Solution Concepts for Cooperative Games with Circular Communication Structure

CentER Working Paper Series No. 2011-100

28 Pages Posted: 5 Sep 2011

See all articles by Dolf Talman

Dolf Talman

Tilburg University - Department of Econometrics & Operations Research

Takamasa Suzuki

Tilburg University - Department of Econometrics & Operations Research

Date Written: September 1, 2011

Abstract

We study transferable utility games with limited cooperation between the agents. The focus is on communication structures where the set of agents forms a circle, so that the possibilities of cooperation are represented by the connected sets of nodes of an undirected circular graph. Agents are able to cooperate in a coalition only if they can form a network in the graph. A single valued solution which averages marginal contributions of each player is considered. We restrict the set of permutations, which induce marginal contributions to be averaged, to the ones in which every agent is connected to the agent that precedes this agent in the permutation. Staring at a given agent, there are two permutations which satisfy this restriction, one going clockwise and one going anticlockwise along the circle. For each such permutation a marginal vector is determined that gives every player his marginal contribution when joining the preceding agents. It turns out that the average of these marginal vectors coincides with the average tree solution. We also show that the same solution is obtained if we allow an agent to join if this agent is connected to some of the agents who is preceding him in the permutation, not necessarily being the last one. In this case the number of permutations and marginal vectors is much larger, because after the initial agent each time two agents can join instead of one, but the average of the corresponding marginal vectors is the same. We further give weak forms of convexity that are necessary and sufficient conditions for the core stability of all those marginal vectors and the solution. An axiomatization of the solution on the class of circular graph games is also given.

Keywords: Cooperative game, graph structure, average tree solution, Myerson value, core stability, convexity

JEL Classification: C71

Suggested Citation

Talman, Dolf J. J. and Suzuki, Takamasa, Solution Concepts for Cooperative Games with Circular Communication Structure (September 1, 2011). CentER Working Paper Series No. 2011-100, Available at SSRN: https://ssrn.com/abstract=1922707 or http://dx.doi.org/10.2139/ssrn.1922707

Dolf J. J. Talman (Contact Author)

Tilburg University - Department of Econometrics & Operations Research ( email )

Tilburg, 5000 LE
Netherlands
+31 13 466 2346 (Phone)

Takamasa Suzuki

Tilburg University - Department of Econometrics & Operations Research ( email )

P.O. Box 90153
Tilburg, DC Noord-Brabant 5000 LE
Netherlands

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