Fuzzy Spatial Models
Most political issues are more than one-dimensional in scope. For example, budget bills contain funding across a number of issue areas, and political parties engaged in cabinet formation must concern themselves with several issue dimensions in determining
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With Peter C. Casey, Christopher W. Cook, Meghan C. Freeman, Alan J. Hickle, Danae D. Mercer, Eric M. Ruchensky, and Raivydas Simenas Most political issues are more than one-dimensional in scope. For example, budget bills contain funding across a number of issue areas, and political parties engaged in cabinet formation must concern themselves with several issue dimensions in determining a government program capable of uniting a legislative majority. Hence, spatial models must incorporate n > 1 dimensions to be useful. Unfortunately, increasing the space to include multiple dimensions opens the possibility of cycling. The conditions under which cycling can occur have been the object of study for decades, and the conclusion is that the possibility of cycling is pervasive. Plott (1967), for instance, found that a maximal set1 exists in two-dimensional space only when ideal points are arrayed symmetrically to one another (the radial symmetry condition); and McKelvey (1976) found that in the absence of a maximal set, cycling is possible over the entire two-dimensional space. In order to reduce the likelihood of cycling, scholars have adopted increasingly restrictive assumptions in their models. While this has permitted the models to predict outcomes, increasingly restrictive assumptions remove the models further from reality, and empirical tests have often falsified the predictions. As a consequence, formal models have come under increasing criticism for the gap between their predictions and their empirical implications. The empirical implications in theoretical models (EITM) movement is one reflection of these criticisms (see Achen et al., 2002; De Marchi, 2005). In what follows we begin by discussing the extent of the cycling problem in two-dimensional spatial modeling. Next we consider an example of how scholars resolve the cycling problem in crisp analysis. We then demonstrate fuzzy set theory’s ability to reduce the cycling problem and strengthen the ability of models to predict more empirically sound outcomes. The demonstration relies partly on the fuzzification of one of the more interesting spatial models in The 1
A maximal set in two-dimensional formal models is most oftent referred to as a core. A core in this sense is not the same as the core of a fuzzy number. To avoid confusion, we will use the term maximal set when referring to a core in formal modeling.
T.D. Clark et al.: Applying Fuzzy Math. to Formal Models, STUDFUZZ 225, pp. 109–135, 2008. c Springer-Verlag Berlin Heidelberg 2008 springerlink.com
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Fuzzy Spatial Models
New Institutionalism’s repertoire, Laver and Shepsle’s (1996) portfolio allocation model. Our general argument is that fuzzy theory offers a more elegant approach to formal modeling. As was the case with one-dimensional models, a fuzzy approach jettisons two common assumptions of spatial models: 1) the assumption of a single, ideal point and 2) the assumption that preferences can be thought of as a function of Euclidean distance from that ideal point. These assumptions impose a set
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