The theory of social choice considers the problem of aggregating the preferences of the members of a given society in order to derive a social preference that represents this society or community. The social preference is to express the general will, the common good as it were. The general will can be viewed as the basis for the very existence of any society. Economists argue that the common good finds its expression in a so-called social welfare function which, described more mundanely, represents a compromise among divergent interests of those who belong to society. The market mechanism cannot be taken as a social welfare function since it is not guided by moral or ethical principles in any deeper sense. A market allocation heavily depends on the initial endowments of the individuals. These possessions determine the power or weakness of the individual agents.
Influences on the development of social choice theory have been manifold over the centuries. Mathematicians, social scientists and philosophers made important contributions of different kinds. The Marquis de Condorcet and de Borda explored the majority rule and ranking methods at the time of the French revolution. Roughly at the same period, Smith elaborated the concept of an impartial observer. The utilitarian philosophy of Hutcheson, Bentham and others has been most influential over several centuries with its modern-day version brought forward by Harsanyi in the middle of the last century. Rawls’ theory of justice from around 1970 became a powerful contestant of utilitarianism over the last few decades. It is probably fair to say that the modern theory of collective choice started with Arrow’s path-breaking work on the non-existence of a social welfare function around 1950. All these works and several more will be discussed in what follows.
If a social welfare function is an expression of the general will of the populace, it should be able to deal with whatever kind of preferences the individual members of a given society have. More technically speaking, and this is Kenneth Arrow’s definition, a social welfare function is a mapping from the set of all logically possible combinations of individual preference relations over a given set of social states or alternatives to the set of all logically possible orderings over these states. This is the requirement of “unrestricted domain”. In other words, it should not be admissible that any individual ranking be a priori excluded, for whatever reason, from the set of preferences of the members of society. The next requirement, called “the weak Pareto principle”, says that, if for any two alternatives x and y, let’s say, all members of society agree that x, for example, is strictly preferred to y, then society should have exactly the same strict preference. Next, information gathering for the aggregation procedure should be parsimonious, that is, if society has to make a decision between two alternatives, let us call them again x and y, the individual preferences with respect to x and y only, and not any other preferences, should be taken into account in order to distil the social ranking between these two alternatives. Arrow calls this condition the requirement of “independence of irrelevant alternatives”. Lastly, we do not want that there exist a particular person in society such that whenever this person has a strict preference for some alternative over another, society “automatically” has the same strict preference, for any two alternatives and any preference profile of the members of society. This is called the “non-dictatorship” condition. For Arrow, these four conditions are necessary requirements for a democratic decision procedure, perhaps not sufficient since there may be other demands as well. Unfortunately, in the case of at least three alternatives, these four conditions cannot be simultaneously fulfilled by any social welfare function. In other words, there does not exist a social welfare function satisfying these four requirements. This is Arrow’s famous “impossibility theorem”.
Clearly, almost everyone will agree that there should not exist a dictator in society who via his or her own strict preference over any pair of social alternatives automatically determines the social preference over these alternatives. Arrow defined social alternatives as very complex social states which can include, among other things, the issue of waging a war against another nation or introducing the death penalty for certain crimes. On the other hand, democracies should allow individuals a certain amount of freedom and autonomy over purely private matters. John Stuart Mill spoke about a circle around every human being that nobody should be allowed to intrude. So “local decisiveness” to a certain degree should be permissible. Amartya Sen was the first to integrate this idea into Arrovian social choice. He formulated that each and every individual be permitted to exercise local decisiveness over at least one pair of social states which differ only with respect to some private matters of that particular person. He then proved another important impossibility result, “the impossibility of a Paretian liberal”. He showed that the requirements of unrestricted domain and the weak Pareto principle are incompatible with the exercise of local dictatorship or individual autonomy over purely private matters. In other words, there does not exist a social welfare function fulfilling these three conditions.
Wulf Gaertner, Prasanta Pattanaik, and Kotaro Suzumura, among others, have argued that the Arrovian set-up may not have the appropriate structure to formulate the exercise of personal rights; the game form structure within which individuals select certain actions that shape private features may be closer to what one observes in real life, where closeness refers to conformity with existing rights systems. Notice, however, that this alternative suggestion by no means denies that there can exist a clash between the exercise of individual rights and the Pareto principle. A reference to findings in non-cooperative game theory may be adequate here, namely the existence of Nash equilibria that are Pareto-dominated.
A third important impossibility theorem in social choice theory refers to the fact that it can be advantageous for individuals to misrepresent their preferences and, by doing so, to achieve an outcome that is more favourable for them than the one that would have come about if they had announced their true or honest preferences. This phenomenon is well known from the allocation of public goods where individuals may want to hide their true willingness to pay in order to achieve a lower contribution fee for themselves. In social choice theory, Allan Gibbard, a philosopher, and Mark Satterthwaite, an economist, were the first to independently provide a deeper analysis of this phenomenon. Of course, there are social choice or aggregation rules that are not manipulable, but these are highly unsatisfactory in the sense that they do not respond at all or only very faintly to changes in individual preferences. The ranking rule proposed by Borda that we shall discuss a bit later in this essay, is highly susceptible to manipulation (Borda was very well aware of this fact so that he declared that his method was only for honest men), since its aggregation scheme employs very detailed information coming from the individual rank orderings. Gibbard and Satterthwaite proved that if individual rank orderings are unrestricted, if furthermore the aggregation method is “monotonic” or responsive to changes in the individuals’ preference rankings and Pareto-efficient, and if there are at least three social alternatives, the only non-manipulable or “strategy-proof” aggregation method is dictatorial.
Though there are lots of other negative results in social choice theory, there is wide agreement among researchers in this area that Arrow’s, Sen’s and the last result about strategy-proofness are the most important. We shall now turn to positive results.
We begin this section by focussing on the simple majority rule which is widely applied in many committees. According to this rule, alternative x, let’s say, is majority-wise weakly preferred to another alternative y, if the number of voters who find x at least as good as y is larger or equal to the number of voters who find y at least as good as x. The simple majority rule is an attractive decision mechanism since it treats both voters and alternatives equally (there is neither a discrimination amongst voters nor amongst issues to be decided upon – the latter property is called “neutrality”). Simple majority decision is also responsive to changes in voters’ rankings, much more than the absolute majority rule, for example. Why is the method of majority decision not a counter-example to Arrow’s famous negative result? The answer is that under an unrestricted domain of individual orderings, the majority rule may generate so-called Condorcet cycles. In their simplest form, in the case of three voters, they come about if one person prefers x to y, and y to z (and also, of course, x to z due to transitivity), the second voter prefers y to z, and z to x, and the third person prefers z to x, and x to y. Simple majority counting establishes a social preference for x over y, for y over z, and for z over x. In other words, there is a preference cycle on the aggregate level so that an Arrow-type social welfare function does not exist. For three alternatives and three voters, the probability for cyclical social preferences to occur is about 5.5 % which is not very large, but this probability increases with the number of voters and the number of alternatives so that it cannot be considered as “une quantité négligeable”.
The Welsh economist and political scientist Duncan Black has shown that if the individual preference rankings are “single-peaked” over each triple of alternatives, the simple majority rule yields a Condorcet winner, i.e., a candidate who is majority-wise at least as good as every other candidate. If the number of voters is odd, the method of majority decision generates a social ordering (so that the rule becomes an Arrow social welfare function). Single-peakedness as a qualitative property on preference profiles can be taken literally. Each and every voter has a most preferred alternative (in terms of ordinal preferences) and on either side of this most preferred object, the person’s preference decreases. Black himself thought in terms of the political spectrum (left-right). Of course, the far left and the far right voters show declining preferences only on one side of their respective peaks.
Amartya Sen generalized Black’s condition to what he called “value restriction”. For each triple of alternatives, there exist one alternative such that all voters agree that it is never worst (case of single-peakedness), or agree that it is never best (case of single-troughed preferences”), or agree that it is never in the middle between the other two. Again, a Condorcet winner exists under the simple majority rule and this rule once more becomes an Arrow social welfare function for an odd number of voters.
Single-peaked preferences have another remarkable property. Under this domain restriction, the method of simple majority decision turns out to be strategy-proof. This was shown by Hervé Moulin very generally – the position of the so-called median voter balances deviating interests to the left and to the right of this voter’s peak. A strategic misrepresentation would be against the very interest of those who are involved in the decision making.
Consider a social choice rule that specifies for any two alternatives x and y that x is socially at least as good as y if and only if it is not the case that all voters find y at least as good as x and there is at least one individual who finds y strictly better than x. Sen called this rule the “Pareto-extension rule”. It fulfils Arrow’s conditions of unrestricted domain, weak Pareto, the independence condition and non-dictatorship but it does not satisfy the requirement of full rationality of the social relation. The Pareto-extension rule is not transitive with respect to the indifference part of the social preference relation. Therefore, this rule is not a counter-example to Arrow’s theorem either; it “only” constitutes a social decision function. The latter is always able to provide a non-empty choice set but not necessarily a social ordering. The reader realizes that subtle differences can matter a lot. The Pareto-extension rule may be viewed as unsatisfactory for social decision making since, whenever there is at least one person who strictly opposes the strict preference of the rest of society for x over y, let’s say, the social outcome between x and y will be an equivalence or indifference. Such a person has been called a “weak dictator”. A consequence of this weak dictatorship is that the Pareto-extension rule is largely unresponsive to changes in the underlying preference profile of the members of society.
The so-called Borda rank-order method which without any doubt is the best known rule from the class of general scoring functions attaches ranks to positions. If there are m alternatives on which a social decision has to be made and if all alternatives are ranked in a strictly descending way by all voters, Borda proposed to attach the rank m-1 to the top-ranked element, m-2 to the second element from the top etc. and, finally, 0 to the bottom-ranked alternative. The element(s) with the highest aggregate rank sum is (are) socially chosen. Clearly, this rule uses much more (positional) information than the majority rule, for example. The latter just registers whether x, let’s say, is preferred to y, or vice versa. The Borda rule also considers the “distance” between options, i.e. the number of positions of alternatives between x and y.
Peyton Young showed that the Borda rule can be uniquely characterized by being neutral, consistent, faithful and having the cancellation property. Neutrality has already been explained in connection with the majority rule. Consistency makes the following requirement.
Imagine that the preference profile of society is split up into two disjoint subprofiles representing two disjoint sets of voters. If the intersection of the set of elements picked by the choice rule from the first subprofile and the set of elements chosen from the second subprofile is non-empty, consistency requires that all elements in this intersection are identical to those that would have been picked by the choice rule if there had only been one profile, viz. the union of the two subprofiles. A scoring function is faithful if “socially most preferred” and “individually most preferred” have the same meaning when society comprises just one person. Note that if a scoring function is consistent and faithful, it satisfies the weak Pareto principle. Finally, the cancellation property requires that given any set of alternatives, if for all pairs of alternatives from this set, the number of voters preferring a to b, let’s say, equals the number of voters with the opposite preference, then all elements from this set are equally chosen.
Clearly, the Borda winner can be different from the Condorcet winner. It can be shown that a Condorcet winner is never bottom-ranked according to the Borda rule and a Condorcet loser, i.e. a candidate that loses in pairwise contest against all other options, is never top-ranked according to the Borda count. The equi-distance between two adjacent scores is typical of the Borda rule but by no means necessary for general scoring functions. Nonlinear transformations of the Borda rank numbers, for example, can be introduced if there are plausible reasons for doing so.
Beyond Ordinal Non-Comparability
Up to this very point, we have only been considering ordinal rankings both on the individual and the societal level, with no trace of interpersonal comparability. John Rawls’s approach to justice, however, presupposes that we can compare levels of well-being (strictly speaking, in terms of so-called primary goods) across persons so that we can, for example, say that person i is better off under state x, let’s say, than person j is under y. This kind of interpersonal comparability is called ordinal level comparability. Utilitarianism which was made well known by Bentham, a law person, postulates the greatest happiness of the greatest number. In more prosaic terms, utilitarianism either maximizes the aggregate sum of utilities over all persons concerned or the average utility per society. In either case, differences in utilities must be a meaningful concept, and these have to be compared across persons. The underlying concept here is the cardinal concept comparable to temperature or weight where we also usually measure differences.
Apart from the informational requirement, namely ordinal level comparability in the Rawlsian approach versus comparability of utility differences in utilitarianism, Rawls focuses in his second principle of justice on the worst-off in society whereas utilitarianism, as just stated, focuses on the aggregate sum of utilities which is to be made as large as possible. Axiomatically speaking, both models of distributive justice have a lot in common. Both maxims satisfy the Pareto principle, independence of irrelevant alternatives and anonymity. The point of bifurcation between both rules is a so-called equity axiom in the Rawlsian set-up with its single focus on the worst-off (strictly speaking in terms of an index of primary goods such as income, wealth, opportunities and self-respect).
The modern-day version of utilitarianism was formulated by John Harsanyi. His model, based on the Bayesian rationality postulates, employs the von Neumann-Morgenstern expected utility hypothesis. In one of his models, an ethical observer evaluates different policies for a particular society in an impartial manner, thereby determining the utilities that accrue to each member of society. This approach is very reminiscent of the role that an impartial spectator plays in Adam Smith’s Theory of Moral Sentiment. Again, as in the case of de Borda and Condorcet, a fundamental idea bridged over two centuries, receiving a modern scaffolding, so to speak.
Finally, we would like to mention Sen’s capability approach. For Sen, what defines freedom, autonomy and well-being are the functionings of a person, her achievements and not just the accumulation of primary goods as in Rawls’ theory. What a person manages to do or to be (for example, being well-nourished, well-clothed, taking part in community life, having access to medical care) are functionings that are important for a person’s life. The total number of functionings that are available to a person or household define the advantages of that person, her real opportunities. These make up the person’s capability set.
Wulf Gaertner See also Bargaining Theory, Capabilities, Collective Rationality, Decision Theory, Judgment Aggregation and the Recursive Dilemma
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