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Operations Research Letters

Volume 46, Issue 6, November 2018, Pages 588-591
Operations Research Letters

Parametric axiom of associated consistency

https://doi.org/10.1016/j.orl.2018.10.005Get rights and content

Abstract

It is proved that the equal allocation of non-separable costs value is characterized if associated consistency is invoked for parametric values λ=1k,k=2,3,…,n−1, even without the axiom of continuity imposed in Hwang (2006).

Introduction

Hamiache [1] axiomatized the Shapley value as the unique solution verifying the inessential game property, continuity, and associated consistency. Later on, Hwang [3] modified the definition of Hamiache’s associated game and axiomatized the EANSC value as the unique solution verifying Pareto optimality, symmetry, translation covariance, continuity, and associated consistency (with respect to the modified-associated game).

The key point behind using associated consistency to axiomatize a solution is to show that the sequence of the repeatedly associated games is convergent. By applying repeatedly associated consistency of the solution for “the same one parametric value”, the sequence of solutions is no difference. Then, by continuity of solution, the sequence of solutions converges to the solution of its limit game.

In this article we turn to a different approach. It is one way to get rid from the axiom of continuity. This approach replaces the continuity with introduction of a finite series of parameters. These values of parameter are involved in the computation of the successive associated games. The successive associated games smash the original game into a “constant game” and it is therefore easy to find the solution.

With respect to other axiomatic characterizations of the EANSC value including an axiom of associated consistency may be found in [[4], [9]] and [5]. The first two articles exploit two different associated games, and the third article relies on the so-called matrix approach.

Section snippets

Definitions and notation

Let U be a non-empty and finite set of players. A coalition is a non-empty subset of U. A game with transferable utility is a pair (N,v) where N is a coalition and v is a mapping such that v:2N⟶R and v(∅)=0. The size of coalition S is denoted by s. For simplification, if no confusion arises, v instead of (N,v) in this note. We denote by GN the set of all games in which the set of players is N. A solution on GN is a function σ which associates with each game v∈GN an element σ(v) of RN. We focus

Main result

For convenience, the zero vector in RN is denoted by 0. Also, we use the following terminology: Let v be a game in GN. Call v is s+symmetric, if v(T)=v(N), for all T⊆N with t≥s, where 1≤s≤n. When v is 1+-symmetric, we also write that v is a constant game. Fig. 1 illustrates how to construct these games v1,v2,v3,…,vn−1 in the proof of Theorem 1. Note that TR means to make a translated game of a game; AG means to make an associated game of a game.

Theorem 1

The EANSC value is a unique solution on GN

Final remark

This article contributes to the growing literature on associated consistency. We explain that the contribution is significant by making a comparison with three closely related articles, [[2], [3]] and [4].

  • •

    To compare with [3]: The difference between two articles is whether to use the axiom of continuity. Two motivations to dispense with the axiom of continuity in this article:

    • 1.

      From the viewpoint of technique:(The referee proposes this viewpoint.)

      Firstly, starting from a given game, one needs an

Acknowledgments

The author is very grateful to the AE and anonymous referees for valuable comments which much improved the paper.

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