A Shapley Value for Games with Restricted Coalitions

Abstract

1. Introduction A transferable utility game is a pair (N,v) where N:= {1,2, … ,n} (n ≥ 1) is the set of players and v : 2^N → ℝ is a map assigning to each coalition S ⊂ N its worth v(S), with v(0) := ∅0. By way of an example, consider the game ({1,2,3},v) with v(1) = 10, v(2) = v(3) = 0, v(12) = 20, v(13) = 30, v(23) = 0, and v(N) = 30 (braces are omitted for simplicity). Here player 1 is the seller of an object who values this object at 10 dollars; 2 and 3 are buyers who value the object at 20 and 30 dollars, respectively. Assuming that the grand coalition N will be formed, the question arises how to divide the worth v(N) among the players. A well-known answer to this question was provided by Shapley (1953), now known as the Shapley value.