On a Competitive Secretary Problem with Deferred Selections

TL;DR

This paper extends the classical secretary problem to a multi-agent setting with endogenous competition, analyzing equilibrium strategies and social welfare when agents can select previously arrived awards with different tie-breaking rules.

Contribution

It introduces a multi-agent secretary model with deferred selections, characterizes equilibria under different tie-breaking rules, and shows near-optimal social welfare in equilibrium.

Findings

Equilibria are characterized for random tie-breaking, with nearly optimal social welfare.

In ranked tie-breaking, equilibria are fully characterized for three agents.

As the number of agents increases, the probability of winning approaches that of immediate selection.

Abstract

We study secretary problems in settings with multiple agents. In the standard secretary problem, a sequence of arbitrary awards arrive online, in a random order, and a single decision maker makes an immediate and irrevocable decision whether to accept each award upon its arrival. The requirement to make immediate decisions arises in many cases due to an implicit assumption regarding competition. Namely, if the decision maker does not take the offered award immediately, it will be taken by someone else. The novelty in this paper is in introducing a multi-agent model in which the competition is endogenous. In our model, multiple agents compete over the arriving awards, but the decisions need not be immediate; instead, agents may select previous awards as long as they are available (i.e., not taken by another agent). If an award is selected by multiple agents, ties are broken either randomly or according to a global ranking. This induces a multi-agent game in which the time of selection is not enforced by the rules of the games, rather it is an important component of the agent's strategy. We study the structure and performance of equilibria in this game. For random tie breaking, we characterize the equilibria of the game, and show that the expected social welfare in equilibrium is nearly optimal, despite competition among the agents. For ranked tie breaking, we give a full characterization of equilibria in the 3-agent game, and show that as the number of agents grows, the winning probability of every agent under non-immediate selections approaches her winning probability under immediate selections.