a weighted game of best choice defined using a prefix equivariant statistic, the optimal strategy is positional.

By Corollary \[c:t1\], there exists some $k$ such that all of the increasing prefixes with size less or equal to $k$ are negative, and all of the increasing prefixes with size greater than $k$ are positive. Applying the $\sigma_q$ isomorphisms, the same $k$ also serves to separate positive and negative eligible prefixes in the rest of the tree by Theorem \[t:pres\]. Hence, the optimal strike strategy coincides with the positional strategy that rejects the first $k$ candidates and accepts the next left-to-right maximum thereafter.

\[r:hist\] It is interesting to compare this proof at ${\theta}= 1$ with arguments from the literature on the secretary problem. The paper of Gilbert–Mosteller [@gilbert--mosteller] is often cited (e.g. by [@ferguson]) as a proof of the result that among all potential strategies, the optimal one must be positional. However, their proof only considers strategies that always stop at a single particular position and does not address the possibility that a player may stop at various positions depending on the relative rankings encountered in the game. The paper of Kadison [@kadison] does provide a complete proof of the desired result 