The Paradox of Voting

Speaking about Obamacare, MIT economics professor Jonathan Gruber said, “We’ve decided as a society that we don’t want people to have insurance plans that expose them to more than six thousand dollars in out-of-pocket expenses.”
What does it mean that “we” decide something “as a society”? It’s an important question: This sort of statement gets used frequently as a justification for government intervention. When, in the same fashion, Obama says “we as a nation,” he is just using a variation of the same expression and talking like the average politician.
“We as a society” or “we as a nation” is generally used as an incantation with no scientific meaning. If it has any ascertainable meaning, it means “we who want to impose our current and perhaps changing whims on others.”
The simplest interpretation of “we as a society” is that it represents what a majority votes for. It would simply mean, we as a majority of 51 percent (or 60 percent, or 30 percent if we are talking of a mere plurality). But how is the majority representative of society? What tells us that another majority wouldn’t vote differently if the issues were presented differently? Whose preferences exactly does the majority represent?

That Median Voter

In certain cases, the majority represents the preferences of a small group of voters, perhaps a single voter. The “median-voter theorem” shows that if you have one voter (or one group of voters), whose preferences are exactly in the middle of the distribution of preferences, he will win elections. 
For example, if the median voter prefers public expenditures to be $3 trillion, no politician can win an election against one who runs on this proposal. Any politician who proposes to spend more or less will lose more than 50 percent of the electorate to the one who stands exactly in the center. The median voter theorem explains why a successful politician has to “hug the center,” as The Economist puts it to explain the recent gubernatorial elections.

Preference Aggregation

When, however, the electorate is polarized around two opposing stances, the median voter theorem does not apply. More diverse individual preferences, and a more diverse society, weaken the median voter’s power. What happens in this case? Who is the majority? How does it behave?
These issues fall under the label of “preference aggregation,” within a field of inquiry called social choice. The broad question is, how can the preferences of voters—or, more generally, of individuals in society—be aggregated to produce social choices?
A little intellectual voyage will help us answer this question.
First, meet Jean-Antoine-Nicolas de Caritat, marquis de Condorcet (1743–1794). Condorcet was a French mathematician, philosopher, and classical liberal. Like many politicians, he became cross with the French authorities under the Terror (the nastier phase of the French revolution), was arrested on March 27, 1794, and died in jail a few days later.
His death, however, had nothing to do with his 1785 book, Essay on the Application of Probability Analysis to Decisions Made with a Plurality of Votes—except perhaps to the extent that he was not an intellectual yes man. Condorcet was the first one to clearly isolate a strange phenomenon that came to be known as the “paradox of voting”: even if each voter is rational, the result of a vote can be irrational.
“Rational” in this context simply means consistent or transitive preferences: If you prefer X to Y, and Y to Z, you will also prefer X to Z. The Condorcet paradox says that even with rational electors, a majority that prefers X to Y and Y to Z can prefer Z to X.
An example will make this easier to grasp. Suppose the issue is whether the president should have more power over the budget (compared to Congress), less power, or the same degree of power as now. Let P represent the status quo, P- mean less power to the president, and P+ more power. Now consider an electorate composed of three voters: Alice, Bob, and Charlie. Suppose that Alice prefers P- to P to P+, which we can write as P->P>P+. We use symbols to economize on words: “>”simply means “preferred to.” Like all other voters, Alice is rational, which implies that she also prefers P- to P+. Assume that Bob’s preferences are P>P+>P-. As for Charlie, his preferences are represented by P+>P->P. Bob and Charlie are also supposed to have transitive preferences.
It is easy to check that if our voters are asked to vote between P- and P, the majority (Alice and Charlie) will choose P-. If the electorate votes between P and P+, the majority (Alice and Bob) will choose P. Since the electorate prefers P- to P, and P to P+, you would think that it would prefer P- to P+ if presented with these two alternatives. But no! You can check that P+ would win over P- with a majority of votes (Bob and Charlie). The electorate is irrational even if each voter is rational.
Other preference orderings will produce a rational electoral choice. But the example shows that the paradox of voting can appear. “We as a society” is more a casino roulette than a rational actor.

Cyclical Majorities

This theory explains many observable phenomena. It explains the inconsistencies we often find in public opinion surveys. It may explain why voters vote both for job creation programs and for minimum wages that destroy jobs. It explains the votes on the Muscle Shoals hydroelectric project in the U.S. senate in 1925. Over less than a week in January of that year, and without any senator changing his mind, the U.S. senate voted to refer the alternatives to a study commission instead of allowing private development, then for private development instead of public ownership, and then again for public ownership instead of a study commission (see John N. Neufeld et al., “A Paradox of Voting: Cyclical Majorities and the Case of Muscle Shoals,” Political Research Quarterly, vol. 47, no. 2, 1994).
This is another example of the paradox of voting, also called “cyclical majorities.” Voters—U.S. senators in this case—cycle between issues without being able to reach a definitive decision.
Mathematician Charles L. Dodgson (1832–1898) rediscovered the phenomenon of cycling a hundred years after Condorcet. Dodgson was also known as Lewis Carroll, author of Alice in Wonderland and other literary works. That such a creative spirit as Dodgson worked on cycling lends more credence to the importance of the topic.
Our intellectual voyage now takes us to Duncan Black (1908–1991), a Scottish economist who rediscovered the paradox in the mid-twentieth century. When a numerical example he was working on showed an irrational electorate made of rational voters, Black was deeply disturbed: “On finding that the arithmetic was correct and the intransitivity persisted,” he later explained, “my stomach revolted in something akin to physical sickness.” He had to admit that his prior intuition—that rational voters produce a rational electorate—was disturbingly wrong.
The final destination in our voyage is Kenneth Arrow, a Stanford University economist who extended the opportunity for nausea to all economists and political scientists who study the issue. In his 1951 book, Social Choice and Individual Values, Arrow mathematically demonstrated that the discovery of Condorcet, Dodgson, and Black was only a special case of a more general theorem: Whatever the decision mechanism used, a social choice cannot be both democratic and rational. If all individual preferences are to count equally (and given a few other axioms), a social choice must be either irrational or imposed by some on others. For his work, Arrow (along with with John Hicks) won the 1972 Nobel Prize in economics.
The political implications are striking. Saying “we as a society” means one of two things: “We who agree with the choice imposed on others,” or, “We are irrational in this choice, and could as well have chosen something else.” In other words, “we as a society” does not really exist, except perhaps with respect to a few fundamental values on which unanimity obtains.