[Paper Review] Complexity of coalition structure generation

Haris Aziz, Bart de Keijzer|arXiv (Cornell University)|May 2, 2011

Game Theory and Voting SystemsEconomics, Econometrics and Finance22 references40 citations

TL;DR

This paper presents a polynomial-time algorithm for coalition structure generation in coalitional games when the number of player types is bounded by a constant, enabling efficient computation of optimal partitions. The key contribution is a general solution applicable to weighted voting games with few weight values and coalitional skill games with few skills, alongside complexity characterizations for compactly represented games.

ABSTRACT

We revisit the coalition structure generation problem in which the goal is to partition the players into exhaustive and disjoint coalitions so as to maximize the social welfare. One of our key results is a general polynomial-time algorithm to solve the problem for all coalitional games provided that player types are known and the number of player types is bounded by a constant. As a corollary, we obtain a polynomial-time algorithm to compute an optimal partition for weighted voting games with a constant number of weight values and for coalitional skill games with a constant number of skills. We also consider well-studied and well-motivated coalitional games defined compactly on combinatorial domains. For these games, we characterize the complexity of computing an optimal coalition structure by presenting polynomial-time algorithms, approximation algorithms, or NP-hardness and inapproximability lower bounds.

Motivation & Objective

• To address the computational challenge of generating optimal coalition structures in coalitional games.
• To identify conditions under which the coalition structure generation problem becomes tractable.
• To provide efficient algorithms for specific classes of coalitional games, such as weighted voting games and coalitional skill games.
• To characterize the complexity of optimal coalition structure computation in combinatorially compactly represented games.
• To establish polynomial-time solvability, approximation algorithms, or inapproximability bounds for various game classes.

Proposed method

• Propose a general polynomial-time algorithm for coalition structure generation when the number of player types is constant.
• Leverage dynamic programming over partitions of player types rather than individual players to reduce complexity.
• Apply the algorithm to weighted voting games by grouping players with identical weights into types.
• Extend the approach to coalitional skill games by treating players with identical skill sets as equivalent types.
• Analyze the complexity of compactly represented games using reductions and hardness results to classify tractability.
• Use approximation algorithms and inapproximability lower bounds to characterize the limits of efficient computation.

Experimental results

Research questions

• RQ1 Under what conditions can coalition structure generation be solved in polynomial time?
• RQ2 Can the problem be efficiently solved for weighted voting games with a constant number of distinct weight values?
• RQ3 Is there a general method to compute optimal coalition structures when player types are bounded?
• RQ4 What is the complexity of coalition structure generation in compactly represented combinatorial games?
• RQ5 Are there approximation algorithms or inapproximability results for hard instances of the problem?

Key findings

• A polynomial-time algorithm exists for coalition structure generation when the number of player types is bounded by a constant.
• The algorithm enables efficient computation of optimal partitions in weighted voting games with a constant number of distinct weight values.
• The approach also yields polynomial-time solutions for coalitional skill games with a constant number of distinct skill sets.
• For compactly represented games, the paper establishes polynomial-time solvability, approximation algorithms, or inapproximability lower bounds.
• The results demonstrate that bounded player types lead to tractable coalition structure generation across multiple game classes.
• The framework provides a unified approach to analyzing complexity across diverse coalitional game types.