Encouraging the Grand Coalition in Convex Cooperative Games

A solution function for convex transferable utility games encourages the grand coalition if no player prefers (in a precise sense defined in the text) any coalition to the grand coalition. We show that the Shapley value encourages the grand coalition in all convex games and the τ -value encourages the grand coalitions in convex games up to three (but not more than three) players. Solution functions that encourage the grand coalition in convex games always produce allocations in the core, but the converse is not necessarily true. 1. Cooperative games We begin by recalling the main concepts and their basic properties. The notation mostly follows [Cur97] and/or [BDT05]. Let N = {1, . . . , n}. The elements of N are called players, its subsets are called coalitions, and the set N is called the grand coalition. A cooperative transferable utility game with n-players is a function v : 2 → R such that v(∅) = 0, where 2 is the set of all subsets of N . For a given game v, we often denote v ({i}) by v(i) or vi. More generally, for any function x : N → R and i ∈ N , we denote x(i) = xi. Thus we (sometimes) think of functions x : N → R as vectors in R. For a function x : N → R and a coalition A ⊆ N , we write x(A) = ∑ j∈A xj . A game v : 2 → R is called super-additive if, for all disjoint coalitions A,B ⊆ N , v(A) + v(B) ≤ v(A ∪B) and is called convex if, for all coalitions A,B ⊆ N , v(A) + v(B) ≤ v(A ∪B) + v(A ∩B). Example 1. Define a 4-player game v onN = {1, 2, 3, 4} by the diagram in Figure 1 (the value of each coalition is provided at the vertex representing the coalition). The same game is given in a tabular form in Table 1. It is straightforward to check that the game v is convex. A way to interpret cooperative games is as follows. Assume that the players in the set N can form various coalitions each of which has value prescribed by v (say v(A) represents the amount the coalition A can earn by cooperating). The superadditivity condition implies that “the whole is larger than the sum of its parts”, i.e., forming larger coalitions positively affects the value. The convexity condition 2000 Mathematics Subject Classification. 91B32, 91B08, 91A12.