

TL;DR

The paper addresses the problem of determining the most exciting game among $d+1$ players by modeling the winner probabilities as a win-martingale on the subprobability simplex and linking the optimization objective to a scaling limit of Shannon entropy. The core approach derives a Hamilton–Jacobi–Bellman equation for the value function, which reduces to an elliptic Monge–Ampère equation $g(x)=\log\det(\tfrac{1}{2}\nabla^2 g(x))$ with infinite boundary data; the resulting optimal diffusion is $dM_t=\sqrt{\dfrac{2 (\nabla^2 g(M_t))^{-1}}{1-t}} dB_t$, called the Aldous martingale. The main contributions include (i) a rigorous, self-contained existence and sharp boundary regularity analysis for the Monge–Ampère equation on the simplex, (ii) the construction and verification of the Aldous martingale as the unique optimizer, and (iii) connections to moment measures and martingale optimal transport, grounding the approach in barrier methods and invariance properties. The results provide a PDE-controlled diffusion representation of optimal win-martingales with implications for information design and financial modelling.

Abstract

We consider a competition between $d+1$ players, and aim to identify the "most exciting game'' of this kind. This is translated, mathematically, into a stochastic optimization problem over martingales that live on the $d$-dimensional subprobability simplex $Δ$ and terminate on the vertices of $Δ$ (so-called win-martingales), with a cost function related to a scaling limit of Shannon entropies. We uncover a surprising connection between this problem and the seemingly unrelated field of Monge-Ampère equations: If $g$ solves \begin{equation*} \begin{cases} g(x)=\log \det\left(\frac{1}{2}\nabla^2 g(x)\right), \quad \, \ \ \ \ \, \, \, \, x \in Δ, \\ g(x)=\infty, \quad \quad \quad \quad \ \ \ \ \ \quad \quad \, \ \ \ \ \ x\in \partial Δ, \end{cases} \end{equation*} then the winning-probability of the players in the most exciting game is described by $$dM_s=\sqrt{\frac{2 (\nabla^2 g(M_s))^{-1}}{1-s} } \, dB_s.$$ To formalize this, a detailed quantitative analysis of the Monge-Ampère equation for $g$ is crucial. This is then leveraged to prove that $M$ is indeed an optimal win-martingale.

Theorems & Definitions (47)

• Definition 1.1
• Theorem 1.1
• Theorem 1.2
• Proposition 3.1
• proof
• Lemma 3.1
• proof
• Proposition 3.2
• proof
• Definition 4.1