A two-player version of the assignment problem

TL;DR

This paper introduces the competitive assignment problem, a two-player game based on the assignment problem, analyzes its computational complexity, and explores special cases where the problem is tractable.

Contribution

It formalizes a two-player competitive variant of the assignment problem and establishes its PSPACE-completeness, while also identifying cases with polynomial or linear solutions.

Findings

The problem is PSPACE-complete even with agents having at most two nonzero efficiencies.

When agents have at most one nonzero efficiency, the problem is in XP parameterized by the number of tasks.

For two tasks, the optimal score can be computed in linear time.

Abstract

We introduce the competitive assignment problem, a two-player version of the well-known assignment problem. Given a set of tasks and a set of agents with different efficiencies for different tasks, Alice and Bob take turns picking agents one by one. Once all agents have been picked, Alice and Bob compute the optimal values $s_A$ and $s_B$ for the assignment problem on their respective sets of agents, i.e. they assign their own agents to tasks (with at most one agent per task and at most one task per agent) so as to maximize the sum of the efficiencies. The score of the game is then defined as $s_A-s_B$. Alice aims at maximizing the score, while Bob aims at minimizing it. This problem can model drafts in sports and card games, or more generally situations where two entities fight for the same resources and then use them to compete against each other. We show that the problem is PSPACE-complete, even restricted to agents that have at most two nonzero efficiencies. On the other hand, in the case of agents having at most one nonzero efficiency, the problem lies in XP parameterized by the number of tasks, and the optimal score can be computed in linear time when there are only two tasks.