In Chapter 3, we outlined the logic of complex policy spaces in terms of “everyday thinking.” In this chapter, we will refine that material, using the language of matrix notation. Instead of two dimensions, however, we will assume that the policy space has n dimensions, where n is an arbitrary number. Technically, the policy space p is the generalized Cartesian product: p = P1 × P2 × … × Pn.

As before, each dimension measures the budget of one project, and members are assumed to have preferences over all the projects. Using matrix notation will allow us to make precise the ideas of salience and nonseparability that we presented graphically in Chapter 3. Mathematical notation will afford us several refinements that make the additional complexity worthwhile. Matrices and vectors are useful notational devices because they save space and clutter in representing organized arrays of characters. Once you get used to the idea of using matrix notation, you will find it easier than using summations cluttered with indexes and brackets.

Matrix notation and definitions

A matrix is an ordered array of numbers or characters. We will use boldface type to distinguish a matrix from scalar variables. A scalar might be thought of as a 1 × 1 matrix, but there is no need to be so fussy.