Monotonicity and dummy free property for multi-choice cooperative games

Abstract

Given a coalition of an n-person cooperative game in characteristic function form, we can associate a zero-one vector whose non-zero coordinates identify the players in the given coalition. The cooperative game with this identification is just a map on such vectors. By allowing each coordinate to take finitely many values we can define multi-choice cooperative games. In such multi-choice games we can also define Shapley value axiomatically. We show that this multi-choice Shapley value is dummy free of actions, dummy free of players, nondecreasing for non-decreasing multi-choice games, and strictly increasing for strictly increasing cooperative games. Some of these properties are closely related to some properties of independent exponentially distributed random variables. An advantage of multi-choice formulation is that it allows to model strategic behavior of players within the context of cooperation.

Consistent extensions and subsolutions of the core of multi-choice NTU games

“…As we know solutions of multi-choice TU games can be applied to many fields, such as economics, political sciences, accounting, and even military sciences. Related results may be found in Hsiao and Raghavan (1992), Derks and Peters (1993), van den Nouweland et al (1995), Klijn et al (1999), Peters and Zank (2005), Grabisch and Lange (2007), Hwang and Liao (2008) and so on.…”

“…Hsiao and Raghavan 1992), each player has several activity levels. Hence, multi-choice TU games constitute a generalization of standard coalition TU games.…”

The generalized symmetric coalitional Banzhaf value for multichoice games with a coalition structure

“…Obviously, (4) degenerates to be the symmetric coalitional Banzhaf value [1] where the range of the mapping v is N , namely, every player has two activity levels 0 and 1. Furthermore, (4) degenerates to be the value for multichoice games introduced by van den Nouweland, et al [5] where h is equal to 1.…”

Multi-choice clan games and their core

“…Further, a clan game whose all subgames are also clan games is called a total clan game. A more sophisticated model of cooperative games, which is a natural extension of the traditional model, called multi-choice game, was introduced by Hsiao and Raghavan (1992Raghavan ( , 1993 and reconsidered by Nouweland et al (1995) in a more general setting. Multi-choice cooperative games have become a useful tool for modeling interaction of players in situations in which they may have different options for cooperation, varying from non-cooperation (participation level 0) to a maximal participation level which is greater than or equal to 1.…”