Encouraging Cooperation in Sharing Supermodular Costs

Abstract Consider a situation where a group of agents wishes to share the costs of their joint actions, and needs to determine how to distribute the costs amongst themselves in a fair manner. For example, a set of agents may agree to process their jobs together on a machine, and share the optimal cost of scheduling these jobs. This kind of situation can be modelled naturally as a cooperative game. In this work, we are concerned with cooperative games with supermodular costs. A set function r : 2N → R is supermodular if r(S ∪ {i}) − r(S) ≤ r(T ∪ {i}) − r(T ) ∀S ⊆ T ⊆ N\{i}. Our primary motivation behind studying these games is that many problems from combinatorial optimization have optimal costs that are supermodular. In particular, we show the following theorem (for notational convenience, for any vector x we define x(S) := i∈S xi ): Theorem. Let N be a finite set, and let r : 2N → R be a supermodular function such that r(∅) = 0. If dj ≥ 0 for all j ∈ N, t...

• Type: Conference
• Year: 2007
• Where: APPROX
• Authors: Andreas S. Schulz, Nelson A. Uhan
• Keywords: Algorithms, APPROX 2007, Cooperative Games, Optimal Costs, Optimization Problem