those interview rank orders having $p$ as a prefix. Since each denominator of $f(\pi)$ is just a normalizing constant, we may cancel it obtaining $${\mathcal{S}}(p) = \left( \sum\limits_{p\text{-winnable } \pi \in {\mathfrak{S}}_N} \theta^{c(\pi)} \right)\Big/\left( \sum\limits_{p\text{-prefixed } \pi \in {\mathfrak{S}}_N} \theta^{c(\pi)} \right)$$ In particular, ${\mathcal{S}}_N(p)$ is $0$ unless $p$ ends in a left-to-right maximum; for this reason we refer to a prefix as [**eligible**]{} if it ends in a left-to-right maximum or has size $N$ (included for completeness). Hence, it suffices to restrict our attention to strike sets

1.  consisting of prefixes that are eligible, and

2.  having no pair of elements such that one contains the other as a prefix flattening, and

3.  such that every permutation in ${\mathfrak{S}}_N$ contains some element of the strike set as a prefix flattening.

Given this setup, we let ${\mathcal{S}}(p^{\circ})$ be the “open” probability of a win if we play optimally, using any strike set consisting of prefixes that contain (but are not equal to) $p$. Similarly, let ${\mathcal{S}}(\bar{p})$ be the “closed” probability of a win if we play optimally, using any strike set consisting of prefixes that contain (and may include) $p$. Let each of these be conditional probabilities with [**standard denominator**]{} $\sum_{p\text{-prefixed 