Peter C. Casey

The uncovered set and indifference in spatial models: A fuzzy set approach

The uncovered set was developed in order to predict outcomes when spatial models result in an empty core. In contrast to conventional approaches, fuzzy spatial models induce a substantial degree of individual and collective indifference over alternatives. Hence, existing definitions of the covering relationship return differing results. We develop a definition for a fuzzy covering relation. Our definition results in an uncovered set that is, in most cases, contained within the Pareto set. We conclude by characterizing the exceptions.

The Beginnings of a Weighted Model and New Frontiers

We present some preliminary ideas on the inclusion of a weighted maximal set public choice model. We then compare the results of the tests of the models developed and tested in this book. We conclude with a consideration of fuzzy social choice functions as a means for predicting outcomes in public choice models.

Fuzzy Preferences: Extraction from Data and Their Use in Public Choice Models

We describe a method for extracting fuzzy preferences from the Comparative Manifesto Project (CMP)Comparative Manifesto Project data that makes use of the bootstrap procedure designed by Benoit et al. (2009). We argue that fuzzy preferences are a better representation of the abstract concept of a player’s preferences in public choice models. Instead of representing preferences as precise points, our fuzzy approach maps them as bounded areas in a subset of \({\mathbb {R}}^{k}\). In so doing, we eschew the conventional assumption that political actors have precise policy positions. Instead, fuzzy preferences permit us to conceive of actor’s preferences as vague, but communicated accurately. We conclude the chapter by introducing our basic approach to using fuzzy preferences in fuzzy public choice modelsPublic choice!fuzzy model. We argue that a fuzzy public choice model satisfies some of the intuitive and practical problems faced by the conventional model. Moreover, a fuzzy public choice model allows us to shed the assumption that actors perceive shifts in utility in infinitely precise increments at the same granularity across and infinite policy space.

Fuzzy Single-Dimensional Public Choice Models

We consider the problem of intransitivity in collective preference in the presence of which there is no maximal set upon which to base a prediction of an outcome founded on collective preference. We give special attention to the conditions identified by Black’s Median Voter Theorem that guarantee against intransitivity and assure a maximal set. We then argue that the theorem holds when preferences are fuzzy.

Predicting the Outcome of the Government Formation Process: Fuzzy Single-Dimensional Models

We are now in a position to present a one-dimensional fuzzy public choice modelPublic choice!fuzzy model designed to predict the outcome of the government formation process in parliamentary systems. Such a model allows us to represent flexibility in actors’ preferences and predict when those actors may make allowances for minor policy shifts as well as when they may prefer major policy shifts. This is because the fuzzy public choice model allows for broad areas of indifferenceIndifference in actor’s preference profiles. Moreover, a fuzzy model is more likely to predict stable outcomes by avoiding the intransitivity problem that plagues traditional models. We present two approaches to such a model. The first makes use of the fuzzy maximal set; the second makes use of the fuzzy Pareto set. We test both models using fuzzy preferences derived from the Comparative Manifesto Project (CMP) data.

Predicting the Outcome of the Government Formation Process: A Fuzzy Two-Dimensional Public Choice Model

Under the most basic of assumptions of Euclidean preferences, majority rule erupts into cycling in two or more dimensional space, and no alternative remains undefeated. The resulting McKelvey’s Chaos TheoremMcKelvey’s Chaos theorem (McKelvey 1976) forces scholars to reconsider basic assumptions about the rational behavior of political actors and their attempts to form coalitions. The government formation literature remains divided on how to best solve the problem. More recently, proposed models either assume cabinet ministers are virtual dictators over their policy jurisdiction (Laver and Shepsle 1996) or rely on complex game-theoretic arguments, which do not lend themselves to empirical verification (Baron 1991; Diermeier and Merlo 2000). This chapter builds on the fuzzy maximal set model developed in Chap. 4. It presents a fuzzy maximal set multi-dimensional model to predict the outcome of the government formation process. We conclude by comparing the predictions made by the model using CMP against actual governments formed after European Parliamentary elections between 1945–2002.

Fuzzy Social Choice Models

This book explores the extent to which fuzzy set logic can overcome some of the shortcomings of public choice theory, particularly its inability to provide adequate predictive power in empirical studies. Especially in the case of social preferences, public choice theory has failed to produce the set of alternatives from which collective choices are made. The book presents empirical findings achieved by the authors in their efforts to predict the outcome of government formation processes in European parliamentary and semi-presidential systems. Using data from the Comparative Manifesto Project (CMP), the authors propose a new approach that reinterprets error in the coding of CMP data as ambiguity in the actual political positions of parties on the policy dimensions being coded. The range of this error establishes parties fuzzy preferences. The set of possible outcomes in the process of government formation is then calculated on the basis of both the fuzzy Pareto set and the fuzzy maximal set, and the predictions are compared with those made by two conventional approaches as well as with the government that was actually formed. The comparison shows that, in most cases, the fuzzy approaches outperform their conventional counterparts.

A Fuzzy Public Choice Model

Public choice models can be a powerful tool for the explanation and prediction of political phenomena in the discipline of political science. However, the predictions of such models depend to a considerable extent on whether the preferences of political actors can be properly estimated. Fuzzy mathematics offers one possibility for dealing with this challenge. A fuzzy approach permits us to consider the preferences of individuals as ambiguous, marked by a considerable degree of indifference related to actors’ uncertainty about the exactitude of their ideal points. Moreover, fuzzy sets also permit us to reconsider approaches to making predictions on the basis of those fuzzy preferences. One particularly interesting possibility is offered by a fuzzy maximal set.

Issues in Fuzzy Multi-dimensional Public Choice Models

Conventional multi-dimensional public choice models are notoriously unstable. Under all but the most restrictive assumptions, they fail to produce a maximal set under majority rule. We demonstrate that fuzzy multi-dimensional public choice models offer a wider degree of stability without resort to highly complex mathematical calculations.

A CONSIDERATION OF DIFFERENT DEFINITIONS OF FUZZY COVERING RELATIONS

Fuzzy spatial models map a substantial degree of preference indifference. It has been shown that different definitions of covering result in different elements in the uncovered set when preference indifference is present. We consider several of the most frequently used definitions of covering relations found in the literature. The first definition that we examine yields an uncovered set, some of the elements of which are not Pareto efficient. Given that there is no reason to expect a set of players comprising a majority to settle for a Pareto deficient outcome, the remainder of the paper considers the ability of alternative definitions to avoid such a result.