Applying Fuzzy Mathematics to Formal Modeling in Comparative Politics

In spite of substantial criticism (e. g., Shapiro & Green 1994), a great number of spatial models in comparative politics are grounded in the rational choice assumption that political actors are self-interested, utility maximizers. As a consequence, an individual’s preference order over a set of alternatives is transitive. A further assumption commonly adopted by these models is that political actors possess perfect information; they know their own preferences as well as that of others. Hence, an individual’s preference order possesses the additional characteristic of completeness. Given any set of choices, she can state de…nitively whether she prefers one to the other or she is indi¤erent between them. In application, however, the latter possibility is often minimized or altogether ignored. In e¤ect, a crisp set assumption is imposed. An option in relation to another is either strictly preferred or not preferred by a political actor. Only those points lying on a circle equi-distant from an individual’s ideal point (at the center of the circle) are equally preferred. Points further from the circle, no matter how in…nitesimally small the distance, are less preferred, and those within the circle are more preferred. This reduction in ambiguity signi…cantly e¤ects conclusions derived from the models. Were there substantial congruence between fact and theory, criticism of the assumptions behind spatial models might be more easily dismissed. However, the persistent disconnect between the predictions made by these models and political reality, to which the empirical implications of theoretical models (EITM) movement has emerged in (see Achen, et.al. 2002, de Marchi 2005), argues that the criticism should be taken seriously. The most well-known incongruence is the majority cycling problem, a problem that is particularly problematic for spatial models. McKelvey (1976) found that in the absence of Plott’s (1967) radial symmetry, outcomes of majority vote games can lead anywhere. In other words, there is no a set of majority preferred outcomes, or maximal setM(R;X). Formally, M(R;X) = fx 2 Xj8y 2 X;xRyg (Austen-Smith & Banks 1999, p. 4), where R is a binary relationship, X is the set containing all alteratives over which political actors are making binary choices, and R denotes weak preference (i.e., x is at least as good as y). If there is only one alternative in the maximal set, then it is the Condorcet winner. In the absence of a maximal set, there is no equilibrium, majority cycling results, and formal models in comparative politics cannot predict outcomes. Rational choice formal modelers have taken two approaches to address the gap between theory and empirical evidence. The …rst is to study the e¤ect of di¤erent assumptions about the voting behavior of political actors. The most promising of these is encompassed in a research agenda that is focused on the uncovered set (see McKelvey 1986, Miller 1980, Miller 2007, and Shepsle & Weingast 1984). While the uncovered set reduces the outcome to a region in two-dimensional space, its shape and location are non-intuitive and cumbersome to calculate. A second, more elegant approach is to consider the e¤ect of varying institutional designs. Kenneth A. Shepsle is among the pioneers of this approach, which most often takes the form of considering institutional rules that induce outcomes at the dimensionby-dimension median (the intersection of the medians on each dimension). His groundbreaking research on the U.S. Congress argues that work on legislative bills is divided among committees and that the germaneness rule constrains committees to deal only with