Springer  2008
Models in Cooperative Game Theory (Hardcover)
by Rodica Branzei (Author), Dinko Dimitrov (Author), Stef Tijs (Author) 
f7 P
Hardcover: 203 pages 2 j; n/ n0 L8 I! ]6 z! [
Publisher: Springer; 2nd ed. edition (April 1, 2008)
Language: English

Contents
 Part I Cooperative Games with Crisp Coalitions& K$ I  o7 r, r
1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5' M2 s$ i3 F  e+ \7 L+ _' G5 R
2 Cores and Related Solution Concepts . . . . . . . . . . . . . . . . 13% Z+ |. c1 i* G* G# U# e# y0 a
2.1 Imputations, Cores and Stable Sets . . . . . . . . . . . . . . . . . . . 13
2.2 The Core Cover, the Reasonable Set and the Weber Set . 20
The Shapley Value, the τ -value, and the Average
Lexicographic Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.1 The Shapley Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.2 The τ-value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.3 The Average Lexicographic Value . . . . . . . . . . . . . . . . . . . . 338 o% ?. ?/ L! n% v+ p1 ^, q" |6 N
4 Egalitarianism-based Solution Concepts . . . . . . . . . . . . . . 37
4.1 Overview. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.2 The Equal Split-Off Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4.2.1 The Equal Split-Off Set for General Games . . . . . . 39
4.2.2 The Equal Split-Off Set for Superadditive Games . 417 ]- X* I# D) P; L
5 Classes of Cooperative Crisp Games . . . . . . . . . . . . . . . . . 433 T' M5 ^# ]) O$ \
5.1 Totally Balanced Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435 Z  ~7 |, d& k
5.1.1 Basic Characterizations and Properties of
Solution Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
5.1.2 Totally Balanced Games and Population
Monotonic Allocation Schemes . . . . . . . . . . . . . . . . . 45
5.2 Convex Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462 R4 p% T& x+ S- h& H* X/ j, L" {
5.2.1 Basic Characterizations . . . . . . . . . . . . . . . . . . . . . . . 46/ X; g. A6 e3 j' o
5.2.2 Convex Games and Population Monotonic0 w9 S' @, Q# `
Allocation Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . 497 h$ Z9 O4 E4 h8 h4 K( |
5.2.3 The Constrained Egalitarian Solution for Convex# N( a4 S: u" s
Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
5.2.4 Properties of Solution Concepts . . . . . . . . . . . . . . . . 53
5.3 Clan Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
5.3.1 Basic Characterizations and Properties of
Solution Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
5.3.2 Total Clan Games and Monotonic Allocation" f0 G% Z$ |3 h3 G! D
Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623 L' Z5 ~: P$ H  d4 q) B
5.4 Convex Games versus Clan Games . . . . . . . . . . . . . . . . . . . 65* {. c9 e3 z9 g, S- |
5.4.1 Characterizations via Marginal Games . . . . . . . . . . 661 B5 @5 y: H) z! \2 J3 v- ^
5.4.2 Dual Transformations . . . . . . . . . . . . . . . . . . . . . . . . . 689 G. |" j# X6 X
5.4.3 The Core versus the Weber Set . . . . . . . . . . . . . . . . . 70
Part II Cooperative Games with Fuzzy Coalitions
6 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77+ z& G! D5 b9 k" J6 G) V7 C
7 Solution Concepts for Fuzzy Games . . . . . . . . . . . . . . . . . . 83
8 X9 O1 h( }! i7.1 Imputations and the Aubin Core . . . . . . . . . . . . . . . . . . . . . 83
7.2 Cores and Stable Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
7.3 Generalized Cores and Stable Sets . . . . . . . . . . . . . . . . . . . . 89. W" U# [3 [1 S, ^
7.4 The Shapley Value and the Weber Set . . . . . . . . . . . . . . . . 945 R% U$ k& c) t8 A! V) O7 O% I
7.5 Path Solutions and the Path Solution Cover . . . . . . . . . . . 96
7.6 Compromise Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1002 R% e- n) v- V+ F  q8 \1 l
8 Convex Fuzzy Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103( O( \. e; B7 o
8.1 Basic Characterizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1035 O- m% x) x& L) j; w  w' M. C0 U, Y
8.2 Egalitarianism in Convex Fuzzy Games . . . . . . . . . . . . . . . 1109 c& V5 W9 b* G
8.3 Participation Monotonic Allocation Schemes . . . . . . . . . . . 116+ i* |: H  n  q
8.4 Properties of Solution Concepts . . . . . . . . . . . . . . . . . . . . . . 119
9 Fuzzy Clan Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
9.1 The Cone of Fuzzy Clan Games . . . . . . . . . . . . . . . . . . . . . . 127# g0 R$ [7 @2 p/ h
9.2 Cores and Stable Sets for Fuzzy Clan Games . . . . . . . . . . 1316 h; l! A! a  {& z) ^& A2 _
9.3 Bi-Monotonic Participation Allocation Rules . . . . . . . . . . . 135
Part III Multi-Choice Games4 q3 V, z2 a% H9 c5 v
10 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
11 Solution Concepts for Multi-Choice Games . . . . . . . . . . 149
11.1 Imputations, Cores and Stable Sets . . . . . . . . . . . . . . . . . . . 149
11.2Marginal Vectors and the Weber Set . . . . . . . . . . . . . . . . . . 155
11.3 Shapley-like Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
11.4 The Equal Split-Off Set for Multi-Choice Games . . . . . . . 163; _1 n  c. ~) T# T! z
12 Classes of Multi-Choice Games . . . . . . . . . . . . . . . . . . . . . . 165
12.1 Balanced Multi-Choice Games . . . . . . . . . . . . . . . . . . . . . . . 165, \& @& }' a: c3 c' o5 g
12.1.1 Basic Characterizations . . . . . . . . . . . . . . . . . . . . . . . 165) O9 ^6 u: a6 v" W0 X5 r
12.1.2 Totally Balanced Games and Monotonic7 N7 u/ f, o3 V5 s5 Z
Allocation Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
12.2 Convex Multi-Choice Games . . . . . . . . . . . . . . . . . . . . . . . . . 170
12.2.1 Basic Characterizations . . . . . . . . . . . . . . . . . . . . . . . 170
12.2.2 Monotonic Allocation Schemes . . . . . . . . . . . . . . . . . 173
12.2.3The Constrained Egalitarian Solution . . . . . . . . . . . 174
12.2.4 Properties of Solution Concepts . . . . . . . . . . . . . . . . 180; |5 l  i" T' a/ h# i% m
12.3Multi-Choice Clan Games . . . . . . . . . . . . . . . . . . . . . . . . . . . 182" r$ o% N: e7 ?0 m
12.3.1 Basic Characterizations . . . . . . . . . . . . . . . . . . . . . . . 182
Bi-Monotonic Allocation Schemes . . . . . . . . . . . . . . . 186  r0 d1 o. }+ n, u
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1935 g9 }' a+ a$ W. c
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201