The spatial model of voting has its roots in work by Hotelling (1929), Black (1958) and Downs (1957). In the time since these early works, spatial models of voting have proven to be extremely useful in understanding a wide variety of social phenomena, including legislative behavior, mass voting, judicial behavior, popular referenda, and inter-institutional bargaining, among others. It has been known for some time that stable outcomes to a voting procedure generally cease to exist when dimension of the issue space is greater than or equal to 2 and the number of voters is 3 or more (Plott 1967, McKelvey 1976, McKelvey 1979, McKelvey and Schofield 1979). The fact that such instabilities do not appear frequently in most deliberative bodies has lead several groups of scholars to reexamine the assumptions underlying the chaos results. Of particular interest to us in this paper is the work on structure induced equilibrium (SIE) (Shepsle 1979, Shepsle and Weingast 1981). The key insight behind this line of research is that instabilities result from the ability of an agenda setter to propose an alternative that shifts policy on several dimensions simultaneously. If, on the other hand, changes to the status quo must be decided one dimension at a time, then a stable outcome will generally exist.
We propose a statistical measurement model that can be used to estimate voter ideal points when (probabilistic) voting is occuring on 1 dimension at a time in a multidimensional issue space. The model we employ turns out to be a finite mixture of item response models in which the mixing probabilities depend on a vector of observed covariates. We illustrate the model with data from U.S. Supreme Court.
