“A ‘social welfare function’ representing a set of ‘social indifference curves’ belongs not to ‘we’ but to whoever writes down, or imagines, the preferences of ‘society.'”

One can barely read a newspaper or listen to a politician’s speech without hearing the standard “we as a society” or its derivatives. “You know, we’re going to have to make some choices as a society,” said President Barack Obama about NSA surveillance. Even some economists, who should know better, fall into this trap. Jonathan Gruber, an MIT economist and an architect of Obamacare, declared: “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.”

The same problem exists in expressions such as “from society’s point of view” or “society as a whole,” and in the personification of countries, as in “America thinks, or does, this or that.” When an Afghan deputy minister for social affairs said that “Afghanistan is an Islamic country and we want our children to be raised in an Islamic way,” he was making the same sort of blanket statement, just draped in a different flag.

The truth is that this collective “we” has no scientific meaning.

The collective “we” can make sense when it refers to a contractual group (“We at Ford produce great cars”) or to a constructed or imagined group with which one wants to identify. But for a whole society made of different individuals with different preferences, if all individuals are to count equally, the collective “we” makes no sense.

In the history of economic thought, two main strands of analysis support this conclusion. One was meant to criticize economists’ use of “social indifference curves” (also called “community indifference curves”), which generally appeared in the welfare analysis of international trade between personalized trading countries. Countries were personalized in the sense that they were assumed to have preferences, just as an individual does. In a famous 1956 article, Paul Samuelson definitively demonstrated that such social indifference curves, analogous to individual indifference curves, do not exist.

The argument runs like this. Assume a society with two individuals in which only two goods are produced. If social indifference curves existed analogously to individual indifference curves, one could derive from them a social demand curve for the two goods as a function of their prices. (The only use of indifference curves is to derive demand curves, at least conceptually.) Now, assume that the relative prices of the two goods change. Keeping “social utility” constant, society would move along one indifference curve in order to reach a new equilibrium—buying less of one good and more of the other. If the two individuals have different preferences for the two goods, their real incomes change, and so does their demand. This, in turn, induces a change in relative prices, which means that society is not really on its starting indifference curve. (Students of economics will recall that, in equilibrium, the slope of an indifference curve must be equal to the ratio of prices.)

Put differently, at every point in commodity space, there are many social indifference curves, each corresponding to a different income distribution. Therefore, there is no way to construct a social demand curve from these unstable social indifference curves. Thus, they are not really social indifference curves. As welfare economists would come to realize, a “social welfare function” representing a set of “social indifference curves” belongs not to “we” but to whoever writes down, or imagines, the preferences of “society.”

Strangely, some economists continue using social indifference curves, but they often feel obliged to qualify their use with footnotes that, if written and interpreted properly, invalidate their analysis.

A second strand of analysis leads to a similar but more general conclusion. The problem has come to be known as the “preference aggregation” issue: how can we aggregate—”add up” as it were—individual preferences? Can we fuse them into social preferences—a “social welfare function”—that would equally represent all individuals? This second tradition of analysis follows a long and broken line of theorists. The Marquis de Condorcet in the 18th century, Charles Dodgson (a.k.a. Lewis Carroll) in the 19th, and economist Duncan Black in the 20th all discovered independently that majority voting does not provide an acceptable aggregation mechanism.

Consider the following illustration. Assume an electorate of three voters—Alice, Bob, and Charlie—and three political issues or candidates, P, Q, and R. Suppose that Alice prefers P to Q to R, which we can write P>Q>R. Alice is rational in the sense that her preferences are consistent or transitive, which means that she also prefers P to R (written P>R). Assume that Bob’s preferences are Q>R>P and that Charlie’s are R>P>Q. Bob’s and Charlie’s preferences are also transitive. It is easy to verify that if our voters are asked to choose between P and Q, the majority (Alice and Charlie) will choose P. If the electorate votes for either Q or R, the majority (Alice and Bob) will choose Q. If the electorate has transitive preferences, it will choose P over R. But, in fact, if this same electorate is presented with the alternative P or R, the majority of votes (Bob and Charlie) will go to R. The electorate is irrational even if each voter is rational!

This result is known as the phenomenon of cyclical majorities, or cycling. In one election, the electorate would choose P over Q, and in another, Q over R; but then, if there is a third election, the voters choose R over P. Intuitively, what happens is that different majorities prevail on each pair of issues because the voters use different ranking criteria. With voters’ preferences different than those assumed in our example, the reader can verify that cycling need not occur. In the real world, the more issues the voters face and the more heterogeneous their preferences are, the more likely cycling is to occur. At the other end of the probability spectrum, the more similar are individual preferences on a small set of issues, the more intransitivity becomes an exception. But the problem is lurking and cannot be assumed away.

When Duncan Black independently rediscovered cycling in 1942, he was deeply troubled: “On finding that the arithmetic was correct and the intransitivity persisted,” he later explained, “my stomach revolted in something akin to physical sickness.”

Cyclical majorities may often explain why public opinion appears inconsistent—for example, when voters support both measures that destroy jobs, such as minimum wages, and policies aimed at creating jobs. Smaller assemblies are also subject to cycles. One well-documented case occurred in 1925 in the U.S. Senate. Over less than a week in January of that year, and with only one senator being inconsistent (or changing his mind), the U.S. Senate voted to support private development of the Muscle Shoals hydroelectric project instead of government ownership; then, to refer the problem to a study commission instead of allowing private development; and, finally—and inconsistently—to have government ownership instead of a study commission.

Who makes up the political “we”? It is, at best, a majority, but this majority or plurality changes and contradicts itself, even if each voter has transitive preferences and doesn’t change his mind. As an analyst puts it, “[F]or any given alternative, there is always another alternative that is more preferred.” The only way to be sure that the paradox of voting does not materialize is if voters have the same preferences or, at least, rank alternatives according to the same criterion. In this last case, “we” is really only one voter—or one class of voters: the so-called median voter, who stands in the middle of the distribution of preferences.

The last major step in the second strand of analysis came when Kenneth Arrow generalized the paradox of cycling and showed that, under reasonable conditions, individual preferences cannot be aggregated without either producing inconsistencies or imposing some individuals’ preferences on others. This impossibility theorem (also called the Arrow Theorem) applies to any political choice and to any aggregation of individual preferences (including by the market). In 1972, Arrow earned a Nobel Prize, in part for this work.

Arrow’s proof uses symbolic logic. Only an intuitive sketch of his reductio ad absurdum proof can be given here. Consider a society of individuals, each of whom has his own utility function for at least three possible states (configurations) of society. Let a social welfare function hypothetically aggregate all these individual utility functions so as to give all individuals an equal weight. We don’t exclude any individual preferences (otherwise, of course, we could obtain any social welfare function we want), which implies that individuals may use different criteria to rank their preferred states of the world. Consequently, inconsistencies (intransitivities) will appear in the social welfare function. In order to avoid this result, some individual (or group of individuals) must resolve the inconsistencies and break the cycles. Thus, the preferences of these individuals (it may be a minority or a majority) weigh more than those of other individuals, which implies a contradiction in the hypothesized welfare function. Therefore, the hypothesized social welfare function does not exist.

In other words, individual preferences cannot be aggregated into consistent and “democratic” social preferences. The political “we” is, indeed, either inconsistent or dictatorial. “We” does not exist, or if it does, it’s the royal “we.”

One objection to Arrow’s theorem came from James Buchanan, who claimed that cyclical political choices are preferable to the continuous exploitation of the same minority. He also argued that the very idea of a social welfare function assumes that society is an independent organic entity, which contradicts methodological individualism. “[T]he individual,” he wrote, “is the only entity possessing ends or values. In this case no question of social or collective rationality may be raised.” Perhaps he did not take seriously enough the normative arbitrariness of a cycling political majority. At any rate, he only confirmed the non-existence of the collective “we.”

For Gordon Tullock, Arrow’s results are irrelevant because public choices are filtered by institutions that make cycles rare in practice. For example, electors are seldom allowed to revisit the exact issues they previously voted on; logrolling generates stable coalitions; and so forth. Tullock may be right, but this does not change the conclusion that, when a cycle is possible, an arbitrary or random majority wins.

Except, arguably, at the very abstract level of a social contract, expressions such as “we as a society” have no ascertainable meaning, except to say “people who, like me, impose their preferences and choices on others.” In any diversified society, the scope of unanimity is restricted to very general and abstract rules, an idea we also find in Frederick Hayek’s “Great Society.”

Any aggregation problem disappears if a social choice is made unanimously. But it can be argued that, when individual preferences differ, unanimity can never be attained, however abstract is the goal or rule proposed. Imagine, for example, a single misanthropic individual who is intent on committing suicide and who would rather see mankind disappear with him than sign on to a project to intercept earth-bound asteroids. To avoid this sort of problem, the social contract is always framed in the context of some original position in which individuals face a somewhat uncertain future. At least, near-unanimity should be feasible if the goal or rule proposed is general enough. But this is not sufficient to justify invoking “we as a society” in public policy debates.

Like graphs of “social indifference curves,” the “we as a society” and similar expressions should be definitively buried. Unlike an individual, “society” does not have preferences on the basis of which it makes consistent and non-dictatorial choices. Society is not a big individual.