Exciting games and Monge-Amp\`ere equations

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 $\Delta$ and terminate on the vertices of $\Delta$ (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-Amp\`{e}re equations: If $g$ solves \begin{equation*} \begin{cases} g(x)=\log \det\left(\frac{1}{2}\nabla^2 g(x)\right), \quad \, \ \ \ \ \, \, \, \, x \in \Delta, \\ g(x)=\infty, \quad \quad \quad \quad \ \ \ \ \ \quad \quad \, \ \ \ \ \ x\in \partial \Delta, \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-Amp\`{e}re equation for $g$ is crucial. This is then leveraged to prove that $M$ is indeed an optimal win-martingale.