Cost Sharing

The objective of cooperative game theory is to study ways to enforce and sustain cooperation among agents willing to cooperate. A central question in this field is how the benefits (or costs) of a joint effort can be divided among participants, taking into account individual and group incentives, as well as various fairness properties. In this chapter, we define basic concepts and review some of the classical results in the cooperative game theory literature. Our focus is on games that are based on combinatorial optimization problems such as facility location. We define the notion of cost sharing, and explore various incentive and fairness properties cost-sharing methods are often expected to satisfy. We show how cost-sharing methods satisfying a certain property termed cross-monotonicity can be used to design mechanisms that are robust against collusion, and study the algorithmic question of designing cross-monotonic cost-sharing schemes for combinatorial optimization games. Interestingly, this problem is closely related to linear-programming-based techniques developed in the field of approximation algorithms. We explore this connection, and explain a general method for designing cross-monotonic cost-sharing schemes, as well as a technique for proving impossibility bounds on such schemes. We will also discuss an axiomatic approach to characterize two widely applicable solution concepts: the Shapley value for cooperative games, and the Nash bargaining solution for a more restricted framework for surplus sharing. 15.1 Cooperative Games and Cost Sharing Consider a setting where a setA of n agents seek to cooperate in order to generate value. The value generated depends on the coalition S of agents cooperating. In general, the set of possible outcomes of cooperation among agents in S ⊆ A is denoted by V (S), where each outcome is given by a vector in R , whose i’th component specifies the utility that the agent i ∈ S derives in this outcome. The set A of agents along with the functionV defines what is called a cooperative game (also known as a coalitional game) with nontransferable utilities (abbreviated as an NTU game). A special case, called a cooperative game with transferable utilities (abbreviated as a TU game), is when the