Rival Strategies Mapped in New Game Theory Challenge for Resource Allocation

Researchers have investigated a novel take on the classic assignment problem, formulating it as a competitive two-player game. Florian Galliot from Aix-Marseille Université, Nacim Oijid from Umeå University, and Jonas Sénizergues from LaBRI, Université de Bordeaux, et al. present a scenario where two players alternate selecting agents to complete tasks, subsequently optimising their individual assignments to maximise or minimise a shared score. This work is significant because it models strategic resource allocation found in diverse contexts, ranging from sports drafts to competitive card games, and demonstrates a surprising computational complexity. The team proves the problem is PSPACE-complete, even with limited agent capabilities, yet reveals tractable solutions exist for simplified instances, offering valuable insights into the boundaries of computational feasibility.

This new framework models scenarios where two entities compete for a set of agents to perform tasks, ultimately impacting their respective efficiencies.

The research details a game where Alice and Bob alternately select agents, each aiming to maximize their subsequent assignment score, the difference between their optimal task assignments. This innovative approach provides a mathematical model for understanding optimal strategies in competitive resource allocation, with implications for diverse fields.
The core of the work lies in analyzing the computational complexity of this competitive assignment problem. Researchers demonstrate that the problem is PSPACE-complete, even when restricted to agents with limited task proficiency, specifically, those capable of performing at most two tasks efficiently.

This establishes a fundamental limit on the tractability of finding optimal solutions as the problem scales. However, a significant finding emerges when considering agents with even more limited capabilities; if each agent excels at only one task, the problem falls into the complexity class XP, parameterized by the number of tasks.
Furthermore, the study reveals that for instances with only two tasks, the optimal score can be computed in linear time. This simplification offers a pathway for solving small-scale instances efficiently and provides insights into the problem’s structure. The competitive assignment problem has direct relevance to scenarios like sports drafts and card games, where entities vie for resources before competing against each other.

It also extends to broader applications, such as competitive business environments where companies prioritize talent acquisition and partnerships. This research not only defines a new game-theoretic model but also establishes its computational boundaries and potential for practical application. By framing resource allocation as a competition between two players, scientists offer a fresh perspective on optimization problems and open avenues for developing strategies in scenarios where achieving a relative advantage is paramount. The work builds upon decades of research into the assignment problem, including the influential Hungarian algorithm dating back to 1955, which originally solved the problem in O(n4) time.

Competitive assignment game formulation and computational complexity are central to resource allocation problems

The competitive assignment problem, a two-player variant of the assignment problem, served as the focus of this work. This game involves two players who alternate selecting agents, each possessing varying efficiencies across a set of tasks. Once all agents are chosen, each player independently assigns their selected agents to tasks, aiming to maximise the sum of efficiencies for their assignments, with the constraint of one agent per task and one task per agent.

The game’s score is calculated as the difference between the two players’ maximum assignment values, where the first player seeks to maximise this score and the second player attempts to minimise it. Researchers demonstrated the problem’s computational complexity by proving its PSPACE-completeness, even when restricted to agents with at most two non-zero efficiencies.

This establishes a firm upper bound on the difficulty of finding optimal solutions as the number of agents and tasks increases. Further analysis revealed a more tractable case when agents have at most one non-zero efficiency, placing the problem within the XP complexity class parameterised by the number of tasks.

Specifically, with only two tasks, the optimal score could be computed in linear time, indicating a significant simplification under these conditions. This work builds upon existing research in positional games, exemplified by Milnor’s 1953 study of sums of positional games, and extends the understanding of complexity in combinatorial game theory.

Computational complexity of competitive assignment with drafted agents remains largely unexplored

The draft game, a competitive assignment problem, reveals that optimal scores can be computed in linear time when only two tasks are considered. This work introduces a two-player version of the assignment problem where agents are drafted and then assigned tasks to maximize individual scores, resulting in a score defined as the difference between Alice’s and Bob’s assignment efficiencies.

Researchers demonstrate that the decision problem associated with this game, determining if the optimal score is greater than or equal to a given integer, is PSPACE-complete, even when restricted to agents with at most two nonzero efficiencies. In instances involving agents with at most one nonzero efficiency, the study establishes that the problem falls into the complexity class XP parameterized by the number of tasks.

Specifically, the optimal score can be determined in linear time with only two tasks present. This finding is supported by the game’s membership in Milnor’s universe, a class of combinatorial games exhibiting dicotic and nonzugzwang properties. The research proves the draft game is dicotic, meaning each player has a legal move if and only if the other does, and nonzugzwang, indicating that the optimal score is greater or equal regardless of who moves first.

A key result is that the optimal score for any instance of the draft game is greater than or equal to zero, derived from the game’s nonzugzwang nature and the ability to switch player roles. The study defines an agent with at most one nonzero efficiency as an OTP and an agent with at most two nonzero efficiencies as a TTP, providing a framework for analyzing agent characteristics. This framework allows for a deeper understanding of the game’s complexity and potential applications in scenarios like sports drafts or resource allocation.

Computational complexity and tractability in competitive assignment games remain challenging research areas

The competitive assignment problem, a two-player variant of the assignment problem, concerns the alternating selection of agents to perform tasks, followed by individual assignment of chosen agents to maximise efficiency. This framework models competitive scenarios such as sports drafts or resource contention, where entities vie for resources before utilising them in competition.

Analysis reveals the problem is PSPACE-complete, even when restricted to agents with limited task proficiency, demonstrating its inherent computational complexity. However, the problem simplifies under specific conditions; when agents possess proficiency in only one task, it falls into the XP complexity class parameterised by the number of tasks, and becomes solvable in linear time with only two tasks.

The research establishes that the game belongs to Milnor’s universe, meaning it is both dicotic, ensuring equal turns for both players, and non-zugzwang, where the first player can always achieve a score at least as high as the second. Furthermore, the optimal score for any instance of the draft game is guaranteed to be non-negative.

The authors acknowledge that their model assumes any agent can attempt any task, even with zero efficiency, and allows for unassigned tasks, which may not reflect all real-world scenarios. They also note a technical detail regarding the representation of agents, choosing to treat instances as sets rather than multisets for notational simplicity. Future research could explore variations of the problem with more restrictive agent capabilities or task assignments, and investigate the development of efficient approximation algorithms for larger instances where finding the optimal solution is computationally prohibitive.