On the set of imputations induced by the k-additive core

TL;DR

This paper explores the $k$-additive core in cooperative game theory, analyzing the set of classical imputations derived from $k$-additive imputations via sharing rules, highlighting its properties and implications.

Contribution

It characterizes the set of classical imputations obtainable from the $k$-additive core through sharing rules, extending understanding of $k$-additive game structures.

Findings

The $k$-additive core is non-empty for $k \\geq 2$.

It provides a framework to derive classical imputations from $k$-additive imputations.

The analysis clarifies the relationship between $k$-additive and classical cores.

Abstract

An extension to the classical notion of core is the notion of $k$-additive core, that is, the set of $k$-additive games which dominate a given game, where a $k$-additive game has its M\"obius transform (or Harsanyi dividends) vanishing for subsets of more than $k$ elements. Therefore, the 1-additive core coincides with the classical core. The advantages of the $k$-additive core is that it is never empty once $k\geq 2$, and that it preserves the idea of coalitional rationality. However, it produces $k$-imputations, that is, imputations on individuals and coalitions of at most $k$ inidividuals, instead of a classical imputation. Therefore one needs to derive a classical imputation from a $k$-order imputation by a so-called sharing rule. The paper investigates what set of imputations the $k$-additive core can produce from a given sharing rule.