A two-player version of the assignment problem

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.

💡 Research Summary

The paper introduces the Competitive Assignment Problem, a two‑player game that extends the classic assignment problem. An instance consists of t tasks and n agents, each agent being a t‑dimensional vector of efficiencies (non‑negative values). Alice and Bob alternately pick agents, Alice moving first. After all agents are allocated, each player solves a maximum‑weight bipartite matching on his/her own set of agents to obtain values s_A and s_B; the game score is s_A − s_B. Alice tries to maximize this difference, Bob to minimize it, modeling drafts in sports, fantasy leagues, or any situation where two parties compete for the same resources and then use them against each other.

The authors first establish basic game‑theoretic properties. The game is dicotic (both players always have a legal move) and non‑zugzwang (going first is never disadvantageous). Consequently, the optimal score from the starting position is always non‑negative, and the mean score over many independent copies of the game is zero.

Complexity results are the core contribution. When each agent has at most two non‑zero efficiencies (called TTP agents), the decision problem “does Alice have a strategy guaranteeing a score at least s?” (DraftGame) is PSPACE‑complete. The proof reduces from a generic PSPACE‑hard two‑player game while preserving the two‑non‑zero‑efficiency restriction, showing that the game tree depth remains linear in the number of agents.

In contrast, if every agent has at most one non‑zero efficiency (OTP agents), the problem becomes tractable when parameterized by the number of tasks t. An XP algorithm runs in time O(f(t)·poly(n)), essentially by separating agents per task and applying dynamic programming to decide how many agents each player should claim for each task. For the special case t = 2, the authors give a linear‑time greedy algorithm: sort agents by their single efficiency, let Alice take the higher half and Bob the lower half; this yields the optimal score.

The paper also connects the game to Maker‑Breaker positional games. By constructing a hypergraph whose hyperedges are subsets of agents that would give Alice a score ≥ s, the DraftGame instance corresponds to a Maker‑Breaker win condition. Since Maker‑Breaker games are known to be PSPACE‑complete, this provides an alternative hardness perspective.

A domination lemma is proved: if one agent’s efficiency vector component‑wise dominates another’s, then picking the dominating agent is at least as good for the player whose turn it is. This structural insight enables pruning of the search space in algorithmic implementations.

Finally, the authors discuss extensions such as multi‑round drafts, more than two players, or allowing negative efficiencies, and suggest that the competitive assignment framework could serve as a tool for analyzing balanced partition problems in combinatorial optimization. The work thus establishes both theoretical hardness and concrete algorithmic avenues for a natural two‑player extension of a classic optimization problem.