Jo Nov 10, 2023

Core and Weber set are the most important set-valued solution concepts in TU-games. The core is a set of efficient payoff vectors that satisfy coalitional rationality i.e., a coalition receives at least its own worth, while the Weber set is a convex hull of all marginal vectors. In TU-games, core is always contained in Weber set. Furthermore, a game is convex if and only if its core coincides with its Weber set.

Inclusion of Weber set into core plays an important role in studies of 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 marginal vectors.

O Un Suk, a lecturer at the Faculty of Applied Mathematics, has proved the stability of the extent/intent Shapley value using the inclusion of the extent/intent Weber set into the extent/intent core in games on concept lattices.

In games on extents, she introduced extent Weber set, newly defined strong convexity, and proved the inclusion of the extent Weber set into the extent core under this condition. Similarly, she proved that the intent Weber set is included into the intent core under the weak concavity in games on intents. Finally, she studied the relation between the game on extents and the game on intents, and derived a sufficient condition for stability of two Shapley values.