Applying Fuzzy Mathematics to Formal Models in Comparative Politics

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Springer Science & Business Media, Mar 20, 2008 - Computers - 214 pages
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This book explores the intersection of fuzzy mathematics and the spatial modeling of preferences in political science. Beginning with a critique of conventional modeling approaches predicated on Cantor set theoretical assumptions, the authors outline the potential benefits of a fuzzy approach to the study of ambiguous or uncertain preference profiles. While crisp models assume that ambiguity is a form of confusion emerging from imperfect information about policy options, the authors argue instead that some level of ambiguity is innate in human preferences and social interaction. What fuzzy mathematics offers the researcher, then, is a precise tool with which he can model the inherently imprecise dimensions of nuanced empirical reality. Moving beyond the limited treatment fuzzy methodologies have received in extant political science literature, this book develops single- and multidimensional models of fuzzy preference landscapes and characterizes the surprisingly high levels of stability that emerge from interactions between players operating within these models. The material presented makes it a good text for a graduate seminar in formal modeling. It is also suitable as an introductory text in fuzzy mathematics for graduate and advanced undergraduate students.

 

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Contents

Applying Fuzzy Set Theory to Comparative Politics
1
11 Comparative Politics
2
12 Fuzzy Mathematics and Political Science
4
13 The New Institutionalism
8
14 Single Dimensional Models
11
15 Spatial Multidimensional Models
12
16 Democratic Consolidation
15
17 Modeling the Problem of Presidentialism
16
45 Fuzzifying Kiewiet and McCubbins Presidential Veto Model
94
46 Discrete Fuzzy Numbers
100
47 A OneDimensional Model with Discrete Fuzzy Numbers
104
Fuzzy Advantages
106
References
107
Fuzzy Spatial Models
108
51 The Cycling Problem in Crisp TwoDimensional Spatial Models
110
52 A Fuzzy Set Theory Approach to TwoDimensional Models
119

18 Veto Player Theory
21
References
24
Fuzzy Set Theory
28
21 Fuzzy Sets
30
22 Membership Functions
31
23 AlphaCut or αCut
34
24 Fuzzy Numbers
37
241 Triangular Fuzzy Numbers
39
242 Trapezoidal Fuzzy Numbers
40
243 Differentiable Piecewise Quadratic Fuzzy Numbers
41
246 Impulse Fuzzy Numbers
42
247 SShaped Fuzzy Numbers
43
25 Constructing Fuzzy Sets
44
251 General Views of Fuzzy Data
45
252 Granularity
46
254 Automatic Methods
53
255 Adaptive Methods
55
26 Fuzzy Set Operations
57
261 Subsets
58
263 Union
59
264 Complement
60
27 Metrics for Fuzzy Numbers
61
28 Fuzzy Geometry
62
References
63
Fuzzy Geometry
65
32 Circles and Polygons
72
33 Looking Ahead
79
References
80
Fuzzy OneDimensional Models
81
41 Crisp OneDimensional Models
82
42 Modeling the Presidential Veto Game
88
43 The Case for a New Approach
90
44 Toward a Fuzzy OneDimensional Model
91
521 Separability and Finite Sets of Alternatives
120
522 Separability and Bargaining over Outcomes
123
523 Nonseparability
129
53 Implications
133
References
134
Estimating Fuzzy Policy Preferences
137
61 Information Conveyed in Membership Grades
138
Membership as Utility and Intensity
140
63 Aggregating Fuzzy Preferences in Spatial Analysis
142
64 Aggregation Operators
143
65 Summary
149
References
150
Fuzzy Operators
151
A11 Fuzzy Complement
153
A12 Fuzzy Set Intersections
155
A 13 Fuzzy Set Unions
157
A2 Generating Functions
158
A3 Combinations of Operations
163
A4 Averaging Operator
166
A41 Aggregation Operations
167
A42 OWA Operators
168
Cycling in Fuzzy Spatial Models
169
72 Modeling Consensus
171
73 Cycling in Spatial Models
175
74 Conclusions
181
References
182
List of Symbols
183
List of Tables
186
List of Figures
189
Annotated Bibliography
192
Author Index
209
Index
214
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About the author (2008)

TERRY D. CLARK is Associate Professor of Political Science and Director of the Graduate Program in International Relations at Creighton University. He has published extensively on emerging democratic institutions in postcommunist Europe. His articles have appeared in Slavic Review, East European Politics and Societies (EEPS), PS, Nationalities Papers, and the Journal of Policy Studies. Clark is a past recipient of several IREX research grants and was a Fulbright Fellow at Vilnius University, Vilnius University.

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