Public Interest, Democracy, and Condorcet Jury’s Theorem
A Failed Justification of the Majority Rule
The coming months will be busy for me. On top of finishing writing my book – which by the way is too long and will demand a significant effort to reduce its length – I’m working on several related papers to be hopefully presented at conferences this spring and summer. A recurrent theme is the issue of preference aggregation within a democratic setup: what should be aggregated and how to make democratic social choices?
Well, this is a vast question but rereading some classic texts has made me discover some interesting – if not new – ideas. A first classic is of course Kenneth Arrow’s Social Choice and Individual Values. It is well-known because this is where Arrow establishes his “general possibility theorem”, more commonly called Arrow’s impossibility theorem. In a nutshell, the theorem establishes that there is no procedure based on which it is possible to derive a social ordering of alternatives from all logically possible profiles of individual orderings of those same alternatives that satisfies a weak unanimity condition (if everyone orders x strictly above y, then x is socially ranked above y) and that makes the social ordering of every subset of alternatives depends only on the individual orderings of those alternatives without being systematically identical to the ordering of a single individual. Or, said otherwise, if you aggregate individual preferences through a rule that it is Paretian, independent from irrelevant alternatives, and non-dictatorial, then you will occasionally find out that it generates non-transitive and even cyclical social preferences.
All this is well-known. What is less known however is Arrow’s ensuing discussion of the way to circumvent the theorem. While nowadays most of the discussions focus on the condition of the irrelevance of independent alternatives (which excludes for instance the use of the Borda rule) or on the possibility to enrich the informational basis by including cardinal and interpersonally comparable utility representations of preferences, Arrow was mostly concerned with the unrestricted domain condition. More specifically, in chapters 6 and 7, Arrow reviews three plausible restrictions over the set of profiles of individual orderings. The first restriction consists in imposing “economic” restrictions on the set of possible individual preferences.[1] As it happens, Arrow shows that the theorem still holds with these restrictions and from that conjectures that it also holds for the so-called “Bergson-Samuelson” social welfare functions used in the new (at that time) welfare economics. This will spark an obscure and long controversy about the implications of social choice theory for welfare economics and the announced “death” of the latter.[2] Arrow then considers the possibility to restrict our attention to single-peaked individual preferences, i.e., preferences that are decreasingly monotonous at the left and right of individuals’ best alternatives, when alternatives are ordered along a unique axis. In this case, Arrow’s impossibility result no longer holds. The third and last restriction is the most radical one. Following what he dubs the “idealist school” of political philosophy (in which Arrow includes T.H. Green, Rousseau, and Kant), Arrow considers the possibility that individual orderings may express the “general will”:
“The idealist doctrine may be summed up by saying that each individual has two orderings, one which governs him in his everyday actions and one which would be relevant under some ideal conditions and which is in some sense truer than the first ordering. It is the latter which is considered relevant to social choice, and it is assumed that there is complete unanimity with regard to the truer individual ordering.”[3]
Though Arrow leaves unexplained how this “truer” individual ordering can be formed and does not give any indication about its content,[4] the implication is straightforward: if individuals are unanimous regarding what the general will is, then the problem of social choice disappears. Of course, this unanimity assumption has no empirical value and is also doubtful from a normative point of view. Nonetheless, in the added chapter to the second edition published in 1963, Arrow briefly returns to the idealistic view while discussing Condorcet’s jury theorem. He characterizes this theorem as “a stochastic version of an idealistic position.”[5]
This may sound enigmatic, but it actually fits very well with another classic economic account of social choice and democracy, Anthony Downs’s An Economic Theory of Democracy (1957). In this book, Downs famously uses rational choice theory to study the behavior of voters and political parties. His theory emphasizes the importance of uncertainty and the imperfection of information, in particular, to account for the role of “ideologies” in the democratic political system. However, more than his 1957 book, it is an article that Downs published a few years later on the concept of public interest that is particularly interesting.[6] In this paper, Downs develops an analysis that is very in line with Arrow’s discussion of the idealistic position, though he doesn’t mention it. Basically, Downs assumes that individuals vote and assess government performance against their conception of the public interest. The latter expresses a minimal consensus among the members of a democratic society:
“This consists of an implicit agreement among the preponderance of the people concerning two main areas: the basic rules of conduct and decision-making that should be allowed in the society; and general principles regarding the fundamental social policies that the government ought to carry out.”[7]
This minimal consensus is actually constitutive of a democratic society. Presumably, that implies that citizens’ conceptions of the public interest sufficiently overlap such as to eliminate a significant number of what I would call “democratically non-admissible social orderings”. As an aside, it is interesting to note at this point that Downs suggests that not only preferences over policies but over aggregation rules can be deduced from the overlapping conceptions of the public interest.[8] Indeed, he claims that a key principle that can be deduced from the minimal consensus is that “anything that is in the long run detrimental to the majority of citizens cannot be in the public interest, unless it is essential to the protection of those individual rights included in the minimal consensus.” What Downs is here suggesting is that preferences that run contrary to liberal democracy are non-admissible, pure and simple.
Let’s return to the level of agreement between the various conceptions of the public interest that individuals may entertain in a democracy. Downs is of course aware that once we exit the idealistic setting, citizens do not vote uniquely based on their view of the public interest; private interests are obviously also relevant. While one’s conception of the public interest and private interests will in general be divergent, Downs suggests that the divergence increases with the imperfection of the information:
“Because acquiring information is costly, men are never fully informed about any issue: there is always more to know that might influence each decision. The resulting ignorance creates a potential gap between what a man perceives as private and public interests and what he would perceive them to be if he had perfect information.”[9]
Let’s suppose for a moment that the “minimal consensus” is actually maximal or that, leading to the same result, the issues on which voters have to express their views all belong to the minimal consensus. That means that there is a unique conception of the public interest that is relevant. Imperfect information however implies that actual conceptions will not be fully convergent. This is related to the role that Downs confers to ideologies in his theory of democracy: ideologies are not all located at the same spot on the political spectrum and reflect partially the imperfection of information. Ideologies just helped not well-informed voters to form views at a moderate cost for themselves and to participate in the democratic endeavor. It is at this juncture that a link with Arrow’s enigmatic claim can be made. Ideologies are biased renderings of the public interest and the underlying consensus. That means that most of the time, individuals’ social preferences will not be “correct”. Nonetheless, Condorcet’s jury theorem tells us that under specific (and demanding) conditions, the majority rule will lead to a social choice that is almost systematically the correct one, i.e., reflecting the authentic public interest. This seems to provide an important argument in favor of the majority rule.
However, the limits of this argument are so obvious that it is not worth discussing them in detail. Besides the fact that the conditions for the jury theorem to apply are never satisfied in a democratic context, the main problem is that the idealistic position is false, even in its stochastic version. As Downs himself points out, people largely disagree about what the public interest is, and not only because the information is imperfect or because their views are polluted by their personal interests. Even though a minimal consensus exists, the range of conceptions is still fairly large, such that it hardly makes sense to speak of the “true” conception of the public interest.[10] Admittedly, this is not a surprising conclusion. But once we acknowledge it, we may rightly wonder why scholars in democratic theory are so keen on relying on the jury theorem as part of an “epistemic” account of democracy.[11] The theorem just has little if any relevance in a pluralistic society where the minimal consensus is, indeed, minimal.
[1] Basically, individual preferences are exclusively defined on and positively monotonous in the space of goods.
[2] Readers who are interested in this controversy should consult two articles by estimated French colleagues: Herrade Igersheim’s “The Death of Welfare Economics: History of a Controversy” (History of Political Economy, 2019) provides a detailed historical account of the controversy, while Marc Fleurbaey and Philippe Mongin’s “The News of the Death of Welfare Economics is Greatly Exaggerated” (Social Choice and Welfare, 2005) offers an insightful analysis of its conceptual and theoretical intricacies.
[3] K. Arrow, Social Choice and Individual Values (Yale, 3rd edition, 2012), p. 83. The emphasis in the quote is mine.
[4] Even though he will never articulate them with Arrow’s work on social choice, John Harsanyi’s utilitarian theorems can be viewed as detailed normative accounts of how rational individuals should form their “true” or “social” preferences.
[5] Ibid., p. 95.
[6] A. Downs, “The Public Interest: Its Meaning in a Democracy”, Social Research, 1962.
[7] Ibid., p. 5.
[8] Here again, Downs is following Arrow’s footsteps, as the latter also has considered the possibility of including decision rules in the domain over which individual preferences are defined. I have written a post on this blog about this interesting idea.
[9] Ibid., p. 20.
[10] To this, I would add also that a “populist” democratic regime that automatically identify the public interest with the majority is susceptible to destabilize the minimal consensus if there is no filter regarding the conceptions that can legitimately expressed.
[11] See for instance, R. Goodin and K. Spiekermann, An Epistemic Theory of Democracy (OUP, 2018) and H. Landemore, Open Democracy: Reinventing Popular Rule for the Twenty-First Century (Princeton University Press, 2020).