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 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 set 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).