strategy**]{} in which the player rejects the first $k$ candidates and accepts the next left-to-right maximum thereafter. From our point of view, this seems to be a minor miracle as there are [*many*]{} antichains in the partial order of prefix-containment that would serve as potentially optimal strike sets but only $N$ possible positional strategies. In this section, we give a concrete explanation for this and generalize it to a class of weighted games.

The key idea is to use a fundamental bijection in order to transport structure around the prefix tree. To define it, let ${\mathcal{T}}(p^{\circ})$ be the “open” subforest of prefixes containing (but not equal to) $p$, and let ${\mathcal{T}}(\bar{p})$ be the “closed” subtree of prefixes containing $p$, including $p$ itself.

Suppose that $p = [12 \cdots k]$ and let $\sigma_q$ be the permutation that rearranges the prefix $p$ to give some other prefix $q$ of size $k$. We can extend this to an action on ${\mathcal{T}}(\bar{p})$, denoted $\sigma_q \cdot \pi$, by similarly permuting the first $k$ entries and fixing the last $i-k$ entries of $\pi$, where $i$ is the size of $\pi \in {\mathcal{T}}(\bar{p})$. Then $\sigma_q$ is a bijection from ${\mathcal{T}}(\bar{p})$ to ${\mathcal{T}}(\bar{q})$.

Suppose that the statistic $c$ 