The most exciting game

Abstract

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

$$d{M}_t=\frac{sin(\mathrm{\pi }{M}_t)}{\mathrm{\pi }\sqrt{1-t}}d{B}_t.$$

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.