Sunday, May 28, 2006


I'll have to come back to IIA, which has always seemed (to me) the most confusing of Arrow's conditions. So here's the third in my series of posts, on non-dictatorship:

(5.8) Non-Dictatorship

5.8.a Plausibility

Arrow’s Condition 5 is that the social welfare function is not to be dictatorial. He elaborates this by adding “A social welfare function is said to be dictatorial if there exists and individual i such that, for all x and y, x Pi y implies x P y regardless of the orderings R1, … , Rn of all individuals other than i, where P is the social preference relation corresponding to R1, … , Rn”[1].

As long as we are dealing with democracy, this seems uncontroversial – indeed, it seems definitional that democracy (rule of the people) involves many, as opposed to monarchy. Once one person always determines the ‘social’ ordering – like an absolute monarch – we seem to have left behind the area we’re interested in[2]. There might be reasons why a dictatorship may be advantageous, at least in certain circumstances (e.g. emergency powers), but it is clear that we have left behind the arena of collective choice – or, more precisely, we would be dealing only with one individual making choices for a collective, rather than decisions by a collective. Yet if one individual was to determine the final outcome, there would be no point in asking all the others[3].

The key point to note is that an Arrovian dictator decides for all x and y, that is her preferences determine all issues. One may perhaps want to say that if i can solely determine the choice over a single issue (say, kPil implies kPl even if for all other individuals lPjk) then i is a dictator over that issue, but Arrow does not use the term that way. MacKay introduces the term ‘vicious’ to describe such rules, that allow one person’s preferences to prevail, even over the opposition of all others[4]. But despite this morally-loaded term, it is not clear ‘vicious’ rules are necessarily a bad thing. Indeed, those who accept some form of liberal private sphere or self-regarding rights[5] generally accept that each should be decisive over some set of decisions. In any case, this is not a dictatorship in the technical sense used here. A rule is only dictatorial if one specified individual decides all issues. This is actually very weak – it technically leaves open the possibility that i decides for all x and y, except for k and l.

Two further details remain: Firstly, what if everyone wants a dictator, and secondly how is the conditional implication to be understood? On the first, Arrow says, “the desires of those individuals [can be said to] include a liking for having social decisions made by a dictator or at least a liking for the particular social decisions which they expect the dictator to make”[6]. In Chapter VII, he goes on to briefly consider the possibility of higher order preferences for a decision-mechanism conceived of as a value itself – for example, whatever our preferences between x and y, we might prefer x-arrived-at-democratically to y-arrived-at-in-other-ways. In such circumstances, we cannot rule out that “the desire for a dictatorship or for a particular dictator may be overwhelming”, in which case “our social welfare problem may be regarded as solved since the unanimous agreement on the decision process may resolve the conflicts as to the decisions themselves”[7]. It is, of course, quite possible that all would prefer to defer to an authority they accept, e.g. the pope[8], and this can be understood either as them having a higher order preference for what he decrees, or as them revising their lower order preferences between x and y in light of what he says (‘I must have been wrong, x is better for me after all’). I think Arrow is right to regard such as an acceptable solution to the social choice problem, however. While they may formally appear to be dictatorships – because the pope’s preference over x and y determines the social ordering – this is so only because others (unanimously) accept the authority. As such, it seems a consequence of Pareto that the social choice must conform to these preferences[9].

This brings us to the related point, how one should understand Arrow’s “if there exists and individual i such that, for all x and y, x Pi y implies x P y”[10]. One natural understanding is as a material conditional: it is true iff in all circumstances where xPiy then xPy. This, however, seems to miss something from an ordinary, intuitive understanding of implication – one that, at least implicitly, involves the idea of causation. We think i is a dictator if it is because xPiy that xPy. Conversely, we do not necessarily think there is anything untoward simply because whenever xPiy it is also the case that xPy. Consider the example of an individual with higher order democratic preferences. It may be that she is able to transform her preferences, perfectly and instantaneously, so that whenever xPy (as the democratic outcome of social choice), her preferences are xPiy. In this case, it will also be true that whenever xPiy so also xPy, but this does not seem to be a case of dictatorship. I think the implication Arrow is talking about has to be a stronger sense. It has to be that in any possible world[11] where xPiy then it follows that xPy, which comes pretty close to saying xPy because xPiy[12]. Understanding dictatorship as requiring something stronger, of course, makes the requirement of non-dictatorship on our choice procedure weaker. I think it is clear, however, that – subject to the considerations of the previous paragraph – a social choice procedure must be non-dictatorial, or it is simply not a social (collective) choice at all.

5.8.b Lottery-Voting Assessed

In lottery-voting, a single person is picked to decide a given issue. Since this effectively disregards all other preferences, the rule is what MacKay would call vicious. Despite the fact it is often informally called a ‘random dictator’ method[13], it is not, however, dictatorial in Arrow’s sense. There is no individual whose preferences automatically determine all social choices. While i might be drawn to decide this issue, next time it might be j. The randomness means that, even in the sense in which each is a quasi-dictator, each has an equal chance to be so on each occasion.

Random selection is not, however, taking turns. While over a large number of persons and issues, it is likely power will change hands regularly – even if not everyone gets a turn – we might worry that one individual might receive more than her fair share of power as a result of mere fluke. For instance, suppose we have ten individuals (a-j) and ten issues to resolve. It might be convenient if each individual got to decide one of the issues[14]. If we leave who decides to random chance, however, then it is logically possible that an individual – say, d – may get to decide all ten issues[15]. Does this mean that because, on each issue, xPdy determines the social choice, d is a dictator? No, this is merely a case where d’s preferences happen to coincide with the social choice, but that does not make d a dictator – for d’s preferences would also coincide with all the social choices if there was unanimity over each issue. For d to be a dictator, her preferences have to prevail in all possible worlds. Even if it happens that d’s preferences do determine each and every social choice, it is not the case that they always do so simply because they are d’s. It could have been that someone else’s vote had been drawn on any of the issues. Thus even in this vanishingly rare possibility, lottery-voting does not violate non-dictatorship; for while only one person’s vote ultimately counts, it gives each an equal chance of being the decisive one over any given issue.

[1] Arrow (1963) p.30
[2] T. Pratchett (1987) Mort “Ankh-Morpork had dallied with many forms of government and had ended up with that form of democracy known as One Man, One Vote. The Patrician was the Man; he had the Vote” p.176, fn.
[3] MacKay (1980)
[4] MacKay (1980) p.105.
[5] c.f. Sen ‘Paretian liberal’
[6] Arrow (1963) p.30.
[7] Both Arrow (1963) p.90.
[8] This example comes from Barry (). ‘Authority’ can be understood in terms of Raz (?); but note deference occurs to those people think are authorities, it does not necessarily follow that the chosen dictator is better at decision-making, merely accepted as so.
[9] Arrow (1963) p.74 endorses that “the condition of nondictatorship loses its intrinsic desirability” in certain conditions, e.g. complete unanimity. Thus his objection to dictatorship is not absolute. And, in any case, the ‘random dictator’ of lottery voting is not a dictator in the Arrovian sense, for he holds if an individual is decisive on a pair of issues they will be on all (pp.99-100), but this isn’t so if the decisive individual is randomly determined for each vote.
[10] Arrow (1963) p.30.
[11] I’m worried about the possible world formulation. Of course there are possible worlds where i is not a dictator. I don’t mean to claim that to be a dictator i must be necessarily a dictator. Can I appeal to something like relativised possible worlds?
[12] Note we do not need to require that not-xPiy implies not-xPy, since if not-xPiy then i must have some other preference, e.g. xIiy. (But can we assume one has prefs over all issues?)
[13] e.g. Estlund and others?
[14] Though, of course, different issues might differ in importance to different people; and some may be more concerned simply that they get their way (e.g. x) and not that they get it because it is their way (or their vote is decisive).
[15] The probability is (0.1)10.


At 6:09 pm, Anonymous John Lawrence said...

Hi Ben. There is also the possibility of a proxy vote which I don't think is the same as a dictatorsip. An individual, j, could let some other individual, i, vote for him regarding one or more issues. Presumably the individual j respects i's judgment and considers i to have more knowledge regarding the isssue so he delegates his vote to her. Other individuals could do the same delegating the same or different individuals to vote for them. If it turned out that one individual ended up voting for a majority of the electorate, would she be considered a dictator? I don't think so.

There are also various degrees of dictatorship. You could postulate that if the election turned out to be a tie, one individual would be designated to break the tie. You could call that individual a dictator of the tie. Also let's say that an individual determined the election, xPy iff xPiy as long as not more than 2/3 of the electorate had yPjx. So i would be a dictator unless 2/3 of the electorate voted against his choice. In that case yPx despite the fact that xPiy. Any fraction other than 2/3 could be used as long as it was greater than or equal to 1/2. Presumably, if more than 1/2 the electorate voted in the same way as the dictator, he need not be considered a dictator.

At 2:43 pm, Blogger Ben said...

Hi John

"If it turned out that one individual ended up voting for a majority of the electorate, would she be considered a dictator? I don't think so."

Remember, a dictator is only a dictator if they decide over all issues. Therefore j would have to let i decide for him over all issues, which isn't particularly likely - other than if you see representation on this (trustee) model.

If j does decide that i can decide all issues on his behalf, however, I think you could well consider it a dictatorship, in which others have a higher-order preference for a dictator. I think this would be normatively ok, however.


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