Summary: Motivated by a problem posed by Aldous [2, 1] 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 let \(M_t\) be the probability at time \(t\) that the home team wins. Mathematically, \(M = (M_t)_{t \in [0, 1]}\) is modelled as a continuous martingale. We consider the problem to find the most random martingale \(M\) of this type, where ‘most random’ is interpreted as a maximal entropy criterion. In discrete time this is equivalent to the minimization of relative entropy w.r.t. a Gaussian random walk. The continuous time analogue is that the max-entropy win-martingale \(M\) should minimize specific relative entropy with respect to Brownian motion in the sense of Gantert [20]. We 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.