Consistency and the graph Banzhaf value for communication graph games

doi:10.1016/j.orl.2022.01.017

Operations Research Letters

Volume 50, Issue 2, March 2022, Pages 190-194

https://doi.org/10.1016/j.orl.2022.01.017 Get rights and content

Abstract

The graph Banzhaf value was introduced and axiomatically characterized by Alonso-Meijide and Fiestras-Janeiro (2006). In this paper we propose the reduced game and consistency of the graph Banzhaf value for communication situations. By establishing the relationship between the Harsanyi dividends of a coalition in a communication situation and the reduced communication situation, we provide a new axiomatization of the graph Banzhaf value by means of the axioms of consistency and standardness.

Introduction

A cooperative game with transferable utility (shortly a TU-game) describes the situation in which any subset of the player set is able to form a coalition and to generate the corresponding worth by cooperating. The objective is to find an allocation rule that rewards the players for participating in the TU-game with a certain payoff. The Banzhaf value [3] is a well-known allocation rule for TU-games according to the marginal contribution of players so as to reasonably allocate the utility.

However, in many practical situations the generation of worth is affected by way that the players are organized. Myerson [9] introduced TU-games with communication graph structure, in which only coalitions of connected players are feasible. Such a TU-game is called a communication situation or a (communication) graph game. The Myerson value is defined as the Shapley value of the so-called graph-restricted game and characterized by component efficiency and fairness.

Following this line of thought in the definition of the Myerson value, Alonso-Meijide and Fiestras-Janeiro [1] extended the Banzhaf value to communication situation, and defined the graph Banzhaf value. They proposed several characterizations by employing the properties of fairness, isolation, pairwise merging, component total power and balanced contribution. Recently, Shi and Shan [12] further extended the Banzhaf value to probabilistic communication situations.

Consistency is an internal stability requirement and plays a significant role in the axiomatizations of some allocation rules. It connects the allocation rules of a game with the allocation rules of a reduced game when some players leave, and requires that when the allocation rules are applied to any reduced game, the payoffs are the same as the original game. It has been successfully applied to characterize a variety of allocation rules, such as the Shapley value in Hart and Mas-Colell [8], Calleja and Llerena [4], [5], the Owen value in Winter [13], the Banzhaf value in Dragan [6], Sánchez-Pérez [11] and the Myerson rule in Albizuri and Zarzuelo [2], etc.

This paper is devoted to characterize the graph Banzhaf value by consistency for communication situations. We first introduce the reduced game, reduced graph and consistency of the graph Banzhaf value for communication situations. In order to show that the graph Banzhaf value satisfies consistency, we establish a key formula on the relationship between the Harsanyi dividends of a coalition in a communication situation and the reduced communication situation. This avoids the use of the potential function to obtain consistency of the graph Banzhaf value. In the previous literature, the consistency of values is proved by virtue of potential functions. Finally, we show that the graph Banzhaf value is uniquely determined by consistency and standardness.

The setup of this paper is as follows. Section 2 provides some basic notation and preliminary definitions. In Section 3, we define the reduced game, reduced graph and consistency, and we give an axiomatization of the graph Banzhaf value via consistency.

Section snippets

Preliminaries

A TU-game is a pair $(N, v)$ , where $N = {1, 2, \dots, n}$ is the set of players and v is a characteristic function defined on $2^{N} = {S | S \subseteq N}$ such that $v (\emptyset) = 0$ . For any $S \subseteq N$ , $v (S)$ represents the worth of the coalition S, and the cardinality of the coalition S is represented by $| S |$ or s. The set of all TU-games $(N, v)$ on N is denoted by $G^{N}$ . For notational simplicity, we abbreviate $v ({i, j, \dots, k})$ , $S \cup {i}$ and $N ∖ {i}$ as $v (i, j, \dots, k)$ , $S \cup i$ and $N ∖ i$ , respectively.

An allocation rule or a value on $G^{N}$ is a mapping $φ \in R^{n}$ , in which $φ_{i} ($

Consistency and a new axiomatic characterization

Consistency describes the invariance of allocation rules by a subgroup of individuals. Consistency is a basic attribute, which is always applies in the same form in the literature. However, the definitions of the reduced games are diverse. Defining a appropriate reduced game is the key to apply consistency to axiomatizations of allocation rules.

In this section, we first define the reduced game applicable to the graph Banzhaf value. Since the graph is involved, we must also define the reduced

Concluding remarks

We introduce the property of consistency of the graph Banzhaf value for communication situations and provide an axiomatization of this value by consistency and standardness. The relationship between the dividends of a coalition in the original communication situation and reduced communication situation is established (Lemma 1) and the formula is applied to show that the graph Banzhaf value satisfies consistency. Unlike in Hart and Mas-Colell [8], Dragan [6] and Albizuri and Zarzuelo [2], the

Acknowledgement

The authors are grateful to an anonymous referee and the area editor for valuable comments and suggestions that contributed greatly to the improvement of this paper. This work is supported by the National Natural Science Foundation of China (No. 11971298).

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Consistency and the graph Banzhaf value for communication graph games

Abstract

Introduction

Section snippets

Preliminaries

Consistency and a new axiomatic characterization

Concluding remarks

Acknowledgement

Discrete Math.

Math. Soc. Sci.

Eur. J. Oper. Res.

Games Econ. Behav.

The Banzhaf value and communication situations

Nav. Res. Logist.

Weighted voting doesn't work: a mathematical analysis

Rutgers Law Rev.