Soft cooperation systems and games

Abstract

A cooperative game for a set of agents establishes a fair allocation of the profit obtained for their cooperation. In order to obtain this allocation, a characteristic function is known. It establishes the profit of each coalition of agents if this coalition decides to act alone. Originally players are considered symmetric and then the allocation only depends on the characteristic function; this paper is about cooperative games with an asymmetric set of agents. We introduced cooperative games with a soft set of agents which explains those parameters determining the asymmetry among them in the cooperation. Now the characteristic function is defined not over the coalitions but over the soft coalitions, namely the profit depends not only on the formed coalition but also on the attributes considered for the players in the coalition. The best known of the allocation rules for cooperative games is the Shapley value. We propose a Shapley kind solution for soft games.

Keywords:

• Game theory
• soft sets
• Shapley value
• cooperation systems

Notes

No potential conflict of interest was reported by the authors.

1 In the classical cooperative game theory all the subsets of the set of players are coalitions.

2 The number of chains in the subposet SC(F, A).

3 The number of chains in the subposet $[(F,A),(F_0,A_0)]$.

4 Full soft sets and therefore full soft games can be introduced in another equivalence way. Let $(F_0,A_0)$ be a soft set of players. A soft coalition (F, A) is spanning if $A=A_0$. A soft game $(F_0,A_0,v)$ is spanning if $v(F,A)=v((F,A)\cup (\mathrm{\varnothing },A_0))$ for all $(F,A)\in SC(F_0,A_0)$. It is easy to test that $(F_0,A_0,v)$ is full if and only if it is spanning.