Why the count de Borda cannot beat the Marquis de CondorcetAbstractIntroductionCondorcet and BordaSaari’s objections to the Condorcet proposalArguing for the Borda countReferencesDOI 10.1007/s00355-005-0045-3ORIGINAL PAPERMathias RisseWhy the count de Borda cannot beatthe Marquis de CondorcetReceived: 20 October 2003 / Accepted: 15 April 2004 / Published online: 9 November 2005© Springer-Verlag 2005Abstract Although championed by the Marquis the Condorcet and many others,majority rule has often been rejected as indeterminate, incoherent, or implausible.Majority rule’s arch competitor is the Borda count, proposed by the Count de Borda,and there has long been a dispute between the two approaches. In severalpublications, Donald Saari has recently presented what is arguably the mostvigorous and systematic defense of Borda ever developed, a project Saari hassupplemented with equally vigorous objections to majority rule. In this article Iargue that both Saari’s objections to majority rule and his positive case for the Bordacount fail. I hold the view that defenders of Condorcet cannot muster arguments toconvince supporters of Borda, and vice versa, but here I am only concerned to showthat the Count de Borda cannot beat the Marquis de Condorcet. Saari ’s approachdisplays what I take to be widespread fallacies in reasoning about social choiceworthy of closer analysis. This debate bears on important questions in thephilosophy of social choice theory.1 Introduction1.1 Majority rule is simple if groups choose between two alternatives. The onlycomplication is that there may be a tie. Matters are more complicated if groupschoose among more than two alternatives. Suppose Tom, Dick, and Harry rank A,B, and C as follows: Tom ranks them (A, B, C), Dick (C, A, B), and Harry (B, C, A).(I write “(A, B, ...)” for rankings, and “{A, B, ...}” for sets. By “rankings,” I mean“ordinal rankings,” rankings that do not convey any information about alternativesother than to identify alternatives to which they are preferred—cf. [17] and [8],chapter 1 for more precise treatment.) Suppose they opt to determine a ranking bytaking pairwise majority votes. Yet since A beats B, B beats C, and C beats A, noM. Risse (*)John F. Kennedy School of Government, Harvard University, Cambridge, MA 02138, USAE-mail: [email protected] Choice Welfare 25:95–113 (2005)ranking emerges; instead, we obtain a cycle. This is the Condorcet paradox.Majority rule as sketched is indeterminate: it does not always deliver a result.Arrow’s Impossibility Theorem, in one way of thinking about it, generalizes thisphenomenon, isolating those features of majority rule that imply that we sometimesneed to ascribe preferences to the group that do not constitute a ranking. In light ofthese results, some (such as [4] and [18]) have argued that majoritarian democracy isconceptually flawed, insisting that we do not have a coherent majoritarian way ofmaking decisions if there are more than two alternatives.Either because of these troubles or because they find it independently moreplausible, some support majority rule’s arch-competitor, the Borda count. Suppose agroup must rank m candidates, given that each individual has already ranked them.Borda has each individual assign 0 to her last-ranked candidate, 1 to the second-to-the-last-ranked, until she assigns n− 1 to her top-ranked, and then, for each of thecandidates, sums over those numbers to determine the group ranking. The questionis: should we abandon majoritarian decision making as incoherent or otherwiseimplausible and adopt the Borda Count instead?Donald Saari, for one, thinks we should, and has defended this view forcefullyin recent publications (such as [10, 11, 12, 15, 16]). Saari is one of the mostdistinguished mathematical contributors to voting theory, and his is conceivablythe most vigorous and sophisticated defense of Borda ever undertaken. Much is atstake. If there is an overwhelming case for Borda, then, whenever we are usinganother method for aggregating preferences, we make some people “losers” thoughthey would have been “winners” had the most defensible method been used, andmutatis mutandis for majority rule. At the same time, if there is no case cham-pioning one method over others, there will always be people who are “losers”though they would have been “winners” had another, equally reasonable methodbeen used. This debate is old: the Marquis de Condorcet and the Count de Borda,French noblemen in troubled times, debated these questions already in the late 18thcentury, a golden age of reflection on group decision making. As far as legitimacyof collective decisions is concerned, Saari’s defense of Borda, if successful, wouldconstitute a great insight.1.2 However, Saari’s defense fails, and my goal is to show why. In earlier work([7]) I have defended a conception of majoritarian decision making that de-monstrates that, contrary to the kind of criticism mentioned in the very firstparagraph, there is indeed a coherent majoritarian decision rule that solves theindeterminacy problem. Since my proposal (which qua aggregation mechanismhad already been discussed elsewhere in the literature) bears affinities to ideas ofCondorcet, I call it the Condorcet Proposal (and here refer to it simply as “ theProposal”). I have also argued there for the multiplicity thesis, the claim that in thesame situation different methods may be reasonable. I hold this view especially forpreference aggregation: it is indeed the case that in such scenarios, some peoplewill be “losers” although they would not have been had another, equally reasonabledecision rule been adopted. So while majoritarian decision making is conceptuallysound, it never is the uniquely reasonable method. Saari, as far as I can tell, doesnot think that majoritarian decision making as captured by the Proposal is in-coherent. He does, however, think that it has several features that render it deeplyimplausible, and that, at any rate, the Borda Count can be derived from ideas sobasic and compelling that even prima-facie supporters of the Proposal have toendorse them.96 M. RisseMy goal is to refute that view. The Count cannot beat the Marquis: defenders ofthe Borda Count have no arguments that should make supporters of the Proposalchange their minds. The Marquis cannot beat the Count either, but that aspect of thedebate is not my concern here. Even the first direction (“the