A value for partially defined cooperative games (Q2366101)

The Shapley value provides a method, which satisfies certain desirable axioms, of allocating benefits to the players of a cooperative game. When there are \(n\) players and \(n\) is large, the Shapley value requires a large amount of accounting because the number of coalitions grows exponentially with \(n\). This paper proposes a modified value (which the author calls the reduced Shapley value) that shares some of the axiomatic properties of the Shapley value yet allows the consideration of games that are defined only for certain coalitions (a coalition list). Its formula resembles one of the familiar formulas for the Shapely value. When all possible coalitions are permitted, the reduced value coincides with the standard Shapley value. The major theorem is the following: Suppose that \(C\) is a symmetric list of coalitions, which includes the coalition of all players. There exists one and only one allocation procedure for partial games defined on \(C\) that is linear, symmetric, and monotone. This unique allocation procedure is the reduced Shapley value. The author also provides another axiomatization of the reduced Shapley value and a probabilistic interpretation of it.