A dynamic process to the core for multi-choice games
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 be a set of players and for , let be an action set of player which means player has activity levels at which he or she can play, where the action 0 means a player does not participate, here we denote as . The set of action vectors is denoted by , where is called an action vector of .
For convenience, we assume for all . A multi-choice game is a triple , where is a set of players,
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 be a game, be an arbitrary payoff configuration in , be an action vector in and , then we use the following terminologies:
- 1.
is said to result from by a transfer of size from
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:
- •
as player set;
- •
the same values
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