Aggregating sets of judgments: two impossibility results compared

Abstract

May’s celebrated theorem (1952) shows that, if a group of individuals wants to make a choice between two alternatives (say x and y), then majority voting is the unique decision procedure satisfying a set of attractive minimal conditions. The conditions are (i) universal domain: the decision procedure should produce an outcome (x, y or tie) for any logically possible combination of individual votes for x and y; (ii) anonymity: the collective choice should be invariant under permutations of the individual votes, i.e. all individual votes should have equal weight; (iii) neutrality: if the individual votes for x and y are swapped, then the outcome should be swapped in the same way, i.e. the labels of the alternatives should not matter; (iv) positive responsiveness: supposing all other votes remain the same, if one individual changes his or her vote in favour of a winning alternative, then this alternative should remain the outcome; if there was previously a tie, a change of one individual vote should break the tie in the direction of that change. May’s theorem is often interpreted as a vindication of majoritarian democracy when a collective decision between two alternatives is to be made. Many collective decision problems are, however, more complex. They may not be confined to a binary choice between two alternatives, or between the acceptance or rejection of a single proposition. Suppose there are three or more alternatives (say x, y and z). In that case, it may seem natural to determine an overall collective preference ranking of these alternatives by applying majority voting to each pair of alternatives. But, unfortunately, pairwise majority voting may lead to cyclical collective preferences. Suppose person 1 prefers x to y to z, person 2 prefers y to z to x, and person 3 prefers z to x to y. Then there are majorities of two out of three for x against y, for y against z, and for z against x, a cycle. This is Condorcet's paradox. But a greater number of alternatives is not the only way in which a collective decision problem may deviate from the single binary choice framework of May's theorem. A collective decision problem may also involve simultaneous decisions on the acceptance or rejection of multiple interconnected propositions. For instance, a policy package or a legal decision may consist of multiple propositions which mutually constrain each other. To ensure consistency, the acceptance or rejection of some of these propositions may constrain the acceptance or rejection of others. Once again, a natural suggestion would be to apply majority voting to each proposition separately. As we will see in detail below, however, this method also generates a paradox, sometimes called the 'doctrinal paradox' or 'discursive dilemma': propositionwise majority voting over multiple interconnected propositions may lead to inconsistent collective sets of judgments on these propositions. We have thus identified two dimensions along which a collective decision problem may deviate from the single binary choice framework of May’s theorem: (a) the number of alternatives, and (b) the number of interconnected propositions on which simultaneous decisions are to be made. Deviations along each of these dimensions lead to a breakdown of the attractive properties of majority voting highlighted by May's theorem. Deviations along dimension (a) can generate Condorcet's paradox: pairwise majority voting over multiple alternatives may lead to cyclical collective preferences. And deviations along dimension (b) can generate the 'doctrinal paradox' or 'discursive dilemma': propositionwise majority voting over multiple interconnected propositions may lead to inconsistent collective sets of judgments on these propositions. In each case, we can ask whether the paradox is just an artefact of majority voting in special contrived circumstances, or whether it actually illustrates a more general problem. Arrow's impossibility theorem (1951/1963) famously affirms the latter for dimension (a): Condorcet's paradox brings to the surface a more general impossibility problem of collective decision making between three or more alternatives. But Arrow's theorem does not apply straightforwardly to the case of dimension (b). List and Pettit (2001) have shown that the 'doctrinal paradox' or 'discursive dilemma' also illustrates a more general impossibility problem, this time regarding simultaneous collective decisions on multiple interconnected propositions. The two impossibility theorems are related, but not identical. Arrow's result makes it less surprising to find that an impossibility problem pertains to the latter decision problem too, and yet the two theorems are not trivial corollaries of each other. The aim of this paper is to compare these two impossibility results and to explore their connections and dissimilarities. Sections 2 and 3 briefly introduce, respectively, Arrow's theorem and the new theorem on the aggregation of sets of judgments. Section 4 addresses the question of whether the two generalizations of May's single binary choice framework -- the framework of preferences over three or more options and the framework of sets of judgments over multiple connected propositions -- can somehow be mapped into each other. Reinterpreting preferences as ranking judgments, section 5 derives a simple impossibility theorem on the aggregation of preferences from the theorem on the aggregation of sets of judgments, and compares the result with Arrow's theorem. A formal proof of the result is given in an appendix. Section 6 discusses escape-routes from the two impossibility results, and indicates their parallels. Section 7, finally, explores the role of two crucial conditions underlying the two impossibility theorems -- independence of irrelevant alternatives and systematicity --, and identifies a unifying mechanism generating both impossibility problems.