Positive Political Theory II: Strategy and Structure

1. Preliminaries

Chapter 1 Preliminaries Positive Political Theory I: Collective Preference concerns the problem of aggregating individual preferences directly into some sort of collective preference or choice. The current book concerns the problem of aggregating individual preferences indirectly, through individual actions, to arrive at a collective choice. These two perspectives, as we argue in Positive Political Theory I and elaborate in this text, are far from mutually exclusive. Inter alia, much of the analytical apparatus of, and many of the lessons from, the theory of collective preference proves important for the theory of indirect preference aggregation. Before moving on to develop this theory, therefore, it is useful first to review the salient notation, concepts and results from Positive Political Theory I (where a more detailed discussion of the material can be found) and to introduce the necessary decision theory for the chapters to follow. 1.1 Review Let X be a (finite or infinite) set of alternatives and let X be the family of all nonempty subsets of X. A binary preference relation R on the set X describes the relative merits of pairs of outcomes in X: for any two alternatives x, y E X, the statement xRy means that (with respect to R) "x is at least as good as y". Define the asymmetric part of R, the strict preference relation, P, by xPy ("x is strictly better than y") if and only if xRy and'" [yRx], all x, y E X; similarly, define the symmetric part of R, the indifference relation, I, by xIy ("x and yare at least as good as each other") if and only if xRy and yRx, all x, y E X. The relation R is a weak preference order on X if R is reflexive (for all x E X, xRx), complete (for 2 CHAPTER 1. PRELIMINARIES all x, y E X, xRy or yRx or both) and transitive (for all x, y, Z E X, xRy and yRz implies xRz). Let n (respectively, P) denote the set of all weak (respectively, strict) preference orders on X. Suppose the set of feasible alternatives X is a subset of some k-dimensional Euclidean space, X C ~kj assume X is convex (for all x, y E X and t E [0,1]' [tx+(l-t)y] E X) and compact (X is closed and bounded). Fix a preference relation R on X and, for all x EX, define the sets P(x) P-1(x) {y EX: yPx} {y EX: xPy}. Thus P(x) is the set of alternatives that are strictly preferred to x and P-1(x) contains all those alternatives to which x is strictly preferred; these are sometimes referred to as the (strict) upper and lower contour sets, respectively , associated with R. The weak upper and lower contour sets, R(x) and R-1(x) respectively, are defined analogously. A binary relation R is said to be continuous on X if, for all x E X, both P(x) and P-1(x) are open relative to X. A virtue of continuous preference orders on convex and compact sets of alternatives is that they admit continuous numerical representations . A utility function u : X --+ ~ represents R if and only if, for all x,y EX, xRy {:} u(x) ~ u(y). In other words, u represents R if and only if it assigns higher numbers to strictly more preferred alternatives and assigns the same number to any pair of alternatives that are indifferent to each other. As defined, u is an ordinal representation; all that matters is whether, say, u(x) is bigger or smaller than u(y) and their particular cardinal values have no meaning whatsoever. A preference ordering R on a convex set of alternatives X ~ ~k is convex if xRy and x =j:. y imply [tx+(l-t)y]Ry for all t E (0,1) j if [tx+(l-t)y]Py for all x =j:. y and all t E (0,1), we say R is strictly convex. Every continuous and convex preference relation R has a continuous quasi-concave representation, Uj that is, for all x, y E X and all t E (0,1), u(tx + (1 - t)y) 2:: min{u(x), u(y)}. If R is strictly convex, the weak inequality can be replaced by a strict inequality and u is said to be strictly quasi-concave. An important feature of quasi-concave utilities is that, for all r E ~, the upper contour sets...