Coalition Structures in Weighted Voting Games

Weighted voting games are a popular model of collaboration in multiagent systems. In such games, each agent has a weight (intuitively corresponding to resources he can contribute), and a coalition of agents wins if its total weight meets or exceeds a given threshold. Even though coalitional stability in such games is important, existing research has nonetheless only considered the stability of the grand coalition. In this paper, we introduce a model for weighted voting games with coalition structures. This is a natural extension in the context of multiagent systems, as several groups of agents may be simultaneously at work, each serving a different task. We then proceed to study stability in this context. First, we define the CS-core, a notion of the core for such settings, discuss its non-emptiness, and relate it to the traditional notion of the core in weighted voting games. We then investigate its computational properties. We show that, in contrast with the traditional setting, it is computationally hard to decide whether a game has a non-empty CS-core, or whether a given outcome is in the CS-core. However, we then provide an efficient algorithm that verifies whether an outcome is in the CS-core if all weights are small (polynomially bounded). Finally, we also suggest heuristic algorithms for checking the non-emptiness of the CS-core.

[1]  Alexander Schrijver,et al.  Combinatorial optimization. Polyhedra and efficiency. , 2003 .

[2]  Roger B. Myerson,et al.  Game theory - Analysis of Conflict , 1991 .

[3]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[4]  Sarvapali D. Ramchurn,et al.  Anytime Optimal Coalition Structure Generation , 2007, AAAI.

[5]  Craig Boutilier,et al.  Bayesian reinforcement learning for coalition formation under uncertainty , 2004, Proceedings of the Third International Joint Conference on Autonomous Agents and Multiagent Systems, 2004. AAMAS 2004..

[6]  Samir Aknine,et al.  Multi-agent models for searching Pareto optimal solutions to the problem of forming and dynamic restructuring of coalitions , 2002, ECAI.

[7]  J. Neumann,et al.  Theory of games and economic behavior , 1945, 100 Years of Math Milestones.

[8]  R. J. Aumann,et al.  Cooperative games with coalition structures , 1974 .

[9]  Krzysztof R. Apt,et al.  Stable partitions in coalitional games , 2006, ArXiv.

[10]  Ulrich Schwalbe,et al.  Dynamic Coalition Formation and the Core , 2002 .

[11]  Onn Shehory,et al.  Anytime Coalition Structure Generation with Worst Case Guarantees , 1998, AAAI/IAAI.

[12]  Victor R. Lesser,et al.  Coalitions Among Computationally Bounded Agents , 1997, Artif. Intell..

[13]  E. Rowland.  Theory of Games and Economic Behavior , 1946, Nature.

[14]  Michael Wooldridge,et al.  Computational Complexity of Weighted Threshold Games , 2007, AAAI.

[15]  William S. Zwicker,et al.  Simple games - desirability relations, trading, pseudoweightings , 1999 .