Pareto

Part two in my series on Arrow's conditions (following this one):

5.6.a Plausibility

Also embedded in Arrow’s citizen sovereignty is the idea that if all citizens share a preference xPiy this should be sufficient for the social choice to be xPy[1]. This is generally termed the weak Pareto condition. It actually requires something quite strong before it comes into play – namely that all citizens prefer x to y. It is because this requirement on being Pareto-satisfying is strong, however, that respecting Pareto-satisfying preferences is a weak requirement on the decision-making procedure. A decision-procedure is not required to yield xPy even in cases where all bar one vote xPiy and that one is indifferent (xIjy), only when there is unanimity.

Since the Pareto requirement merely means that unanimous preferences should be respected, it might seem so weak as to be unobjectionable. If everyone prefers x to y, and society’s preferences are to be a democratic function of individual preferences, then it would be odd for society not to prefer x to y. To use MacKay’s comparison, if one competitor is better in every respect, she must be better overall[2]. One note of caution must be sounded, however. It may be natural to assume that this will lead to better outcomes, but this is not necessarily so. Not only can preferences fail to track what is actually good but, if individuals are to vote for what is good for them, then their votes will fail to reflect ways in which the world may be better without being better for any individual[3]. The idea that outcomes can be better or worse independently on how they affect individual well-being is controversial amongst those who accept the ‘principle of personal good’[4], but there are others who argue the world can be made better though not better for any individual – for example, by being more equal or in keeping with desert[5]. Sen also argues that we should reject some Pareto improvements in order to respect individual liberty[6]. Now it seems far less obvious that Pareto is always something to be respected.

It must be remembered, however, that we are not concerned with how to achieve the best outcomes by any independent standard; we are concerned merely with a democratic decision-procedure that respects individual preferences equally. A decision-procedure, such as flipping a coin between two tied or incommensurable options, x and y, is only supposed to tell us what to do, not what is best. So it is with voting, conceived of as simply a decision-procedure. From the fact that xPiy and yPjx, we cannot infer that either x or y is ‘socially better’, but – at best – that x is better for individual i and y is better for individual j[7]. Similarly, even from the fact that xPiy and xPjy we cannot strictly infer that x is actually better than y, but what we do have is agreement. We can eliminate the Pareto-dominated option (y) because this can be done without conflict, or anyone’s objection[8]. If all prefer x to y, no one opposes the move from x to y[9].

There may be cases where we might question whether a Pareto improvement – in terms of improving an individual’s well-being – is actually an improvement in the total value of the world. Suppose, for example, we had a world consisting of Hitler and Mother Teresa, who had 4 and 5 units of utility, respectively. Now suppose we could bestow two more units of well-being on Hitler (but not Mother Teresa), moving from (4,5) to (6,5). This is a Pareto improvement, in that it makes Hitler better-off and no one worse off, but we might not think the resultant world is actually better[10]. These counter-examples are based, however, on goodness rather than preference. If we suppose Hitler and Mother Teresa (out of generosity) both prefer that Hitler receives the extra well-being, then it is unclear what democratic objection we might have. Of course, if others were added to the world, who preferred that Hitler be worse-off, then it is no longer the case that (6,5) is Pareto-preferred to (4,5)[11]. Provided we keep the focus on what is preferred, rather than what we may think is ‘objectively better’, it is hard to see on what grounds we should ignore a unanimity of preferences. While I have expressed some scepticism about when majorities should overrule minorities, if everyone prefers x to y, then this seems an easy case for collective decision-making: x is socially preferred.

5.6.b Lottery-Voting Assessed

It is easy to show that lottery-voting respects the weak Pareto condition. If all votes are for x, then necessarily any randomly selected vote will be for x. If we know everyone is in agreement, we need only ask a single random person, to give us a representative sample of the population[12]. Indeed, suggests such in his A Theory of Justice; assuming rational convergence (and hence unanimity) behind the veil of ignorance, he says, “we can view the agreement in the original position from the standpoint of one person selected at random”[13]. This is, to my knowledge, the closest he ever comes to endorsing a system like lottery-voting, and he does so only on the condition of unanimity (Pareto).

I would argue that, were we to allow for rational disagreement, and thus the possibility that individuals in the Original Position could reach different conclusions – all equally permissible by justice – it would still be fair to select and implement one such conception of justice at random. This need not be argued here, however. For now, all we need to note is that if there is universal agreement, lottery-voting will respect it, because the randomly-chosen individual must be part of a truly universal consensus. Of course, lottery-voting does not meet any stronger Pareto requirement. It may be that 999,999 individuals rank xPiy and one ranks xIjy; and yet in this case lottery-voting can (by selecting the one) report social indifference. I do not intend to discuss the plausibility of any stronger version of the Pareto requirement, however. I hope the considerations of 5.6.a have shown it is not obviously a necessary requirement on any decision-procedure. For now, I am only arguing that lottery-voting meets minimal conditions, and so it need only satisfy a form of Pareto weak enough to be uncontroversial.


[1] Arrow p.28.
[2] MacKay (1980) p.18.
[3] c.f. MacKay (1980) p.24: “however many individuals happen to prefer X to Y, that does not tend to establish that X is better than Y in any interesting sense”; Bordes () p.196.
[4] Who does? Raz? Broome?
[5] Consider, for example, a modification of Larry Temkin’s ‘sinners and saints’ example. In world one, sinners have 2 units of good and saints 10. In world two, sinners have 12 and saints 11. Everyone is better off in world two, but we might all things considered prefer world one.
G. E. Moore’s view is that anything intrinsically valuable is valuable independently of human interaction or appreciation. Thus the world would be a better one if there was breathtaking natural scenery on Pluto, even though no sentient creature would ever see it. check
[6] Sen ‘Paretian liberal’. I think he is right that no one should be forced to read a book to satisfy others’ preferences; but given there is a Pareto improvement to be had we could allow Lewd and Prude to sign an enforceable contract such that Prude will agree to read the book if Lewd agrees not to. This is no violation of individual liberty, since each agrees to give up discretion over whether he reads the book in order to satisfy a stronger preference over what the other reads, and this is done by mutually voluntary contract. We limit our own (lower-order) freedom by promises or contracts all the time, in order to achieve higher-order goals (and freedom).
[7] ‘At best’ because a preference for x does not imply that x actually is better for one. C.f. MacKay (1980) p.52.
[8] Bordes () p.196-end
[9] Note, however, that this reasoning would support a stronger Pareto requirement on the decision procedure: One that says so long as there exists an i such that xPiy, and for all others xRjy, then xPy. That is, one would only need a single individual to prefer x, so long as all others were at least indifferent – no one prefers y. Such a stronger requirement may be controversial, however: Would we really want to say a single person’s preference should decisively dictate a social preference for x if there were millions of other, all indifferent? I need not resolve this issue here; any reasoning that supports a stronger Pareto requirement will a fortiori support a weaker one.
[10] This obviously draws on Temkin’s sinners and saints example.
[11] Note that this counts these people’s external preferences, in line with the Universal Domain.
[12] Indeed, in such cases, one may think we don’t need randomness: if everyone prefers x to y, then there is no problem with a dictator – she will prefer, and therefore implement, x, and so everyone will get what they want.
[13] Rawls (1999 [1971]) p.120.