The most exciting game

Motivated by a problem posed by Aldous, our goal is to find the maximal-entropy win-martingale: In a sports game between two teams, the chance the home team wins is initially $x_0 \in (0,1)$ and finally 0 or 1. As an idealization we take a continuous time interval $[0,1]$ and consider the process $M=(M_t)_{t\in [0,1]}$ giving the probability at time $t$ that the home team wins. This is a martingale which we idealize further to have continuous paths. We consider the problem to find the most random martingale $M$ of this type, where `most random' is interpreted as a maximal entropy criterion. We observe that this max-entropy win-martingale $M$ also minimizes specific relative entropy with respect to Brownian motion in the sense of Gantert and use this to prove that $M$ is characterized by the stochastic differential equation $$dM_t = \frac{\sin (\pi M_t )} {\pi\sqrt {1-t}}\, dB_t.$$ To derive the form of the optimizer we use a scaling argument together with a new first order condition for martingale optimal transport which may be of interest in its own right.