A Stackelberg Game based on the Secretary Problem: Optimal Response is History Dependent

TL;DR

This paper analyzes a two-player game based on the secretary problem, where the second player's optimal response depends on the history of observed objects and incomplete information, leading to bounds on expected rewards.

Contribution

It introduces a novel Stackelberg game framework based on the secretary problem, deriving bounds for the second player's history-dependent optimal strategy under incomplete information.

Findings

Optimal strategy depends on the number of objects and probability estimates.

Lower bounds are established by limiting Player 2's memory.

Upper bounds are derived with additional information at key moments.

Abstract

This article considers a problem arising from a two-player game based on the classical secretary problem. First, Player 1 selects one object from a sequence as in the secretary problem. All of the other objects are then presented to Player 2 in the same order as in the original sequence. The goal of both players is to select the best object. The optimal response of Player 2 is adapted to the optimal strategy in the secretary problem. This means that when Player 2 observes an object that is the best seen so far, it can be inferred whether Player 1 selected one of the earlier objects in the original sequence. However, Player 2 cannot compare the current object with the one selected by Player 1. Hence, this game defines an auxiliary problem in which Player 2 has incomplete information on the relative rank of an object. It is shown that the optimal strategy of Player 2 is based on both the number of objects to have appeared and the probability that the current object is better than the object chosen by Player 1 (if Player 1 chose an earlier object in the sequence). However, this probability is dependent on the previously observed objects. A lower bound on the optimal expected reward in the auxiliary problem is defined by limiting the memory of Player 2. An upper bound is derived by giving Player 2 additional information at appropriate times. The methods used illustrate approaches that can be used to approximate the optimal reward in a stopping problem when there is incomplete information on the ranks of objects and/or the optimal strategy is history dependent, as in the Robbins' problem