The Intermediate Set and Limiting Superdifferential for Coalition Games: Between the Core and the Weber Set

TL;DR

This paper introduces the intermediate set as a new solution concept in coalitional game theory, bridging the core and Weber set through the limiting superdifferential of the Lovasz extension, and provides explicit characterizations.

Contribution

It defines the intermediate set using variational analysis, explores its properties, and computes explicit forms for various classes of games, enriching the solution hierarchy.

Findings

The intermediate set is non-convex and contains Pareto optimal payoffs.

Explicit formulas for the intermediate set are derived for all games.

Simplified characterizations are provided for specific game types like simple and glove games.

Abstract

We introduce the intermediate set as an interpolating solution concept between the core and the Weber set of a coalitional game. The new solution is defined as the limiting superdifferential of the Lovasz extension and thus it completes the hierarchy of variational objects used to represent the core (Frechet superdifferential) and the Weber set (Clarke superdifferential). It is shown that the intermediate set is a non-convex solution containing the Pareto optimal payoff vectors that depend on some chain of coalitions and marginal coalitional contributions with respect to the chain. A detailed comparison between the intermediate set and other set-valued solutions is provided. We compute the exact form of intermediate set for all games and provide its simplified characterization for the simple games and the glove game.