Welcome to the third installment in our series of discussions of the Most Insightful Articles in economics. Today we are discussing Ken Arrows’s 1950 article A Difficulty in the Concept of Social Welfare.

If you’re interested in politics, you may have done the following thought experiment. Suppose there are three voters—1, 2, and 3—and three alternatives—A, B, and C. Voter 1 prefers A to B to C. Voter 2 prefers B to C to A. Voter 3 prefers C to A to B. By a vote of 2-1, “society” prefers A to B. It also prefers B to C. If a rational person prefers A to B and B to C, then that person prefers A to C. But in this example, “society” prefers C to A! Is society irrational? Is this just a problem with majority rule? To cut to the chase, in this paper Arrow shows that it is a general problem. Any method of aggregating individual preferences is either irrational or has some other properties that are arguably undesirable.

One can envision collective decision-making as selecting a social welfare function and then using it to compare states of the world. Arrow defines five conditions to which a social welfare function ought to hold:

1. The function is universally defined. There are not certain preferences that people can have that cause the function to be indeterminate (though the function is allowed to express indifference).
2. If someone decides that X is more desirable than he initially thought, and nothing else changes, the social welfare function cannot penalize X.
3. The function should provide the same ranking for a subset of options as it would for that subset within a complete set of options. Adding or removing an option should not affect the ranking of other options.
4. No preferences are taboo. Any social preference ordering is possible if people have the right individual preferences.
5. The function must be collective in the sense that it does not simply mimic one person’s preferences.

Seems reasonable, no? Arrow’s proof proceeds by contradiction. He assumes that there are two individuals, three alternatives, and a social welfare function that satisfies the above conditions. These assumptions lead to three consequences:

1. If both individuals prefer one outcome to another, then the social welfare function prefers it as well.
2. “[I]f in a given choice, the will of individual 1 prevails against the will of individual 2, then individual 1′s views will certainly prevail if 2 is indifferent or if he agrees with 1.”
3. If individual 1 prefers X to Y and individual 2 prefers Y to X, then the social welfare function must be indifferent between X and Y.

Now, suppose that individual 1 prefers X to Y to Z and individual 2 prefers Z to X to Y. By consequence 1, the social welfare function prefers X to Y. By consequence 3, the function is indifferent between Y and Z. Since the function prefers X to Y and is indifferent between Y and Z, the function must prefer X to Z. However, also by consequence 3, the function must be indifferent between X and Z. This is a contradiction, which means that one of our assumptions cannot hold. In Arrow’s words:

If we exclude the possibility of interpersonal comparisons of utility, then the only methods of passing from individual tastes to social preferences which will be satisfactory and which will be defined for a wide range of sets of individual orderings are either imposed or dictatorial.

What does this mean? There is no “will of the people,” at least not if that phrase is to encompass what we normally think of as rationality. Rousseau is incoherent. Democracy can be manipulated by agenda control, as Levine and Plott show in a hilarious article in Virginia Law Review.

How can we get around this conclusion? One way is if preferences are unidimensional and single-peaked. If all voters think in terms of a single liberal-to-conservative spectrum and prefer candidates that are closer to some optimum point, then you can no longer demonstrate the contradiction. Another way is if you allow some way of expressing intensity of values (interpersonal comparisons of utility), such as by bidding with dollars, though this may undermine some of Arrow’s conditions.

For discussion: Voter preferences are shockingly unidimensional, are they not? People who favor high taxes tend also to be pro-choice on abortion. What do these have to do with each other? Does this correspondence save democracy from the charge of irrationality? Or is it further evidence of it? Does Arrow’s result explain why everyone is so dissatisfied with the government basically all of the time? In light of Arrow’s result, are some voting systems better than others? Does the fact that collective choice is incoherent mean that collective morality is also incoherent?

Next time, we will discuss Ronald Coase’s 1960 paper The Problem of Social Cost. If Arrow had read Coase, he would not have made the errors that he made on p. 334. See you in the comments!