## Applying Fuzzy Mathematics to Formal Models in Comparative PoliticsThis 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. |

### What people are saying - Write a review

### 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 |