Positive Political Theory II: Strategy and Structure

10. Summary and Conclusions

Chapter 10 Summary and Conclusions A central task for political science is to understand how the preferences of individuals comprising a society map into the collective choices of that society. The development of such an understanding constitutes the organizing principle for this volume and its precursor, Positive Political Theory I: Collective Preference. In the Preface to the first volume, we distinguished two approaches to thinking about how preferences are aggregated to arrive at collective decisions . The direct (collective preference) approach is motivated by an essentially decision-theoretic methodology: individual preferences are directly aggregated into a collective preference which, as in individual decision theory, is maximized to yield a set of best (relative to the maximand) alternatives, the collective choices. The indirect (game-theoretic) approach recognizes that the preferences individuals choose to reveal need not coincide with the preferences with which they are endowed: an individual may have a clear ranking of candidates for electoral office, for example, yet choose to vote strategically or to abstain. Thus, while individual preferences surely influence individual decisions and actions, it is individual actions (in particular for our purposes, voting and agenda-selection), not individual preferences per se, that are aggregated to arrive at collective choices. Formally, let nn be the usual domain of individual preference profiles over a set of alternatives X for a society N = {I, ... ,n}. Let rp : nn ~ X be a collective choice rule; for any profile pEnn, rp(p) ~ X denotes the outcomes chosen by society. Now define a preference aggregation rule f : nn --+ B, where B is the set of all complete binary relations over X [PPTI, ch.2]' and define the mapping m : B ~ X such that, for all pEnn, m(f(p)) = {x EX: V'y E X, xf(p)y} 419 420 CHAPTER 10. SUMMARY AND CONCLUSIONS is the maximal set of alternatives in X with respect to the binary relation j(p) E B, the core of j at p. Then the direct approach to preference aggregation is summarized as the analysis of the collective choice rule defined by the composition ofm and j, cp == mof. The indirect approach, however, begins with a description of the actions, or pure strategies, Mi an individual i E N can take and a rule g : M -> X that associates an alternative g(s) E X with every feasible profile of actions s E M == TIiENMi. A pair (M,g) is a mechanism. An abstract theory of how individuals choose their respective actions under a given mechanism is a mapping f3 : nn =l M where, for any pEnn, f3(p) ~ M is the set of action profiles consistent with the theory f3 at preference profile p. Defining the composition of g and f3 by g(f3(p)) = {x EX: x = g(s) for some s E f3(p)}, all pEnn, the indirect approach to preference aggregation is then summarized as the analysis of the collective choice rule cp == go f3. Figure 10.1 depicts the various links and suggests, rather than being mutually exclusive, that the direct and indirect approaches are complementary. Figure 10.1: Linking preferences to collective choices Positive Political Theory I explores the direct approach to preference aggregation. In that volume we argue that a key distinction between the collective preference and game-theoretic tacks lies in an implicit tradeoff each makes between the general existence of well-defined collective choices and a mild normative condition, minimal democracy, which requires that no individual i can unilaterally veto an alternative x in favor of an alternative y if all individuals other than i strictly prefer x to y. While insuring general existence proves elusive for collective preference theory (the set m(f(p)) is typically empty under all but the most constrained conditions) but largely unproblematic for game theory (the set g(f3(p)) is typically nonempty in all 10.1. RETROSPECTIVE 421 but the most unstructured settings), the reverse is true in regard to collective choices that respect minimal democracy. Indeed, by [PPTI, Corollary 7.1]' there exists no approach to preference aggregation that, in the absence of additional restrictions on the domain of application, is consistent with insuring the existence of well-defined collective choices that invariably satisfy minimal democracy. Collective preference theory insists on minimal democracy and the argument of Positive Political Theory I is that, as a consequence of such insistence, a general theory of political behavior analogous to the decisiontheoretic model of maximizing individual choice is unavailable. This conclusion both informs and...