A dynamic process to the core for multi-choice games

doi:10.1016/j.orl.2015.11.006

Operations Research Letters

Volume 44, Issue 1, January 2016, Pages 50-53

https://doi.org/10.1016/j.orl.2015.11.006 Get rights and content

Abstract

In the framework of multi-choice games, we propose a dynamic process leading to the core which was introduced by van den Nouweland et al. (1995). Also, we prove that it converges if and only if the core is nonempty.

Introduction

The core of TU-games (games with transferable utility) was introduced by Gillies [5] in 1953. Subsequently, in 1998, Cesco [3] provided a transfer scheme leading to the nonempty core of a TU-game. Also he proved that such a transfer scheme converges if and only if the core of a TU-game is nonempty. In the literature, the core of multi-choice games was first introduced by van den Nouweland et al. [11]. This raises the question whether the result of Cesco [3] extends to multi-choice games. The answer is positive. The aim of this note is to describe a model of dynamic bargaining that converges to the core for multi-choice games. Also, such a transfer scheme converges if and only if the core of a multi-choice game is nonempty. In the framework of TU games, related results may be found in Stearns [10], Justman [6], Maschler and Owen [7], and so on. Stearns [10] described two transfer scheme that converges to the kernel [4] and the bargaining set [1], respectively. Justman [6] provided a transfer scheme that converges to the nucleolus [8]. Maschler and Owen [7] described a dynamic process that converges to the Shapley value.

As Cesco [3] pointed out that Bondareva [2] and Shapley [9] used the duality theorem in linear programming to prove that the core of a TU-game is nonempty if and only if the game is balanced. But, Cesco did not use the linear programming method. Cesco proved that a transfer scheme converges if and only if the core of a TU-game is nonempty. Similarly, van den Nouweland et al. [11] used the duality theorem to prove that the core of a multi-choice game is nonempty if and only if the game is balanced. But our proof does not use it.

The paper is organized as follows. In Section 2, we introduce the definitions and some notations. In Section 3, we present a transfer scheme and prove the convergence result. Also, we provide an algorithm for our convergence result. In Section 4, we provide a solid discussion.

Section snippets

Definitions and notations

Let $N = {1, 2, \dots, n}$ be a set of players and for $i \in N$ , let $M_{i} = {0, 1, 2, \dots, m_{i}}$ be an action set of player $i$ which means player $i$ has $m_{i}$ activity levels at which he or she can play, where the action 0 means a player does not participate, here we denote $M_{i} ∖ {0}$ as $M_{i}^{+}$ . The set of action vectors is denoted by $\prod_{i \in N} M_{i} = {(ρ_{1}, \dots, ρ_{n}) : ρ_{i} \in M_{i} for all i \in N}$ , where $ρ = (ρ_{1}, \dots, ρ_{n})$ is called an action vector of $N$ .

For convenience, we assume $m_{i} = m$ for all $i \in N$ . A multi-choice game is a triple $(N, m, v)$ , where $N$ is a set of players, $m \in$

Transfer scheme and convergence result

In this section, we present a transfer scheme in the setting of multi-choice games, and prove that the maximal transfer scheme converges if and only if the core is nonempty. First, we present a transfer and a transfer scheme in the setting of multi-choice games as follows.

Let $v$ be a game, $x$ be an arbitrary payoff configuration in $E (v)$ , $ρ$ be an action vector in $\prod_{i \in N} M_{i} ∖ {θ, m e^{N}}$ and $e (ρ, x, v) \geq 0$ , then we use the following terminologies:

1.
$y$ is said to result from $x$ by a transfer of size $e (ρ, x, v)$ from $[L$

Discussion

We study multi-choice games and focus on a dynamic process that converges if and only if the core is nonempty and if it converges, it converges to the core. The dynamic process closely resembles the process of Cesco [3]. In fact several results have a direct equivalence in the results of Cesco [3]; this includes a proof approaches with a lot of resemblance.

Besides, consider an arbitrary multi-choice game, one could construct a TU game as follows:

•
${(i, j) ∣ i \in N, 1 \leq m_{i}}$ as player set;
•
the same values

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The bargaining set for cooperative games
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Some applications of the methods of linear programming to the theory of cooperative games
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J.C. Cesco
A convergent transfer scheme to the core of a TU-game
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The kernel of a cooperative game
Nav. Res. Logist. Q.
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D.B. Gillies
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Operations Research Letters

A dynamic process to the core for multi-choice games

Abstract

Introduction

Section snippets

Definitions and notations

Transfer scheme and convergence result

Discussion

The bargaining set for cooperative games

Ann. of Math. Stud.

Some applications of the methods of linear programming to the theory of cooperative games

Probl. Kibernet.

A convergent transfer scheme to the core of a TU-game

Rev. Mat. Apl.

The kernel of a cooperative game

Nav. Res. Logist. Q.

Some theorems on n-person games

Some theorems on $n$ -person games