The Shapley-Solidarity Value for Games With a Coalition Structure

Abstract

A value for games with a coalition structure is introduced, where the rules guiding cooperation among the members of the same coalition are different from the interaction rules among coalitions. In particular, players inside a coalition exhibit a greater degree of solidarity than they are willing to use with players outside their coalition. The Shapley value is therefore used to compute the aggregate payoffs for the coalitions, and the solidarity value to obtain the payoffs for the players inside each coalition.

Cognitive Satellite Spectrum Management Scheme Based on the Cooperative Solidarity Values

“…This means that the contribution of a player to a coalition is replaced by the average contribution over the coalition's members. Therefore, we can think that the Shapley value and the Egalitarian value are extreme points, and the Solidarity value is its center [8], [14].…”

“…Therefore, a poor player can achieve a more favorable status, and be in a social level near to those players who participated in solidarity. So is defined on as follows [8]:…”

“…Step 4: At the first phase, the GEO's spectrum resource (ℜ ) is shared by its associated LEOs. Based on the concept of Solidarity value, the value is calculated according to (3), (6), (7) and (8). Based on this information, the ℜ is distributed, and each LEO obtains its own spectrum resource ( ℒ ) by using the equation (9).…”

Spectrum value for coalitional games

“…In its basic interpretation, if a coalition S is formed then the players who contribute more to S than the average marginal contribution members of S contribute to supporting the 'weaker' partners in S. Consideration of the solidarity value, however, does not involve any restriction on the admissible coalitions that can be formed, in contrast with the model used in this paper. (Calvo and Gutierrez, 2011) start from similar suppositions to those in (Owen, 1977), namely that coalitional games are endowed with a coalitional structure, an exogenously given partition of the players. When coalitions are formed, the players interact at two levels: first, bargaining takes place among the unions and then bargaining takes place inside each union.…”

“…When coalitions are formed, the players interact at two levels: first, bargaining takes place among the unions and then bargaining takes place inside each union. (Calvo and Gutierrez, 2011) make use of the solidarity value in this model: first, unions play a quotient game among themselves according to the Shapley value, and then the outcome obtained by the union is shared among its members by paying the solidarity value in the internal game. The coalitional value that they define is then applied in political situations to explain why government coalitions are sometimes larger than minimal winning coalitions.…”