Universal Domain

Excuse the unfinished footnotes, but I'm currently trying to work my way through Arrow's axiomatic requirements of social choice on a one-by-one basis. All comments very welcome.


5.5.a Plausibility

The condition often referred to as Universal Domain in the secondary literature[1] is actually part of Arrow’s Definition 5 Citizen Sovereignty: “we do not wish our social welfare function to be such as to prevent us, by its very definition, from expressing a preference for some given alternative over another”[2]. This excludes what Arrow calls an imposed ordering, where there is some pair of alternatives such that society can never express xPy even if everyone prefers x[3]. Allowing individuals and the society to express any preference ordering seems prima facie democratic. As Riker says, “Any rule or command that prohibits a person from choosing some preference order is morally unacceptable (or at least unfair) from the point of view of democracy”[4]. If we want society to be responsive to individual preferences, then it seems we cannot legitimately place restrictions on those preferences – or, at limit, if we were to restrict all preferences, society would become completely unresponsive.

There are in fact two problems with the admission of any possible preferences. Firstly, it has often been argued by liberals that there are normative justifications for excluding certain preferences, e.g. those based on religious doctrines that do not pass the standard of ‘public reason’ (Rawls), or those over how other people should behave in self-regarding matters (Mill, Dworkin, Sen). Secondly, it may result in a combination of preferences that seem likely to defeat the project of social choice.

As for the first of these problems, the idea that some preferences are, as Arrow puts it, ‘taboo’ has been a recurring one in objections to utilitarianism. Rawls, for example, contrasts the case of utilitarianism, for which “the satisfaction of any desire has some value in itself” with justice as fairness according to which “[a]n individual who finds that he enjoys seeing others in positions of lesser liberty understands that he has no claim whatever to this enjoyment”[5]. One can equally ask why a democracy should want to count the preferences of sadists, rapists and child pornographers. Nozick makes the point that individual rights restrict social choice[6], and Sen offers a case in which allowing individuals complete sovereignty over their own self-regarding private sphere defeats Pareto improvements[7]. If we think that certain issues, such as fundamental rights, warrant constitutional protection, then we thereby remove them from the domain of democratic social choice. For example, if free speech is guaranteed by the constitution, parliament cannot make a law abridging it, even if all representatives would so prefer[8].

Perhaps the reply to these worries is to insist that democracy does not always mean good government. Maybe if we want laws that ensure maximal compliance with moral demands, we are best to leave them to a group of moral experts[9]. There may be very good reasons why we would want to, for example, exclude sadistic preferences or protect free speech, but they are not democratic reasons. On this line of response, democracy is simply about responding to individual preferences, and if those individual preferences are bad, then the result is that democratic social preferences will also be infected; but the failing is with the people, not the democracy. This argument, however, runs into problems when certain preferences involve undermining the equality of others. If the sadist’s preferences are satisfied, his victim is subjugated, just as the speech of some that puts down others may ‘silence’ those groups, thereby defeating equality[10]. To take matters to a logical extreme, suppose the majority of society think the preferences of some minority should be excluded or discounted – e.g. Jews under the Nazis, or blacks under apartheid – this doesn’t make it democratic[11]. There are some cases where the logic of democracy will have to exclude certain preferences, and thus it is not clear the domain can ever be truly universal.

This brings us to the second problem, that some combinations of preferences may make a coherent social choice a priori less likely. This includes the external preferences mentioned in the last paragraph, but also cases where an individual’s preferences are only self-regarding but themselves inconsistent. Universal domain would allow individuals to express any pattern of preferences they like, including one such as xPy, yPz and zPx, i.e. it permits individual inputs to be intransitive[12], and thus it is hardly surprising that we have difficulty producing a transitive social ordering[13]. Moreover, it is a well-known fact that social intransitivity can occur even if individual preferences are themselves transitive[14]. The problem can be resolved, however, if we place further restrictions on the individual orderings, particularly single-peakedness. So long as all agree that the underlying issue can be mapped on a single dimension (e.g. a left-right scale), then agreement is more likely. We need only insist that each voter has some ideal point (which may be central or extreme), and less-prefers points further from that ideal. This excludes anyone who prefers either extreme to a point in the middle[15]; but since such preferences often seem unnatural, it is not intuitively a great cost. Nonetheless, exclusion of non-single-peaked preferences seems motivated only to preserve the possibility of collective judgement, not on a principled basis. There may be cases where such preferences are reasonable – where the issue to be decided is one of expenditure, and an individual might legitimately prefer it to be high or zero (‘do the job properly, or not at all’). Thus it seems that while we should exclude some preferences on the aforementioned moral grounds, we should reject any attempts to restrict individual preferences solely in order to facilitate social choice.

5.5.b Lottery-Voting Assessed

Now we have seen in what sense a universal domain is desirable, we turn to how well lottery-voting satisfies this axiom. If there is room for some issues to be constitutionally protected, then lottery-voting can allow for them to be removed from the agenda. The issue concerns those that are left in the democratic political domain. For example, suppose z is a protected basic right, whereas x and y are matters for democratic decision. Lottery-voting does not put z to the vote, whereas it allows citizens any combination of preferences between x and y (i.e. xPy, xIy or yPx)[16]. Thus it seems that lottery-voting satisfies universal domain fairly easily.

One might, however, worry whether lottery-voting allows voters to express all their preferences. Recall the example where one may want to spend nothing, but where one has a second preference that if the decision is to go ahead a lot is spent. Here, the decision can be broken down into the yes/no question of whether to spend at all, and then the secondary question of how much to spend. It is unclear how this could be resolved by ordinary voting rules – one might vote in several stages, for example, or take a vote between a range of options one of which was zero. It seems that whatever answer is reached will depend as much on how the vote is set up as on the individual preferences, supporting Riker’s contention that electoral outcomes are likely to be either arbitrary or manipulable[17].

While one could employ lottery-voting at each of several stages, e.g. whether to spend at all, and then (if applicable) how much to spend, the obvious way of implementing it is to simply ask each voter for their first preference – which might be no spending, low spending or high spending – and randomly draw one of these. This fits the basic idea of fairness, giving each an equal chance to deciding the vote in favour of their first preference. On the other hand, however, it is clear it will exclude the expression of certain preference profiles. Because one is only asked for one’s first preference, there will be no way of expressing that one ideally wants no spending, but thinks if spending is to happen it should be at a high level. One only gets to express a single preference, and so one will simply vote for no spending and one’s second preference will be ignored[18]. Is this an objectionable exclusion of preference information?

One might have various reasons to think that we should consider all possible preference information: for example, one may think we could reach a compromise that is reasonably acceptable to all[19]. I do not, however, think that not admitting such second preferences is a violation of universal domain. Everyone had a chance to express their first preference, and everyone had a fair chance for that to win the vote. If some people were now allowed to express another second preference, that would be to give them more weight, on account of them having more preferences. We can think of the two issues as being resolved by a single vote, that determines both whether to spend and how much. Everyone is allowed to vote for their ideal outcome, whether that be no spending, low spending or high spending. Supposing the winning vote is one that mandates a set amount of spending, then one’s vote for no spending has already been accounted for, and legitimately defeated. Why, just because one has had one’s vote counted and lost, should one be entitled to another vote to influence the outcome of how much is spent? If it was the latter issue that mattered more to one, one could have voted on it – using one’s vote for either high or low – but ex hypothesi one preferred to vote for no spending. To demand influence also over this second part of the choice is just to demand more say. There is no more reason why a voter should be able to say ‘none, or if not high’ than to let them say ‘high, or if not high’: it would simply be to give some more influence over the ultimate outcome[20].

Thus, in its purest form, it seems lottery-voting satisfies the requirement of universal domain. While the preference information it uses is limited to first preferences, this is true of many voting systems that take only a single vote rather than a list. The axiom is satisfied when and because it is open to voters to express any first preference, e.g. xP[all alternatives], for any x. As I noted in examining the plausibility of the axiom, this will not always be desirable, and we may want to protect certain rights from any democratic decision. This is not, however, an inherent limit of lottery-voting, but simply the result of another choice on constitutional protection. Lottery-voting per se is consistent with either a limited legitimately political domain, or an unrestricted set of allowable first preferences.

[1] E.g.
[2] Arrow (1963) p.28.
[3] Note that Citizen Sovereignty therefore also includes Pareto, which I address next.
[4] Riker (1982) p.117.
[5] Rawls (1999 [1971]) p.27.
[6] Nozick (1974)
[7] Sen ‘Paretian liberal’
[8] Though in this case, of course, they may be able to change the constitution.
[9] This obviously recalls Plato (1992)’s philosopher-rulers, but also Raz’s conception of authority: see Raz (?).
[10] e.g. Fiss
[11] c.f. Dworkin external prefs TRS
[12] See D. Saari (1994) Geometry of Voting p.327.
[13] Imagine a case where everyone had such intransitive preferences. Now it seems we must choose between social transitivity and Pareto.
[14] This ‘paradox of voting’ was discovered by Condorcet, and is discussed in … It can be illustrated by three people (i, j and k) with orderings xPiyPiz, yPjzPjx and zPkxPky.
[15] For the deliberative response, see D. Miller (1992) ‘Deliberative Democracy and Social Choice’ in Political Studies 40 [reprinted in D. Estlund (ed.) (2002) Democracy]. On homogeneity, see G. Mackie (2003) especially pp.47-55.
[16] One might add that any preferences over z (as a matter of private conscience) are also allowed, they are merely ineffective since it is not voted on.
[17] Riker. Which depends on whether the electoral system is already in place, and so produces one outcome unintentionally, or whether it is up to some chairperson to devise an electoral system, in which case they can rig it to favour their preferred outcome.
[18] It would be logically open to vote for high spending, instead, but there would be no point to this if one really did prefer no spending. I will deal with strategic voting later in this chapter. ‘High spending, or don’t bother’ is, of course, another possible – but different – combination of preferences.
[19] Such thoughts motivate Borda counts and approval voting.
[20] One might object that those who vote for no spending only get a say over the former issue (whether to spend), and not over how much; whereas those who vote for either high or low spending de facto also express their opinion that spending should occur. This is not an objection, because the separation into two issues is somewhat artificial: we could simply represent zero as an amount of spending, and therefore only quantitatively, not qualitatively, different. It will be seen that all cast a single vote for what is most important to them: it merely happens that high spending entails a fortiori spending, while a vote for no spending is simply for no spending, and not also for either a high or low amount of such.

Also, if anyone happens to be an expert on Clausewitz, adjudication might be welcome here.