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?

** 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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