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 \(2\partial _t e(t,x)= \log (-\partial _{xx}e(t,x))\) 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 Beiglböck where the match always runs the full length.
