This possibility, called the Condorcet paradox, was identified in the late 18th century by Marie-Jean-Antoine-Nicholas de Caritat, the Marquis de Condorcet, a colleague and archcritic of Borda. The three rankings—Gore over Bush over Nader, Bush over Nader over Gore, and Nader over Gore over Bush—are collectively called a Condorcet cycle. Our comparison of majority rule and rank-order voting appears to have resulted in a dead heat: majority rule satisfies every principle on our list except transitivity, and rank-order voting satisfies all but neutrality. This conundrum leads us to consider whether some other electoral system exists that satisfies all the principles. Arrow’s celebrated impossibility theorem says no. It holds that any electoral method must sometimes violate at least one principle [see “Rational Collective Choice,” by Douglas H. Blair and Robert A. Pollak; Scientific American, August 1983].
BUT ARROW’S THEOREM is unduly negative. It requires that an electoral method must satisfy a given axiom, no matter what voters’ rankings turn out to be. Yet some rankings are quite unlikely. In particular, the Condorcet paradox—the bugaboo of majority rule—may not always be a serious problem in practice. After all, voters’ rankings do not come out of thin air. They often derive from ideology.
To see what implications ideology holds for majority rule, think about each candidate’s position on a spectrum ranging from the political left to the right. If we move from left to right, we presumably encounter the 2000 presidential candidates in the order Nader, Gore, Bush, Buchanan. And if ideology drives voters’ views, then any voter who ranks Nader above Gore is likely to rank Gore above Bush and Bush above Buchanan. Similarly, any voter who ranks Bush above Gore can be anticipated to rank Gore above Nader. We would not expect to find a voter with the ranking Bush, Nader, Gore, Buchanan.
In a pioneering paper published in the 1940s, the late Duncan Black of the University College of North Wales showed that if voters’ rankings are ideologically driven in the above manner— or at least if there are not too many nonideological voters— majority rule will satisfy transitivity. This discovery made possible a great deal of work in political science because, by positing ideological rankings of candidates on the part of voters, researchers could circumvent the Condorcet paradox and make clear predictions about the outcome of majority rule.
Of course, voters may not always conform to such a tidy leftright spectrum. But other situations also ensure transitivity. For another example, look again at the French election. Although Chirac and Jospin led the two major parties, it seems fair to say that they did not inspire much passion. It was the extremist candidate, Le Pen, who aroused people’s repugnance or enthusiasm: evidence suggests that a huge majority of voters ranked him third or first among the three top candidates; few ranked him second. One can argue about whether such polarization is good or bad for France. But it is unquestionably good for majority rule. If voters agree that one candidate of three is not ranked second, transitivity is guaranteed. This property, called value restriction, was introduced in 1966 by Amartya Sen of Harvard University.
In our research on voting, we say that a voting system works well for a particular class of rankings if it satisfies the four axioms when all voters’ rankings belong to that class. For instance, majority rule works well when all rankings are ideologically driven. It also works well when all rankings are “value restricted.” Indeed, we have found that whenever any voting system works well, so does majority rule. Furthermore, majority rule works well in some cases in which other systems do not. We call this the majority dominance theorem. To illustrate, we will imagine a three-way race between Gore, Bush and Nader. Suppose that every voter in fact ranks the candidates as either Gore, Bush, Nader or Bush, Gore, Nader.