Strategy-indifferent games of best choice

TL;DR

This paper classifies a special class of secretary-like decision games where all strategies have the same success probability, revealing their connection to permutation pattern avoidance and maxima distribution.

Contribution

It introduces the concept of strategy-indifferent games of best choice and characterizes their success probabilities and combinatorial properties.

Findings

Success probability is reciprocal to expected maxima count.

Classified strategy-indifferent games based on permutation pattern avoidance.

Analyzed maxima distribution in pattern-avoiding permutations.

Abstract

The game of best choice (or "secretary problem") is a model for making an irrevocable decision among a fixed number of candidate choices that are presented sequentially in random order, one at a time. Because the classically optimal solution is known to reject an initial sequence of candidates, a paradox emerges from the fact that candidates have an incentive to position themselves immediately after this cutoff which challenges the assumption that candidates arrive in uniformly random order. One way to resolve this is to consider games for which every (reasonable) strategy results in the same probability of success. In this work, we classify these "strategy-indifferent" games of best choice. It turns out that the probability of winning such a game is essentially the reciprocal of the expected number of left-to-right maxima in the full collection of candidate rank orderings. We present some examples of these games based on avoiding permutation patterns of size 3, which involves computing the distribution of left-to-right maxima in each of these pattern classes.