Stability and Arbitration in Cooperative Games with Overlapping Coalitions

1.1 Classic Cooperative TU Games In classic cooperative games with transferable utility [4] (TU games), there is a set of agents N = {1, . . . , n}, where each subset S of N has some value v(S). The goal of the agents is to first form a coalition structure by partitioning N into disjoint sets; second, the value of each set in the partition is divided among its members. The payoff division x = (x1, . . . , xn), often referred to as an imputation, is then analyzed. It is often desirable that x maintains some notion of stability; over the years, several methods of payoff distribution, or solution concepts, have been suggested. Formally, given a game G = (N, v), a solution concept SC (G) is a set of imputations; if SC (G) is a singleton for all games, SC is called a value. One of the most popular solution concepts is the core. We say that an imputation x is in Core(G) if for all sets S ⊆ N , ∑ i∈S xi ≥ v(S). Other popular solution concepts are the Shapley value, the bargaining set, and the nucleolus.