A two-player version of the assignment problem

TL;DR

This work introduces the competitive assignment problem and its two-player draft game where players alternate drafting agents and later solve independent assignment problems to obtain a score $sc(G)=s_A-s_B$, with Alice maximizing and Bob minimizing. It places the game in Milnor's universe and in Maker-Breaker frameworks, proving that the decision problem is PSPACE-complete even when agents have at most two nonzero efficiencies, while showing XP and linear-time results for OTP instances and two-task cases, respectively. The hardness proof relies on a reduction from 3-QBF-3 using gadgets that encode variable assignments and clause satisfaction, complemented by structural lemmas such as dominating moves and paired forcing moves. The results illuminate the complexity of drafting resources for competitive post-draft play and provide a foundation for further exploration of two-player partitioning and balanced allocations in practical settings.

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.

Figures (1)

• Figure 1: The different gadgets used in our reduction. Rows correspond to agents and are written in decreasing order of efficiencies. Columns correspond to tasks. If two agents are separated by a horizontal line, then the top one is always picked before the bottom one.

Theorems & Definitions (25)

• Lemma 1
• proof : of Lemma \ref{['milnor']}
• Corollary 2
• proof : of Corollary \ref{['corollary:score-positif']}
• Lemma 3
• proof : of Lemma \ref{['lemma:mean-zero']}
• Lemma 4
• proof : of Lemma \ref{['lemma dominating agents']}
• Lemma 5
• proof : of Lemma \ref{['lemma:boundscore']}