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.

翻译：我们提出了竞争性分配问题，这是经典分配问题的一个双人博弈版本。给定一组任务和一组智能体（每个智能体对不同任务具有不同的效率值），Alice 和 Bob 轮流逐个选择智能体。当所有智能体被分配完毕后，Alice 和 Bob 分别基于各自拥有的智能体集合计算分配问题的最优值 $s_A$ 和 $s_B$，即他们将自己的智能体分配给任务（每个任务至多分配一个智能体，每个智能体至多分配一个任务），以最大化效率值之和。博弈的得分定义为 $s_A-s_B$。Alice 的目标是最大化该得分，而 Bob 的目标是最小化它。该问题可用于模拟体育选秀或卡牌游戏中的轮选机制，或更广泛地描述两个实体争夺相同资源并随后利用这些资源相互竞争的场景。我们证明了该问题是 PSPACE 完全的，即使限制在智能体至多具有两个非零效率值的情形下依然成立。另一方面，对于智能体至多具有一个非零效率值的情况，该问题属于以任务数量为参数的 XP 复杂度类，且当仅有两个任务时，最优得分可在线性时间内计算得出。