O Un Suk, a lecturer at the Faculty of Applied Mathematics, has studied the relations of core, Weber set and Shapley value to one another under the condition of strong convexity / weak concavity.
A cooperative game with transferable utility, a TU-game for short, is defined as a pair (N, v) , where N is a finite set of players and v is a mapping that assigns a real number v(S) to each coalition S⊆N. v(φ) is set 0.
One of the important problems in the cooperative game theory is to select an efficient payoff vector, i.e., a way of sharing among players the total worth of the game when they all join a grand coalition. For this aim, various solution concepts have been developed.
The solutions playing an important role in selecting efficient payoff vectors include Core, Weber set, and Shapley value.
In many real situations, however, not all subsets of N can be realized as coalitions or they are not feasible. The interest in subcollections of the power set of N is inspired by the fact that in case of restricted communication or cooperation structure, the standard model of a cooperative game is not applicable. A game on concept lattices is a good example.
A game on concept lattices is a cooperative TU-game defined on concepts, where a concept is a pair (S, S') of S, a subset of players or objects, and S' a subset of attributes.
Such games induce a game on extents and a game on intents.
The inclusion of the Weber set in core plays an important role in the studies of the stability of the Shapley value (inclusion of the Shapley value into the core) because the Shapley value is defined as a mean value of all vertices of the Weber set.
The Weber set does not become the subset of the core under convexity in games on concept lattices different from classical cooperative games. Thus, the concept of strong convexity is introduced in games on extents.
Strong convexity means that the larger coalition each player joins in, the greater the average creation value of them gets.
The following theorem shows that the extent Weber set becomes a subset of the extent core in strongly convex games on extents.
Theorem 1. The extent Weber set is included in the extent core if a game on extents is strongly convex.
Corollary 1. The extent Shapley value assigns an element of the extent core if a game on extents is strongly convex.
The condition of strong convexity is a sufficient condition that guarantees the extent Shapley value to be in the extent core in the games on extents. Therefore, in convex games, not strongly convex, the extent Shapley value may belong to the extent core.
Similarly, it becomes possible to define weak concavity in games on intents as they are in dual relation with games on extents. Then, it can be proved that the intent Weber set is the subset of the intent core under weak convexity.
It is concluded that the extent/intent Shapley value can be used as a rational solution to this game if a game on extents/intents is convex/concave.
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