Jiménez-Losada - Models for cooperative games with fuzzy relations among the agents: fuzzy communication, proximity relation and fuzzy permission

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Springer International Publishing AG 2017

Andrs Jimnez-Losada Models for Cooperative Games with Fuzzy Relations among the Agents Studies in Fuzziness and Soft Computing 10.1007/978-3-319-56472-2_1

1. Cooperative Games

Andrs Jimnez-Losada 1

(1)

Departamento de Matemtica Aplicada II, Universidad de Sevilla, Seville, Spain

Andrs Jimnez-Losada

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1.1 Introduction

This chapter corresponds to an introduction about cooperative games with transferable utility, namely the framework within which the book is developed. It does not seek to be fully exhaustive because this book is not a handbook about cooperative games but it sets out to be consistent and comprehensive in order to be useful throughout the rest of the chapters.

The study of the theory of games started in von Neumann and Morgenstern [] the authors described coalitional games in characteristic function form, also known as transferable utility games (games for short). The characteristic function of a game is a real-valued function on the family of coalitions. The real number assigned to each coalition is interpreted as the utility of the cooperation among this group of players. In these cases the worth of a coalition can be allocated among its players in any way. The adjective transferable refers to the assumption that a player can transfer any part of his utility to another player.

Solving a game means determining which coalition or coalitions are formed and obtaining a vector at the end of the game with the corresponding individual payoffs for the cooperation of the players (payoff vector). The classic model of game considers that the grand coalition (the coalition of all the players) will be formed and assumes that there are no restrictions in cooperation, therefore every subset of players can form a different coalition. A value for games is a function assigning a payoff vector for each game. The most known value was introduced by Shapley [], etc. This book is focused on the Shapley value.

This chapter introduces cooperative games, the Shapley value and those results about them that we will use in the next chapters. For further learning the reader can use books about games in general as Driessen [] is a broad vision about this value.

1.2 Games

Cooperative games analyze situations among a finite set of agents where they cooperate to get a determined benefit (profits, debts, costs,...). In this book we consider games in coalitional form with transferable utility. In this kind of games the benefit obtained by the cooperation can be allocated among the agents in any way, so the individual utilities of them have a transfer system permitting that an agent can lose utility to other one in order to keep the cooperation. Hence the benefit is a payoff number for the set of agents. The coalitional form implies that it is known a mapping determining the benefit for any subset of agents. Given a finite set N we denote as its set of power, namely the family of subsets of N , and with the tiny letter n the cardinality of N .

Definition 1.1

Let N be a finite set of elements named players . A cooperative game with transferable utility over N is a mapping which assigns a worth to each subset of players (coalition) , satisfying that . The family of cooperative games with transferable utility over N is denoted as .

From now on we use game instead cooperative game with transferable utility. To define a game we need then two elements: the set of players N and the mapping v (named usually characteristic function). In the next examples we see how the games can modeled different situations.

Example 1.1

We consider a production economy in which there are several peasants and one landowner. This model has been studied in Shapley and Shubik []. The peasants contribute only with their work and they are of the same type. The landowner hires the peasants to cultivate his land. If t peasants are hired by the landowner, then the monetary value of the harvest obtained is denoted by The mapping is named production function where m is the total number of peasants. In what follows, it is required that h satisfies these two conditions:

1. the landowner by himself does not produce anything, i.e.,
2. mapping h is nondecreasing, i.e.,

Both conditions imply that h is a nonnegative mapping. We consider the landowner as player 1 and the peasants as players Then this situation can be modeled as a game with players with characteristic function v given by

The value of any coalition that contains only peasants is 0 because they do not have any land. Even more, the worth of each coalition that contains the landowner is equal to the monetary value of the harvest that is obtained by the peasants that are in that coalition. Obviously .

Example 1.2

Control games were proposed by Feltkamp []. A control situation is defined by a set of agents N who decide about a good. They, as group, have the control of the good but they want to determine the power of each one. The control game is with and

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