Praesidium

Friday, November 04, 2005

Arrow's Theorem 2 - Evaluation

Here's the second part...

Evaluation
Since Arrow’s impossibility theorem is a proven theorem, we can’t say it’s wrong. It can be taken as showing that there’s no such thing as a perfect SWF or choice procedure[1], if we accept Arrow’s four conditions. We can, however, question or weaken these axioms.

Non-dictatorship is also relatively uncontroversial, so long as we are dealing with democracy. 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].

Arrow’s Pareto requirement is so weak that it doesn’t seem objectionable. If everyone unanimously prefers x to y, it would be very strange for this not to be society’s ordering. This, however, assumes that the only way for a social outcome to be better is for it to be better for the people in it. One might argue a situation can be better, though better for no one – for example, by being more equal or in keeping with desert[3]. It’s very hard for a social decision mechanism (as opposed to betterness ranking) to take such factors into account. It would seem to incorporate citizens’ ‘external preferences’ – i.e. those they have over how well off others are. For practical purposes, it seems we must assume if everyone is better off then the outcome is socially preferred, even if it might not be from the external viewpoint of an impartial spectator.

More plausible lines of attack might be found against Arrow’s independence or universal domain requirements.

The latter seems obviously 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]. In fact, however, this allows some seemingly unlikely or ‘irrational’ preferences. In particular, it permits individual inputs to be intransitive[5], thus it is hardly surprising that we have difficulty producing a transitive social ordering, even if there was total unanimity of preference orderings!

Further, liberals have argued 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). Perhaps another possibility is that prior deliberation, bringing out everyone’s preferences and concerns, will allow us to reach acceptable compromises and in effect restrict the domain. Greater homogeneity of preferences makes Arrovian results far less likely[6].

The other option is to weaken the independence of irrelevant alternatives requirement, as we spoke about two weeks ago (Borda vs. Condorcet). It might be fair enough to suppose that choice between x and y should depend only on their relative merits not on, for example, z. This, however, rules out methods like the Borda count that use such information as a proxy for intensity. Arrow restricted information to ordinal rankings out of a scepticism about interpersonal comparison[7]. Thus Sen argues, “the impossibility can be seen as resulting from combining a version of welfarism ruling out the use of non-utility information [see above comments on Pareto] with making the utility information remarkably poor (particularly in ruling out interpersonal comparisons)”[8].

I would contend the dogma that interpersonal comparisons are meaningless or impossible is fairly obviously false. Think of someone you know who’s happy – who has a job they enjoy, plenty of money, a loving family, etc. Now think of a starving, abused orphan child somewhere in the Third World. Ask yourself who has higher utility… Admittedly, comparisons are to a certain extent subjective, and necessarily vague. You certainly can’t quantify how much better off the former is than the latter, but it is still possible to make comparisons.

Sometimes when a small group are making decisions, they can take into account that a certain course has more effect on one than another. This might apply, for example, in a group of three flatmates, or Brian Barry’s example of five people in a train carriage, trying to decide whether or not it allows smoking – if one is asthmatic, his interest might hold sway even if numerically out-voted[9]. This is democratic – it treats each equally, but in doing so recognises inequalities between them.

So it seems sometimes egalitarian decision procedures can operate, on the basis of either a narrower range of preferences than Arrow allows, or if there is consensus over intensities. These are not, however, conditions that necessarily hold when it comes to decision-making in a large, heterogeneous democratic state. What can we do then?

Arrow’s result demonstrates an impossibility. If we find all his axioms normatively compelling, we will be disappointed to learn that we cannot satisfy them together. The search now is not for a ‘perfect’ system, but the best we can do.

In this case, I offer lottery-voting. This satisfies U, P and I, as any preferences can be expressed, and the chance of any option winning depends on the number of votes it gets. This could be called a ‘random dictator’ model, suggesting that it fails condition D. It is, of course, the case that just one person preferring y to x could be picked, yielding such a result against everyone else’s opposition (note this satisfies weak Pareto, as that’s a unanimity requirement). Such random or changing dictatorship avoids what I think is the serious problem of one person always determining the social outcome, however.

The problem with lottery-voting is that it doesn’t produce a stable welfare function. One can get different results from the same preference profiles on different occasions, and intransitivities or inconsistencies in results. Shepsle and Bonchek comment that “There is, in social life, a tradeoff between social rationality and the concentration of power”[10]. While this might be an objection to individual rational choice, it’s not so clear we should expect such consistency from social choices, as societies are essentially pluralistic, not single actors. As Mueller notes, “Although obviously arbitrary, the general popularity of random decision procedures to resolve conflictual issues suggests that “fairness” may be an ethical norm that is more basic than the norm captured by the transitivity axiom for decisions of this sort”[11].

[1] E.g. P. Samuelson (1977) Collected Scientific Papers IV “what Kenneth Arrow proved once and for all is that there cannot possibly be found… an ideal voting scheme. The search of [some] great minds of recorded history for the perfect democracy, it turns out, is the search for a chimera, for a logical self-contradiction” pp.935, 938. Quoted G. Mackie (2003) Democracy Defended p.10.
[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] 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.
[4] Riker (1982) p.117.
[5] See D. Saari (1994) Geometry of Voting p.327.
[6] 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.
[7] G. Mackie (2003) “Historically, Arrow’s theorem is the consequence of noncomparabilist dogma in the discipline of economics, that it is meaningless to compare one person’s welfare to another’s, that interpersonal utility comparisons are impossible” p.8.
[8] A. Sen (1982) p.330.
[9] B. Barry ‘Is Democracy Special?’ in his (1991) Democracy and Power p.38. This fits Fleurbaey’s argument for democratic votes to be weighted according to the interests one has at stake. Note, however, that even if we can agree the relative interests people have at stake, we don’t necessarily want to accord with the person who has most at stake. The convicted criminal, for example, may lose more for going to prison than any of us – individually or collectively – gain by his doing so. (But then, this is not simply a democratic decision; it brings in independent requirements of justice).
[10] K. Shepsle & M. Bonchek (1997) Analyzing Politics: Rationality, behavior and institutions p.67. Quoted in G. Mackie (2003) p.14.
[11] D. C. Mueller (2003) Public Choice III p.588.

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