What is the max-entropy win-probability martingale?


In a sports game between two teams of equality ability, the chance the home team wins is initially \( 1/2 \) and finally \( 0 \) or \( 1 \). As an idealization we take a continuous time interval \( [0,1] \) and consider the process \( (X(t), 0 \le t \le 1) \) giving the probability at time \(t\) that the home team wins. This is a martingale, and if we further idealize by assuming continuous paths, it is mathematically natural to take it to be an (inhomogeneous) diffusion specified by some variance rate function \( \sigma^2(x,t) \). Imposing the "win or lose" condition \[ T := \inf \{t: X(t) = 0 \mbox{ or } 1 \} = 1 \] corresponds by classical theory to a certain two integrals being finite or infinite.

Now a natural way to write down a simple model for a particular sport is to assume (as in most real-world sports) that there is a notion of "point difference", say \( Z(t) \), which for equal-ability teams would have symmetric distribution. A specific model for point difference then implies a specific model for win-probability \[(*) \quad \quad X(t) = \mathbb{P} (Z(1) > 0 \vert \mathcal{F}(t)) . \] It turns out (Mykhaylo Shkolnikov, private communication) that for any \( (X(t), 0 \le t \le 1) \) of the form above, there exists an inhomogeneous martingale diffusion \( (Z(t), 0 \le t \le 1) \) such that (*) holds. Now suppose we are designing a game and its scoring rules, and in principle could arrange to have any \( (Z(t), 0 \le t \le 1) \) and hence any \( (X(t), 0 \le t \le 1) \). What \( (X(t), 0 \le t \le 1) \) should we choose?

The criteria we have in mind is "excitement to spectators". Part of what made a particular match exciting to watch was that the outcome was not very certain until near the end. That is, a match where \( X(0.5) \) was close to 0 or 1 might be considered "over by half-time" and so less exciting to watch. So we don't want it to be common (in the ensemble of matches) that \( X(0.5) \) is close to 0 or 1. On the other hand, we also don't want it to be common that \( X(0.5) \) is close to \( 0.5 \) because then spectators would feel "the first half never matters". So we want the distribution of \( X(0.5) \) to be spread out over \( [0,1] \) and it turns out (by an easy exercise -- see e.g. section 2 of this paper) -- that within a fairly realistic general model for \( Z(t) \), the distribution of \( X(0.5) \) is actually uniform on \( [0,1] \).

So we could stop there and conclude that in fact the usual real-world point-difference schemes are good. However using the implicit criterion "halftime win-probability distribution is uniform" is a little contrary to intuition: wanting the second half to be exciting suggests making the halftime distribution be somewhat biased toward 0.5. Also, the general model does not tell us what the distribution of \( X(t) \) at other times \( t \), such as \( t = 0.75\), should be. Is there a better "excitement to spectators" criterion?

The math problem

The notion of most random subject to given constraints often interpreted as maximal entropy. So what is the max-entropy diffusion of the form above? The difficulty is that what we call entropy is always really relative entropy, relative to some reference measure, and there is no suitable reference measure for our diffusions parameterized by \( \sigma^2(x,t) \). One approach, outlined in these notes is to discretize time and space. The discretized process has finite state space and so there is a well-defined max-entropy distribution. Heuristically, there is a continuous limit, in which there is a function \( e(x, t) \) representing "normalized entropy for the process started at position x at time t", and this function satisfies the PDE \[ e_t = \frac{1}{2} \log( - e_{xx}) \] with natural boundary conditions, and finally the max-entropy distribution for \( (X(t), 0 \le t \le 1) \) is \[ \sigma^2(x,t) = -1/ e_{xx}(x,t) . \]

Problem. Formalize the limit argument; find an explicit solution of the PDE above, or at least prove existence and uniqueness of solution, and find some qualitative properties. In particular, what is the distribution of \( X(0.5) \) ?

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