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

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.

💡 Research Summary

The paper investigates the “k‑additive core”, a generalisation of the classical core in cooperative game theory, and studies which classical imputations can be obtained from it by means of sharing rules. A game is a pair (N,v) with a characteristic function v:2^N→ℝ, v(∅)=0. The classical core C(v) consists of all pre‑imputations x∈ℝ^N satisfying x(N)=v(N) and x(S)≥v(S) for every coalition S⊂N. Unfortunately C(v) is often empty, which limits its practical relevance.

A k‑additive game is defined by the property that its Möbius transform (Harsanyi dividends) m(S) vanishes for all subsets S with |S|>k. The set of all k‑additive games on N is denoted G_k(N). The k‑additive core C_k(v) is the collection of all k‑additive games φ∈G_k(N) that dominate v, i.e. φ(N)=v(N) and φ(S)≥v(S) for all S⊂N. A crucial observation, proved in earlier work and recalled here, is that C_k(v) is never empty as soon as k≥2.

However, an element φ∈C_k(v) does not directly give a classical payoff vector. Its Möbius transform m_φ provides a “k‑order pre‑imputation”: a set of numbers m_φ(S) for every coalition S of size at most k, with the interpretation that the total dividend of S is distributed among its members. To obtain a classical imputation (a vector x∈ℝ^N) one must apply a sharing rule that allocates each coalition dividend m_φ(S) among the players belonging to S.

Two families of sharing rules are considered. A selector α∈A(N) assigns to each non‑empty coalition S a distinguished player α(S)∈S; the corresponding sharing rule gives the whole dividend of S to α(S). A sharing function q∈Q(N) assigns a non‑negative weight q(S,i) to each player i∈S such that ∑_{i∈S}q(S,i)=1. The set Q(N) is a convex polytope whose extreme points are precisely the selectors.

The main results can be summarised as follows.

1. Theorem 1 (Positive sharing functions).
   If q∈Q(N) satisfies q(K,i)>0 for every coalition K and every i∈K, then the image of the 2‑additive core under the sharing map coincides with the whole pre‑imputation set:
   \