Avoiding patterns and making the best choice

Abstract: We study a variation of the game of best choice (also known as the secretary problem or game of googol) under an additional assumption that the ranks of interview candidates are restricted using permutation pattern-avoidance. We develop some general machinery for investigating interview orderings with a non-uniform rank distribution, and give a complete description of the optimal strategies for the pattern-avoiding games under each of the size three permutations. The optimal strategy for the "disappointment-free" (i.e. 321-avoiding) interviews has a form that seems to be new, involving thresholds based on value-saturated left-to-right maxima in the permutation.