On the vertices of the k-addiive core

TL;DR

This paper introduces the concept of the $k$-additive core in cooperative game theory, extending the classical core to nonempty convex polyhedra for certain $k$, and characterizes their vertices.

Contribution

It defines the $k$-additive core, generalizing the classical core, and establishes vertex characterization results similar to those for convex games.

Findings

The $k$-additive core is a nonempty convex polyhedron for high enough $k$.

Vertices of the $k$-additive core are characterized similarly to classical core vertices.

The approach generalizes classical results of Shapley and Ichiishi.

Abstract

The core of a game $v$ on $N$, which is the set of additive games $\phi$ dominating $v$ such that $\phi(N)=v(N)$, is a central notion in cooperative game theory, decision making and in combinatorics, where it is related to submodular functions, matroids and the greedy algorithm. In many cases however, the core is empty, and alternative solutions have to be found. We define the $k$-additive core by replacing additive games by $k$-additive games in the definition of the core, where $k$-additive games are those games whose M\"obius transform vanishes for subsets of more than $k$ elements. For a sufficiently high value of $k$, the $k$-additive core is nonempty, and is a convex closed polyhedron. Our aim is to establish results similar to the classical results of Shapley and Ichiishi on the core of convex games (corresponds to Edmonds' theorem for the greedy algorithm), which characterize the vertices of the core.