Max-consistency, complement-consistency and the core of fuzzy games

doi:10.1016/j.fss.2007.07.016

Fuzzy Sets and Systems

Volume 159, Issue 2, 16 January 2008, Pages 152-163

https://doi.org/10.1016/j.fss.2007.07.016 Get rights and content

Abstract

Recently, Hwang introduced two extensions of the max-reduced game to fuzzy games by reducing the number of the players. Based on these reduced games, he offered analogous axiomatizations of the core related to Serrano and Volij in the context of fuzzy games. Here, we continue and develop Hwang's work on similar topics. However, we turn to a different definition of the reduced game by simultaneously reducing both the number of the players and the participation levels. We propose such extensions of the complement-reduced game and of the max-reduced game to fuzzy games and define the related properties of consistency and converse consistency. Based on these properties, we offer several analogous characterizations of the core related to Peleg, Serrano and Volij and Tadenuma, respectively.

References (27)

R. Branzei et al.
Convex fuzzy games and participation monotonic allocation schemes
Fuzzy Sets and Systems
(2003)
R. Branzei et al.
Egalitarianism in convex fuzzy games
Math. Social Sci.
(2004)
D. Butnariu
Fuzzy games: a description of the concept
Fuzzy Sets and Systems
(1978)
E. Molina et al.
The equalizer and the lexicographical solutions for cooperative fuzzy games: characterizations and properties
Fuzzy Sets and Systems
(2002)
H. Moulin
The separability axiom and equal-sharing methods
J. Economic Theory
(1985)
B. Peleg
An axiomatization of the core of cooperative games without side payments
J. Math. Econom.
(1985)
M. Sakawa et al.
A lexicographical concept in an $n$ -person cooperative fuzzy games
Fuzzy Sets and Systems
(1994)
R. Serrano et al.
Axiomatizations of neoclassical concepts for economies
J. Math. Econom.
(1998)
W.W. Sharkey et al.
Supportable cost functions for the multiproduct firm
J. Econom. Theory
(1978)
M. Tsurumi et al.
A Shapley function on a class of cooperative fuzzy games
European J. Oper. Res.
(2001)

M. Voorneveld et al.

A new axiomatization of the core of games with transferable utility

Econom. Lett.

(1998)

J.P. Aubin

Coeur et valeur des jeux flous á paiements latéraux

Comptes Rendus de l’Académie des Sciences

(1974)

J.P. Aubin

Cooperative fuzzy games

Math. Oper. Res.

(1981)

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Max-consistency, complement-consistency and the core of fuzzy games

Abstract

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