The Central Core and the Mid-central Core as Novel Set-valued and Point-valued Solution Concepts for TU Coalitional Games: Topological and Axiomatic Properties

The present paper proposes two new solution concepts that are Core restrictions: the first, set-valued, is the Central Core, the second, point valued, is the Mid-central Core. The basic idea at the root of the Central Core is to allow such Core elements that grant to each player at least the pay-off obtained as the centroid of the extreme points of the set of endogenous outside options that would emerge from a hypothetical bargaining game over the same coalitions except the grand coalition. The Mid-central Core is simply defined as the centroid of the extreme points of the Central Core. The basic topological properties of the Central Core are then analysed showing that it is a convex polytope with dimensionality equal to, maximum, n-2 and, at most, n extreme points lying on the boundaries of the Core, with $n$ being the number of players in the coalitional game. It is further shown that almost all axiomatic properties of the Core are preserved by these restrictions, except for consistency. The Mid-central Core further satisfies aggregate and weak coalitional monotonicity, but not strong and coalitional monotonicity.