Mathematical utility functions make the idea of preference portable and generalizable. If the “utility function” happens to have nice properties, including differentiability, then calculus can be brought to bear on many problems describing responsiveness and rates of change, and a wide variety of intuitive results can be discussed and debated.

Economic theory gave birth to the basic spatial model of politics. The problem, as we noted in Chapter 4, is that economic utility functions cannot be used for public goods and collective decisions. The idea of a spatial utility function to “represent” political preferences was laid out by Black (1948), Black and Newing (1951), Downs (1957), and Black (1958). However, the modern spatial theory of political competition betrays very little of its mitochondrial DNA from economics. Spatial theory has become a stand-alone tool for representing political preferences, analyzing competition, and predicting outcomes.

The tools we have developed in the previous section, based primarily on weak orderings, are useful if the number of alternatives is small and the choosers are in a “committee” setting. It is now time to develop tools more suitable for analyzing elections or complex committee votes, such as budgets. These are votes where the alternative space is of very large dimension and at least some dimensions are measured as continuous numbers.