Voting theory: Arrow’s impossibility theorem

The transformation of a bundle of more or less deep insights on a few voting procedures, as we saw in a preceding entry, into something like a general logico-mathematical theory about the aggregation of individual preferences into a collective choice was the work of a single man, Kenneth Arrow, in his 1951 book Social choice and individual values (based on his PhD dissertation, that had been defended that same year).

Photo: Mulyadi / Unsplash

Arrow’s idea was to consider the relation between individual and collective preferences as a general mathematical function, a concept for which he employed the term ‘social welfare function’ (SWF) that had been introduced and developed in the preceding years by the economists Abram Bergson and Paul Samuelson. There is a fundamental difference between the Berson-Samuelson notion and Arrow’s: whereas for the former the SWF is a numerical utility function, that takes as its domain the set of numerical utility functions of each individual member of the relevant group, Arrow’s idea was to define the social welfare function just on the basis of the individual comparative (or ‘qualitative’) preferences, i.e., the ranking of alternatives as they are ordered by each individual, without assuming that some numerical values could represent the relative ‘intensity’ of those preferences, and the same for the collective preferences.

In a nutshell, an Arrowian ‘collective choice’ is just a mathematical function that says which ordering of alternatives is collectively preferred given the orderings made by the individual members of the group. The term ‘choice’ is justified because the top alternative in the ordering would be the collectively preferred one, if available, but, if it is not, then the ordering could be used to define, between each pair of available options, which one would be selected by the group. This limitation to purely qualitative or comparative preferences entails significant formal limitations, but is a reasonable answer to the problem of the ‘incomparability of individual preferences’ that had plagued in the previous decades the discussions of the idea of a quantitative SWF. Another terminology that became popular in the next years, and perhaps a more intuitive one, was that of a social ‘aggregation of preferences’.

Arrow proceeded then to reflect on the desirable properties a social welfare function should have. First and foremost, both the individual and the collective preferences must be real orderings, i.e., they should not lead to cycles, but obey the rule of transitivity (if A is preferred to B, and B to C, C must not be preferred to A). Next, the following is a small list of some other reasonable properties, either because of formal, or of ethical reasons:

1) Unrestricted domain: the function should be defined for all possible sets of individual preferences (i.e., the group should not order the members to change their values just because they are incompatible with the way it aggregates them);

2) Non-dictatorship: the SWF should not identify a priori one member of the group in order to mimic his or her preference (this is compatible with the fact that in some cases -i.e., a posteriori-, the preferences of the group coincide with those of one of his members).

3) Pareto efficiency, or unanimity: if all individuals prefer A to B, the SWF should rank A over B.

4) Independence of irrelevant alternatives: the collective choice between options A and B should only depend on the individual preferences between A and B, not of the preferences of the individuals over another alternative like C. This condition, which is perhaps less intuitive than the others, is justified because, if it is not fulfilled, then the person with the power to set the agenda could determine the collective choice between A and B just by adding or deleting ‘irrelevant’ options like C.

And then came the disturbing result with the power of a strict mathematical demonstration:

Arrow impossibility theorem: there is no SWF simultaneously satisfying properties 1 to 4.

This theorem made it clear that the ‘problems’ that other authors had identified in the voting rules previously mentioned are not faults of those particular procedures, but a general ‘defect’ of all possible ways of aggregating individual preferences. Actually, Arrow’s theorem was only the first one in a list of similarly negative results, of which I will mention three of the most important ones (some of them already known to our readers):

Sen’s paradox: there is no SWF that satisfies the Pareto principle (property 3) and ‘minimal liberalism’ principle (i.e., the condition that, for each member of the group, there are at least two options A and B, such that, if the member prefers A, then the group also ‘prefers’ A, or, in other words, the group ‘respects’ the individual choice).

This result was also known as ‘the impossibility of a Paretian liberal’.

Gibbard-Satterthwaite theorem: there is no SWF that is non-dictatorial (property 2), allows the choice between more than two options, and is not strategically manipulable (i.e., some member of the group might modify the collective choice in his favour by misrepresenting his own preferences).

The discursive dilemma is a theorem that refers not to the aggregation of ‘preferences’, but about the aggregation of ‘judgments’, i.e., individual opinions about whether some propositions are true or false; contrarily to the case of a SWF, where the options over which the preferences are defined are different and incompatible ‘states of the world’, a judgment aggregation function (JAF) is defined over sets of sentences, between which there can exist other logical connections besides logical incompatibility (i.e., some propositions may be consistent with others, or even logically entail others).

The discursive dilemma or ‘doctrinal paradox’: there is no JAF that satisfies the following properties: universal domain (applicable to all possible sets of individual self-consistent judgments), anonimity (the outcome should not depend on which particular individuals believe what), systematicity (if the logical relations between a given set of propositions are the same ones than those between another set of sentences, and the individuals have analogous beliefs about the propositions in both sets, then the JAF must deliver analogous collective judgments) and collective self-consistency (i.e., the collective judgments must be internally coherent).

It would not be fair, however, to understand social choice theory as merely ‘the science of the impossibility of (rational) social choice’, for these certainly negative results have served as a strong incentive to find out specific conditions under which some more ‘reasonable’ collective outcomes can emerge, both in the logical and the ethical senses of ‘reasonable’. We can remember that the title of Amartya Sen’s 1998 Nobel Prize conference was “The possibility of social choice”, where he argued in favour of both exploring more realistic procedural systems of collective choice, even combining the formal and the empirical research on their properties and distributional consequences, and studying SWF’s capable of containing more information about the welfare of the individual people, in line, for example, with John Harsany new utilitarianism. In particular, SCT has been extremely productive in discussing and elaborating sophisticated and applicable criteria of economic fairness (e.g., Fleurbaey and Maniquet 2011).

References

Arrow, K. J. (1951). Social choice and individual values. New Haven: Yale University Press.

Fleurbaey, M., and F. Maniquet (2011). A Theory of Fairness and Social Welfare. Cambridge. Cambridge University Press.

Written by

Leave a Reply

Your email address will not be published. Required fields are marked *