Randomness and early termination: What makes a game exciting?

• 成果类型: Article
• 署名作者: Guo, Gaoyue; Howison, Sam D.; Possamai, Dylan; Reisinger, Christoph
• 署名单位: Universite Paris Saclay; Centre National de la Recherche Scientifique (CNRS); University of Oxford; Swiss Federal Institutes of Technology Domain; ETH Zurich
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-024-01340-x
• 发表日期: 2025
• 页码: 135-162
• 关键词: equations

摘要

In this paper we revisit an open problem posed by Aldous on the max-entropy win-probability martingale: given two players of equal strength, such that the win-probability is a martingale diffusion, which of these processes has maximum entropy and hence gives the most excitement for the spectators? Our construction is based on the detailed study of a terminal-boundary value problem for the nonlinear parabolic PDE 2 partial derivative te(t,x)=log(-partial derivative xxe(t,x))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2\partial _t e(t,x)= \log (-\partial _{xx}e(t,x))$$\end{document} derived by Aldous. We prove its wellposedness and regularity of its solution by combining PDE analysis and probabilistic tools, in particular the reformulation as a stochastic control problem with restricted control set, which allows us to deduce strict ellipticity. We establish key qualitative properties of the solution including concavity, monotonicity, convergence to a steady state for long remaining time and the asymptotic behaviour shortly before the terminal time. Moreover, we construct convergent numerical approximations. The analytical and numerical results allow us to characterise the max-entropy win-probability martingale and to highlight the behaviour of this process in the present case where the match may end early, in contrast to recent work by Backhoff-Veraguas and Beiglb & ouml;ck where the match always runs the full length.