Abstract
The spatial models presented in the last two chapters used overlapping α-cuts to determine compromises between individuals. In this chapter, we take a closer look at the assumptions implicit in such a method. In particular, we consider ways in which individuals rank alternatives in multiple dimensions. The chapter begins by returning tothe interpretation of fuzzy membership scores. While in earlier chapters the scores on the interval [0,1] represented the degree of inclusion in the set 'excellent policies,' here we propose that membership scores also convey information about the intensity of preferences. Not only might a score of 1 assigned to a policy mean that a person prefers that policy to one assigned a score of .5; the score of 1 might also mean that the person cares more about the policy assigned a 1. Systematically accounting for intensity may be fruitful for formal models, but we leave the issue open and instead use an understanding of intensity to guide our choice of aggregation operators, discussed in the next section. Spatial models are useful because they provide a visual representation of preferences. To derive predictions from a spatial model, we need to know how actors make compromises. If all actors could have policies exactly at their ideal points, spatial models would be simple. Plotting the ideal points of actors would suffice to show the resulting policies. Politics in the real world, however, do not allow all actors to have their way and so require trade-offs and compromise. One way to model compromise is to use straightforward Euclidean distance, so that two actors would be giving up an equal amount if a chosen policy was equidistant from their ideal points. We have spoken at length about how simple Euclidean distance may be an overly restrictive assumption. In this chapter we present a collection of aggregation operators that each offer a different interpretation of trade-offs an actor is willing to make between dimensions and so each result in a slightly different method of modeling compromise. These operators are considered in more detail in the appendix following the chapter, along with other fuzzy operators.