Fifty years of the Paretian liberal paradox.

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2020 marks exactly half a century from the publication of one of the most important papers in the crossing of economics, philosophy, and political science (and, taking into account that it barely contained six skimpy pages, probably one of the most influential papers per page ever): Amartya Sen’s “The impossibility of a Paretian liberal”1. It was not the first, nor the last one, in the lists of “impossibility theorems” for which the branch of economics (or maths) known as “social choice theory” became famous or infamous; here we have seen some of the other main milestones: Arrow’s paradox, Gibbard-Satterthwait theorem on manipulability of voting mechanisms, or the ‘Discursive Dilemma’ about the inconsistency of rules for judgment aggregation. But Sen’s has been probably the most influential one in terms of political discussion, for it directly pointed to one of the most fundamental ideas in the conceptual schemas with which we, contemporary people, tend to think about political issues: the notion of individual rights.

Sen baptized his result as a ‘liberal’ paradox, or a paradox in ‘liberalism’, in a sense of ‘liberalism’ that now is frequently eclipsed by chit-chat on the spectre of ‘neoliberalism’. The ‘liberalism’ Sen talked about was simply the idea that there are some individual rights, i.e., questions on which the society has to admit the decision and preference of an individual, no matter what decision it is. Being a purely formal mathematical demonstration, Sen theorem didn’t contain any hint or presupposition about the content, nor the number, nor the extent, of those individual rights, it only assumed that we would value that an appropriate ‘social decision function’ should give each individual at least some rights, in the most general sense defined above. Formally stated, this amounts to the fact that, between the complete set of ‘states of the world’ x1, x2, x3, …, to which the social decision function SDF (as well as the individual preferences) are applied, for each individual there must be at least one pair of states of the world, xi and xj, such that, if that individual prefers xi to xj, then SDF must also ‘prefer’ xi to xj. To use one example employed by Sen: if you prefer sleeping on your belly to sleeping on your back, then the society also ‘prefers’ that you sleep on your belly, no matter what the rest of the people think about your sleeping position. To say that you have the right to choose what position you sleep in is equivalent to saying that, even if everybody else preferred (by whatever reasons we might imagine) that you slept on your back, it is ‘better from the society’s point of view’ to let you sleep on your belly, if that’s what you want.

Perhaps talking about ‘social preferences’ is a little misleading, because, in the context of rights, we have not to understand ‘social preferences’ as a kind of ‘aggregate preferences’, but rather in the more abstract sense reflected in the notion of ‘social decision function’: the society has agreed to organised itself in such a way that some issues are decided by ‘collective voting’, and some other issues are decided on the basis of the free decision of the individual which happens to involved. What the society decides is that, about some questions, the choice is only yours, and the preferences of others are just irrelevant.

What Sen demonstrated 50 years ago was that this really minimal notion of ‘individual rights’ can contradict another basic rule of social choice: the Pareto principle. This states that, if every member of the society prefers A to B, then the society also prefers A to B (and hence should opt for A in a social choice between A and B). Many examples of this inconsistency have been offered in the literature, starting from Sen’s paper itself; but this one, from Gibbard 2 is specially clear: there is a woman, Alice, two men, Bob and Charlie, and three possibilities: 1, Alice marries Bob; 2, Alice marries Charlie; 3, everyone remains single. Alice’s preferences are 1>2>3; Bob’s are 3>1>2; lastly, Charlie is indifferent between all the options. The relevant ‘rights’ in this example is that Alice has the right to freely decide between 2 and 3 (marrying Charlie or remaining single), whereas Bob has the right to decide between 1 and 3 (marrying Alice or remaining single). The social decision function, taking into account these rights and the subjects’ preferences, would then determine that Charlie and Alice get married and Bob remains a bachelor (possibility 2). But possibility 1 is at least as good as possibility 2 for everybody, and better for at least two people, Alice (who prefers Bob to Charlie), and Bob (who prefers being a bachelor to marrying Alice, but that would prefer wedding Alice before seeing her married to Charlie); hence, the application of the Pareto principle would entail that 1 should be socially preferable to 2. Giving individuals some rights on the choices they make can lead, then to, results that are ‘non optimal’ from the social point of view, in the sense that everybody would have preferred a different result.

As it is more or less clear from this example, the problem arises due to the fact that people’s preferences tend to be ‘meddlesome’ or ‘interfering’: some people have strong preferences about what others do. One ‘easy’ solution would then be to ‘ban’ in some way this type of preferences, persuading people that they ‘should not care’ about what others do, but this is just a simple-minded utopian option, or no realistic option at all: many, if not most social problems are problems precisely because some people want to do things that other people consider ‘bad’ (having sex with the ‘wrong’ individuals, organise politically in the ‘wrong’ way, doing business about ‘wrong’ issues, etc.). The literature on the topic in the last 50 years is impossible to summarise; I want just to mention a recent result by the philosopher Hun Chung 3, that generalises Sen’s theorem to apply it to cases in which the description of individual rights include not only the reference to ‘states of the world’, but also to the ‘perspective’ of the individual, assuming that each person can describe, understand and conceptualise differently the same events; there was the hope that including this ‘perspective diversity’ would offer some space for smoothing out the paradox, but Chung proves that conferring individual rights might conflict again with the Paretian principle even under those more sophisticated assumptions.


  1. Sen, A., 1970, “The impossibility of a Paretian liberal”, The Journal of Political Economy, 78.1, 152-157.
  2. Gibbard, A., 1974, “A Pareto-consistent libertarian claim”, Journal of Economic Theory, 7, 388-410.
  3. Chung, H., 2019, “The impossibility of liberal rights in a diverse world”, Economics and Philosophy, 35.1, 1-27.

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