The most exciting game

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 ∈ (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 ∈[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 (Formula Present) 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.