PhD_03
Chapter
3: Arrow's Impossibility Theorem
Links to the other chapters:
Title
page, dedication, contents, key words, abstract and acknowledgements
Chapter 1:
Introduction: Holism versus reductionism in economic thought
Chapter
2: the Prisoners' Dilemma
Chapter
3: Arrow's Impossibility Theorem
Chapter
4: The Invisible Hand of God in Adam Smith
Chapter
5: Friedrich Hayek: a Panglossian evolutionary theorist
Chapter
6: Keynes’s methodological standpoint and policy prescription
Chapter 3 Arrow’s Impossibility Theorem[1]
3.1 Introduction
I wrote in the previous chapter that the prisoners’ dilemma had been perceived as a challenge to the Smithian ‘invisible hand theorem’ – the view that individual rationality of economic agents will, in general, lead to collective rationality at the level of the economy as a whole. I also argued that the attempt to explain away the problem, by attributing it to a mistaken extension of the concept of rationality from the individual to the collective sphere, was ultimately untenable. In this chapter attention is turned to the problem of Arrovian impossibility, and, in particular, to the criticism of Arrow by libertarian thinkers. Taking IMD Little as a principal exponent of this view, a parallel is drawn between his rejection of the notion of a social welfare function and the rejection, by writers such as John Searle, of the possibility of artificial intelligence. As in Chapter 2, a principal reference point is the collection of writings on Rational Man and Irrational Society edited by Barry and Hardin (1982).
The background to the debate over the Arrow impossibility theorem is as follows. Consumers have preferences over the alternative consumption bundles with which they are faced. Modern microeconomics is based on the idea that everything we need to know to discuss consumer behaviour can be obtained from ordinal rankings by the consumers of these alternative consumption bundles without any need to know how much they enjoy a bundle, or how much more they enjoy this bundle than that. An older, cardinal, theory required that, in principle, these levels and differences in utility were measurable. The liberal tradition had for long held that alongside all the particular interests in society, there was a general interest, the interest of society itself. With the cardinal utility theory this conception of a general interest received an obvious interpretation: Benthamite utilitarianism said just add up the utility levels of all the individuals in society and that total represents the welfare of the whole of society. Then any policy can be judged according to how it increased or decreased that total of social welfare. Significant difficulties with the measurement and interpersonal comparison of utility levels led to the abandonment – for the most part – of cardinal utilities and the adoption of the weaker, and hence more robust, ordinal theory. Although it was now no longer obvious how to construct a social welfare function (hereafter SWF), it was assumed that this was merely a technical problem and that the task could still be accomplished in principle. Arrow, however, in a number of publications, in particular the two editions of Social Choice and Individual Values (1951, 1963), showed that it was in principle impossible to derive any SWF purely from individual rankings of social alternatives which satisfied certain elementary criteria, such as consistency and non-dictatorship. This was a great shock and, indeed, it would not be an exaggeration to say that some theorists despaired at this result. Plott sets the tone here in Reading 12 (of Barry and Hardin, 1982):
“The subject began with what seemed to be a minor problem with majority rule. ‘It is just a mathematical curiosity’, said some ... But intrigued and curious about this little hole, researchers ... began digging in the ground nearby ... What they now appear to have been uncovering is a gigantic cavern into which fall almost all of our ideas about social actions. Almost everything we say and/or anyone has ever said about what society wants or should get is threatened with internal inconsistency. It is as though people have been talking for years about a thing that cannot, in principle, exist ... “ (231-32)
As the editors say, in their Introduction to the reading, ‘Plott maintains that the Arrow results, and others like it in the theory of social choice, undermine the whole tradition of liberal or individualistic political theory that has been developed in the last few centuries’ (230).
A major purpose of this chapter is to evaluate this response.
3.2 The conditions
Arrow’s theorem says that no social choice procedure can exist which satisfies the five conditions O, U, P, D and I. The argument proceeds via a reductio ad absurdam: we assume that all the conditions hold and then demonstrate that they lead to a contradiction. Relaxing any of the conditions removes the impossibility of deriving a SWF, but, in general, such SWFs will exhibit perverse features. The conditions are expressed in many different ways in the literature and what follows is based on Barry and Hardin’s Introductory essay to Part II. It should, however, be noted that the conditions specified in this form are not genuinely primitive: condition O, for example, contains criteria of both uniqueness and transitivity. Transitivity by itself could be expanded into two (or more) conditions. The uniqueness criterion is replicated in Condition I, rendering the latter partly redundant.
Condition O
The SWF is a unique Ordering of the alternatives facing society based only on individual orderings. An ordering is a consistent ranking. In particular this implies transitivity for both preference and indifference. If society prefers x to y and y to z, then it prefers x to z, and so on.
Condition U
The social choice rule must have Unrestricted domain: it must work for every logically possible combination of individual orderings.
Condition P
The social choice rule must be Pareto-efficient: if one individual prefers x to y and all other individuals either prefer x to y or are indifferent between x and y, then the SWF must prefer x to y.
Condition D
There must not be a Dictator, that is, a person whose preference of x for y is always (in every logically conceivable constellation of preferences) the social preference, for any x and y, regardless of the preferences of others.
Condition I
The social ordering of any pair of alternatives x and y is a function solely of the individual orderings of x and y: it is Independent of irrelevant alternatives – individual orderings of x and z, for example.
3.3 Proof of the theorem
One further concept is needed: that of decisiveness. If a group or individual is decisive over x and y, and prefers x to y, then society prefers x to y, whatever anyone else’s preference may be[2]. We also assume that society consists of a finite number of individuals. The proof then proceeds by showing that (a) if there is an SWF which satisfies conditions O, U, P, and I (that is, all except non-dictatorship), then for some constellation of preferences there must be a decisive individual, and (b) if there is a decisive individual then he is a dictator.
Consider any pair of alternatives, x and y, where society prefers x to y. It cannot be the case that everyone in society prefers y to x, by condition P (Pareto). There must be a set of decisive individuals. If only one person prefers x then the set only contains one person; if everyone prefers x then the set of all the individuals in society is decisive. Normally, it will be a set of intermediate size, but that is irrelevant. There will thus be a (non-empty) decisive set for each pair of alternatives where society is not indifferent between the two. Consider the set of all of these decisive sets. From the assumption that society consisted of a finite number of individuals, this set of decisive sets must have a smallest member, or a subset of equally large smallest members, in which case we pick any member of this subset. We can show that there must be a possible pattern of preferences for which this smallest set of decisive individuals has only one member.
We will suppose initially that this smallest decisive set, V, has more than one member, and show that this leads to a contradiction. Suppose V is decisive over x and y, and that it (and hence society – because V is decisive) prefers x to y. This must lead to a contradiction. Since it consists of more than one member we can divide it into two parts, one, V1, consisting of one member and the other, V2, consisting of all the other members of V. We also give the name V3 to the set of all the members of society not in V. Condition U, unlimited domain, tells us that the SWF must work for any logically possible pattern of preferences. So we can pick any pattern of preferences we like. Suppose the pattern of preferences is that for V1, x > y > z[3], for V2, z > x > y, and for V3, y > z > x. For convenience of reference this information is set out in Table 3.
Table 3
Preferences of the three sets of agents, V1, V2 and V3
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Now, we know that V is decisive over x and y, so for society x > y. this is shown in the column headed S. But where does society rank z? Suppose society preferred z to y. Only V2 prefers z to y – both V1 and V3 prefer y to z – so that would make V2 decisive. But V2 is one person less than V, the smallest decisive set, so that is not possible. So society either prefers y to z (Column S) or is indifferent between y and z (Column S'). Hence society must prefer x to z, given x > y and y ³ z, by transitivity (condition O). These two alternatives (x > y > z, and x > y = z), are shown in columns S and S', respectively, of Table 3. But now V1 is decisive since both V2 and V3 prefer z to x. However, V1 consists of only one person, so the assumption that the smallest decisive set, V, consisted of more than one person turns out to be self-contradictory. We have shown, therefore, what was required, that there is a pattern of possible preferences such that there is a decisive individual. That completes the first part of the proof.
The second part of the proof shows that a decisive individual is a dictator. Suppose A is decisive for x against y and that he also prefers x to z. Also suppose, invoking condition U, that everyone prefers y to z. Condition P says that society prefers y to z. If every individual prefers y to z, it would certainly be Pareto-inefficient for the SWF to prefer z to y, or even to be indifferent between them. Hence, x > y > z for society and so, invoking transitivity, x > z. But condition I, independence of irrelevant alternatives, says that the social choice between x and z is independent of individual preferences over y. Hence, regardless of others’ preferences, x is preferred to z by society if and only if A prefers x to z. Hence, if A is decisive for x against y, we can replace y by any other alternative which A finds less desirable than x. Similarly we can replace x by any alternative which A finds more desirable than y. Hence for any possible pair of alternatives, A’s preference is decisive: he is a dictator. But the first part of the proof showed that there was a possible set of preferences such that there was decisive individual. Hence, for this pattern of preferences there is a dictator. But this violates condition D, of non-dictatorship. The concept of an SWF which simultaneously satisfies all five conditions is therefore inconsistent.
What this shows is that there is a possible pattern of individual preference orderings such that a social ordering derived from them which satisfies the Pareto and independence conditions must be dictatorial if it is to be consistent.
3.4 Scope for relaxing the assumptions
All of the assumptions mentioned are invoked in the proof, so relaxing any will make it possible, in principle, for us to construct a SWF. However, the conditions are generally regarded as quite minimal so the resulting SWFs are unlikely to be attractive. If we are prepared to accept a dictator, for example, there is no problem in constructing an SWF – but now the SWF has nothing social about it. The SWF abdicates before the task of aggregating individual preferences. Again, recall that we commenced the proof by considering any pair of alternatives, x and y, where society prefers x to y. If we are unable to do this, because society is indifferent between every pair of alternatives, then we can derive a SWF, which, however, is completely vacuous: x = y for all x and y. This is going to the opposite, but equally useless, extreme.
We assumed that society consisted of a finite number of individuals, and if we relax this assumption then the proof fails, for there need not be a smallest decisive set. Or, to put it another way, if the smallest decisive set were infinitely large, removing one member would not leave a residue smaller than the original set. Again the proof would fail. Since actually existing societies consist of a finite number of individuals, this is scarcely helpful. To consider infinite societies we would have either (a) to consider ‘individuals’ not to be individual at all but rather infinitely subdivisible, or (b) to regard the individuals composing society to be not just those currently living but also the unborn and/or the dead – and even that would be problematic. In neither case would it be possible to establish individual preferences, let alone aggregate them.
SWFs violating condition P or transitivity are no more attractive than those already considered. An SWF capable of choosing Pareto-inferior outcomes would clearly not be maximising society’s welfare, which would contradict the notion of an SWF. Violations of transitivity would also lead to irrationality at the macro level. Consider the set of preferences presented in Table 3, which was used in the proof of the impossibility theorem. This set will play a significant role in the sequel, when we turn to the ‘Paradox of Voting’. Condition U says that our SWF must apply to this set. Suppose that we examine the three pairs of alternatives, x and y, x and z, and y and z, in turn, and adopt a simple majority voting rule. We will obtain an SWF in which x > y > z > x, in each case by a two-to-one majority. These are referred to as cyclical majorities. Note carefully that this does not say that society is indifferent between the three outcomes – which might be unhelpful but would not be inconsistent. What it says is that every outcome is preferred to both alternatives. Permitting intransitivity means that such instances cannot be excluded. In a formal system, once you can prove a contradiction you can prove any statement. So the contamination of irrationality immediately spreads beyond these three alternatives: every possible alternative will be preferred to every other. This is worse than the violation of condition P.
Can we make more progress by relaxing condition U? Considerable research effort has gone into attempts to obviate the impossibility result in this direction. For example, we could require the SWF only to work where preferences are single-peaked. This would eliminate the possibility of patterns of preferences such as that in Table 3, since, as can be seen from Figure 1, below, which plots rank in the individual orderings against outcome for the three sets of actors, V1, V2 and V3, the set of individuals composing V2 were assumed to have dual-peaked preferences. Since the proof depended on this pattern of preferences, it will not hold if such preferences are not in the domain to be considered. However, the resulting SWF will have nothing to say about the cases where preferences are not single-peaked. Now it is well-known that the latter, far from being an exotic theoretical possibility, is a practical problem of pandemic proportions. No SWF of any interest can simply remain silent on these cases. One way out would be to say that society is indifferent between all the outcomes in a preference cycle. This is referred to as ‘taking the transitive closure’ of the preference relation.
Figure 1
‘Peakedness’ of preferences for the three sets of agents, V1, V2 and V3
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In the proof of the impossibility theorem, the three groups of agents were assumed to have preferences as shown above. The vertical axis shows the ranking the individual ascribes to each of the three outcomes, x, y and z, with 1 indicating the most preferred outcome.
This procedure immediately runs into two major problems: (a) it violates yet another assumption, condition I, and (b) by doing this we have in any case restored condition U and, with it, impossibility. The point here is that it is not possible to tell whether there is a preference cycle without checking preferences between other alternative pairs than the one under consideration. We cannot decide society’s preference between x and y without also knowing that between x and z, and y and z. The social ordering of x and y is no longer independent of ‘irrelevant’ alternatives. Turning now to the second problem, condition U could be satisfied by a rule for one set of circumstances together with a rule for the remaining circumstances plus a rule for deciding when to apply which rule. That is exactly the situation which now obtains. Let us rehearse the relevant part of the proof of Arrow’s impossibility theorem, substituting the new rule.
Suppose that the smallest decisive set, V, has more than one member, and that it is decisive for x against y. This must lead to a contradiction. Since it consists of more than one member we can divide it into two parts, one, V1, consisting of one member and the other, V2, consisting of all the other members of V. We also give the name V3 to the set of all the members of society not in V. Condition U, unlimited domain, tells us that the SWF must work for any logically possible pattern of preferences. We first of all have to decide whether preferences are cyclical or not and then either treat as normal or invoke the new transitive closure rule, as appropriate. Suppose, as before, that the pattern of preferences is that for V1, x > y > z, for V2, z > x > y, and for V3, y > z > x (see Table 3). Clearly we do here have cyclical preferences so we invoke the revised rule and say that society is indifferent between x, y and z. But this contradicts our assumption that V is decisive for x against y. Hence the initial assumption, that the smallest decisive set contained more than one member, turns out to be contradictory. So there is still a pattern of preferences for which there is a decisive individual and, therefore, also, a dictator. The proof still goes through.
Condition I (together with the uniqueness criterion associated with it – and with condition O) has, of all the five conditions, provoked the most controversy, not to say confusion, in the literature. The reason is, perhaps, that, at first blush, is seems utterly counter-intuitive to exclude the non-uniqueness which goes with cardinal individual preferences, and the dependence on ‘irrelevant’ alternatives which goes with Borda-type systems. One’s instinct is to feel that if a SWF can work with relatively information-poor ordinal preferences, then, surely, it should work that much better with relatively information-rich cardinal preferences. Again, Borda systems, which (pace Barry and Hardin p219) are neither purely ordinal nor cardinal, are more information rich than purely ordinal systems. It seems perverse to rule out such procedures ab initio.
To take this view, however, is to lose sight of what Arrow is trying to do. The point is to demonstrate the possibility or impossibility of building SWFs on the basis of purely ordinal individual preference orderings. It is no good showing that such an SWF is possible (or at least not proven impossible) if that possibility was due entirely to leaving the door open for cardinal preferences. That would defeat the object of the exercise. The SWF is impounding into the social ordering some extra information – either an arbitrary element that has nothing to do with social welfare or information about individuals’ cardinal preferences. Either way, for the ordinalist project to succeed, such influences must be excluded. Again, the Borda system allows agents to reveal not just their ordering of two alternatives but also some information about the intensity with which that preference is held. Preferences over third, fourth, etc, alternatives are not ‘irrelevant’ and cannot be ignored. We have some grounds – albeit inconclusive – for believing that the individual who ranks x and y tenth and twentieth respectively, holds his preference for x over y with greater intensity than another individual, who ranks them 16th and 15th, respectively, prefers y over x. Borda impounds this partial evidence on cardinal preferences. Again, if the SWF is only possible because Borda has not been excluded, the ordinalist project fails. It is therefore essential for the conditions O and I to be retained. Dilution here spoils the whole point of the exercise.
3.5 The
libertarian response
We have already seen the seriousness with which many took the Arrow results. Plott, in Reading 12, ‘Axiomatic social choice theory: an overview and interpretation’, spells out the breadth and depth of its consequences, on his interpretation. To understand the extent of the concern Arrovian impossibility has stimulated, we need to get a feel for the scope of the alternatives and processes that Arrow’s theorem is about.
“An option or social alternative,” Barry and Hardin explain, “could be a complete description of the amount of each type of commodity, the amounts of various types of work done by each individual, the production level of each firm, the type of government agencies and the services provided by each, etc ... The set of feasible options could be a consumption possibilities set ...” (244)
As for the processes which might exist to make choices between the alternatives society is faced with,
“the process could be a competitive process, a capitalistic process, a socialistic process, or any other kind of process. There is no need, for example, for the process to be directed in that some judge, administrator, or planner uses the defined social ranking to determine the best option and then directs its implementation. The process could be any type of game, voting process, market process, political process ...” (244)
The very comprehensiveness of the Arrow results has forced some writers into a fundamental re-evaluation. Plott states the case for abandoning the very notion of social preferences:
“Some, like myself (Plott 1972)[4] would claim that the concept of social preference itself must go. Buchanan (1954a, b)[5] was right in his original criticism of Arrow, that the concept of social preference involves an illegitimate transfer of the properties of an individual to the properties of a collection of individuals. For me, the Arrow theorem demonstrates that the concept of social preference involves the classic fallacy of composition, and it is shocking only because the thoughts of social philosophers from which we have developed our intuitions about such matters are subject to the same fallacy.” (242)
‘In order to see how extreme this position is’, Plott continues, we should investigate the ways in which this fallacious concept of social preference continually dogs our thought about our social environment. The idea that ‘No one really accepts or uses the idea of a social preference’ is completely wrong, he says:
“many commonly used concepts are equivalent to the concept of a social ranking ... concepts like social needs, group wants, etc. These are simply expressions of priorities and are thus rankings of options ... Take for example the concept of economic welfare. To different options one attaches a number ... indicating the level of welfare. Certain forms of cost-benefit analysis are attempts to operationalise such a formula. But indicators of social welfare clearly imply a ranking of social options according to the numbers which indicate the levels of welfare. The ranking satisfies all of our principles of social preference, and thus the [Arrow] theorem stands as a criticism of any such formula. The only admissible definitions of welfare are those which are dictatorial.” (244-45)
This is incorrect. It is not ‘the concept of economic welfare’ per se that the impossibility theorem rules out, but comparisons of alternative states of economic welfare based solely on ordinal information about individual agents’ preferences.
Lest the significance of Plott’s final remark about dictatorship should escape the reader, we will see how the theme is developed in Reading 14, ‘Social choice and individual values’ by IMD Little. Little argues that there cannot be any such thing as a social preference (in the sense of a preference on the part of society rather than a preference of an individual about society), a ‘scientific ethic’, or an ‘objective moral code’. ‘We can have an “objective” ethic only when there is universal agreement ... when all moral judgements will [in any case] have become futile and redundant’ (279).
So Arrow is chasing a chimera. ‘The very use of the phrase ‘social welfare function’ suggests ... if such were possible ... an objective moral code’ (278). Not only is the idea false; it is much worse: it is an ideology containing the seeds of dictatorship.
Society, in Little’s view, is riven by the clash of antagonistic interests. Philosophers have interpreted this antagonism to be merely a superficial appearance masking a more essential harmony of interest. This general interest is then identified with the state and used to delude individuals into conspiring against their own freedom to pursue their individual interests. On the one hand we have an identification between the idea of the general interest and Arrow’s conception of an SWF, and on the other between the idea of the general interest and dictatorship. Hence Little is able to locate Arrow in an intellectual tradition leading from Rousseau via Hegel to modern totalitarianism:
“Political philosophers used to worry over the question ‘Why ought I to obey the state (or society, or the general will)?’ The pseudo-puzzle of how one can both be free and be subject to law is a variant of the pseudo-puzzle of how duty and self-interest can be reconciled. The philosophers tried to show that it was always really in one’s own interest to abide by whatever the ‘objective’ code might be. Since, plainly, conflict must arise, at least in the absence of complete initial consensus, and since such consensus was obviously absent, they invented the doctrine of a metaphysical consensus. When people actually (ie really) disagreed about some matter affecting the common interest, they were really (ie metaphysically) agreeing. Rousseau ... was first responsible for this nonsense ... ‘Each of us puts his person and all his power under the supreme direction of the general will.’ (1950: 15)[6] ‘Whoever refuses to obey the general will shall be compelled to do so by the whole body ... he will be forced to be free.’ (ibid: 18) ... It is but a very short step from here (a step which Hegel took) to maintain that acceptance of the social order (or obedience to the state) is really only self-obedience. Arrow’s problem clearly has so much in common with Rousseau’s that it seems worthwhile to point out again the insidious danger of this approach. Modern totalitarian philosophy may be not altogether unjustly fathered on Rousseau.” (278-79)
Little is far from being isolated in these views. We have already seen how close they are to Plott’s. And Barry and Hardin, in the Chicago tradition of political economy, share the view that ‘collective rationality’ is a meaningless term – an illicit extension of the concept of rationality beyond its proper domain. While Little tars Arrow with the brush of totalitarianism, Barry and Hardin subject him to criticism of a severity bordering on the intemperate in their Introduction to Reading 13, by Arrow. Arrow’s views are, they say, ‘extraordinary’, ‘grotesque’, ‘bogus’, ‘clearly ... once again, absurd’ and ‘an illusion’ (249-51).
It is Little, however, who can be taken as a representative of a trend, and it will therefore be worth examining his views in more detail. Later in this thesis, in the chapter on Adam Smith, we shall see that the stick Little uses to beat Arrow, the claim, as Pope puts it, that ‘all discord, [is] harmony not understood’, is far more aptly turned on Smith himself and his followers.
3.6 Little: the argument against the existence of an SWF
Perhaps the first point to make is that Barry and Hardin misunderstand the context of the debate. With cardinal utility theory it was straightforward – in principle – to construct an SWF: just add up all the individual utilities. With ordinal utility this was no longer obvious, and the next step was simply to show whether it was or was not possible to construct an SWF within this paradigm. Arrow showed that it was not. Barry and Hardin, Little and Plott ignore this. Thus, for Barry and Hardin,
“[t]he single most penetrating and fundamental statement in the article by Little comes ... when he says ‘that it is foolish to accept or reject a set of ethical axioms one at a time. One must know the consequences before one can say whether one finds the set acceptable ...’ ... Nobody has any immediate views about the desirability of, say, the independence of irrelevant alternatives, and we should refuse to be bullied by a priori arguments to the effect that would be ‘irrational’ not to accept it.” (265-66)
But this is to ignore the purpose of Arrow’s work. The acceptance of condition I, as I have argued above, is essential if we are to carry out Arrow’s research agenda. If we leave it out and we find that an SWF is not in principle impossible, but that possibility is in fact a consequence of the possibility of obtaining more than bare ordinal information about individual preferences, then we simply have not answered the question we set out to address. It has nothing whatsoever to do with the ‘ethical desirability’ of condition I. A similar argument applies to the other four axioms. They do not need to be accepted ‘one at a time’ – whatever that might mean – they have to be taken all together or Arrow simply hasn’t answered the – perfectly legitimate – question he has set himself.
A primary objective of Little’s paper is to distinguish between two interpretations of an SWF: (a) as a social decision procedure (legitimate), and (b) as a judgement about social welfare (illegitimate). He criticises Arrow on two grounds (a) to the extent that Arrow’s theorem is an objection to social decision procedures, Little complains that it is too extreme – viable social decision procedures can and do exist, and (b) to the extent that an SWF is considered as a social preference, which according to Little cannot in principle exist, it is, he thinks, misleading and dangerous to talk about it as though it could.
Little’s major point, therefore, is to argue that a SWF, if it means anything at all about preferences, merely reflects someone’s opinion about what is good for society, not what the society wants:
“the so-called ‘social welfare function’, postulated by welfare economists, should on my view be regarded as a social ordering only in the sense that it orders states of society ... Instead of writing, with Bergson, W = W(U1, ..., Un), we can write Wi = Wi(U1, ..., Un) (i = 1, ..., n). There is no need ... to introduce a further (social) welfare function of the form W = W(W1, ..., Wn). We can deduce the whole effective corpus of welfare economics from, say, W10 = W10(U1, ..., Un) – remembering only to put ‘in the opinion of individual No. 10’ after ‘welfare’ whenever we use the term.” (272)
Because each individual ordering of the social alternatives is now to have the status of a candidate social ordering – a possible ordering for society – Little must now address the issue of whether, and to what extent, the welfare of others is impounded into one’s own ordering:
“In the preceding section it was supposed that each individual ordered states solely according to what he himself would get ... But of course there is no need to suppose this ... In fact, quite generally, we may suppose that they arrange all states in order of what they regard as ultimate desirability, taking everything they know and feel into account ... The problem, then, is to form a ‘master’-ordering; in Rousseau’s language the problem is the well-known one of discovering the general will.” (275)
Actually, it is completely irrelevant, in the aggregation procedure assumed by Arrow, whether individuals take other individuals’ welfare into account or not. The SWF is a purely formal procedure aggregating individual preferences without regard to their construction or content. Indeed, if we look at the outcome, there is a potential perversity here. Suppose there are two individuals, A and B, with their own sets of preferences and that the possible outcomes are points in n-dimensional space (where n is the number of issues), designated by a, b and s, representing the two individual interests and the general interest respectively. The Arrow impossibility theorem says that no ordinalist procedure can guarantee to find s. Suppose that we have a constellation of preferences such that an SWF is, in fact, able to find s – the theorem by no means prohibits this. If A and B represent their preferences accurately the outcome is s. If, however, one party, say B, instead of representing just his own preferences, puts forward a compromise between his own and A’s preferences, the outcome will be somewhere between s and a. B is penalised for being ‘reasonable’. By taking others’ interests into consideration, B is left worse off and, possibly, there is a loss to society as well. On the other hand, if B were to misrepresent his preferences as being further away from A’s than in fact they are, he would be rewarded for this by drawing the social outcome further towards b.
Little makes use of his assumption that individual orderings impound opinions about what is good for society, as we shall see. He points out, quite fairly, that an Arrovian SWF must be a formal procedure and hence capable of being mechanised. He asks us to imagine a machine into which we feed all the individual preferences which the machine aggregates according to the formula in order to output a social preference printed on a card. This result could not, he says, consistently be accepted by anyone whose own ordering differed from it. To the potential objection that the individual orderings are just that, individual orderings, and therefore that anyone may accept the SWF as such although it diverge from their own preferred ranking, Little responds ‘[t]his objection does not, however, apply in the case under consideration, because in this most general version it is presumed that the individual orders take the welfare of others into consideration’ (276), which, as we have just seen, is simply incorrect.
“Now if I think x is better than y,” continues Little, “and the machine announces that y is better than x ... I can say ‘to hell with the machine; x is better than y whatever it says.’ If I always did this (and how could I do anything else, since, remember, we are supposing that all other people’s values are already known; indeed I have taken into account everything I think significant!) then I should naturally have refused to accept the condition of non-dictatorship. It is in the nature of value judgements that the only order which I can fully accept is one that coincides with my own, regardless of the orders of other people. In other words, no one can consistently accept the condition of ‘non-dictatorship’.” (277)
There are two remarks to be made about this passage. Firstly, there is an ambiguity here about the meaning of ‘accepting’ an SWF – between (a) adopting it as one’s own ordering, and (b) understanding that it is society’s ordering. The only order one can ‘fully accept’, that is, adopt as one’s own, is one that already coincides with one’s own ordering. But one is not asked to ‘fully accept’ the SWF, simply to understand that it is the social ordering. This Little cannot understand, since he is unable to conceive of a social ordering in the first place. The second point is that the last sentence, and, indeed the whole drift of the article, is that, if they are to be consistent, everyone must wish to be a dictator. This merely shows that the logical consequence of consistent individualism is solipsism. It is also somewhat ironic, given his remarks linking Arrow and Hegel with totalitarianism.
Let us return to the argument about mechanising the SWF. Little begins by citing Arrow on the benefits of the axiomatic procedure: ‘one of the greatest advantages of abstract postulational methods is the fact that the same system may be given different interpretations’ (Arrow, cited in Little, 275). But Little says that by interpreting his decision-making process as a SWF he has given it a ‘nonsensical interpretation’:
“Imagine the system as a machine which produces a card on which is written ‘x is better than y’ or vice versa, when all individual answers to the question ‘is x better than y?’ have been fed into it. What significance can we attach to the sentence on the card, ie, to the resulting ‘master’-order? First, it is clear that the sentence, although it is a sentence employing ethical terms, is not a value judgement. Every value judgement must be someone’s judgement of values. If there are n people filling in cards to be fed into the machine, then we have n value judgements, not n + 1. The sentence which the machine produces expresses a ruling, or decision, which is different in kind from what is expressed by the sentences fed into it. The latter express value judgement; the former express a ruling between these judgements. Thus we can legitimately call the machine, or function, a decision-making process.
“But what would it mean to call the machine a social welfare function? One would be asserting, in effect, that if the machine decided in favour of x rather than in favour of y, then x would produce more social welfare than y or simply be more desirable than y. This is clearly a value judgement, but it is, of course, a value judgement made by the person who calls the machine a SWF. Thus, in general, to call the machine a SWF is to assert that x is better than y whenever the machine writes the sentence ‘x is better than y’. Now we may suppose that the individual who calls the machine a SWF is one of those who has fed his own value order into it. It is clear that this person must be contradicting himself unless the ‘master’-order coincides with his own ordering ... In other words it is inconsistent both to call the machine a social welfare function and to accept the condition of non-dictatorship.” (275)
I have already commented on the solipsism Little expresses in the final sentence of this passage. He is simply incapable of raising his eyes above the limited horizon circumscribing ‘the individual’. We have also seen how vacuous is Little’s assertion that anyone accepting an SWF which was not identical with his individual ordering would be acting inconsistently. He is not asked to accept it instead of his individual preferences, merely to understand that it is what society prefers. But what is interesting about the passage is the assertion, as something so obvious, once stated, that it needs no supporting argument, that society cannot form and hold a preference. Little says that there are n preferences not n + 1. In fact, there are n + 1 preferences, considered formally: n personal preferences and one social one. The latter cannot really be considered as extra, however: it has more the relationship of a whole to its individual parts.
What is interesting here is the privileging of the level of the individual person and the denial that the society or the machine or, in general, the system, can have preferences or intentions. This is very reminiscent of John Searle’s denial – particularly in his notorious ‘Chinese room’ thought experiment – of the possibility of machine ‘intentionality’. It is to Searle’s standpoint we now turn, in the next section, in order to shed additional light on Little’s argument.
3.7 Searle and Little
In setting up his by now well-known Chinese Room thought experiment, Searle’s objective is to attempt to discredit artificial intelligence (AI) theory and the computational theory of mind. An AI program, for example one for reading stories in Chinese and answering questions about them, is a formal procedure and can be written down in English. Searle, who knows no Chinese, asks us to imagine him in a sealed room with such a program (he doesn’t know what it is) written down in the form of a book of rules. Chinese texts – a story and then questions about it, are passed under the door, although Searle has no knowledge of what they are. He manipulates the symbols on the paper according to the instructions and passes the results back to the Chinese people on the other side of the door. Unknown to him they are answers to the questions about the stories. They are good enough to pass the Turing Test, that is, to convince his Chinese audience at least 50 per cent of the time that the room contains a native-language Chinese speaker. But Searle still doesn’t understand Chinese. Hence – and this is the punch-line – even if we admit AI programs powerful enough to pass the Turing Test, they still won’t be intelligent because they won’t understand what they are doing. However good AI programs and machines get at mimicking intelligent life, it still won’t be intelligent because when we, humans, do something it means something to us – but when machines do exactly the same thing it means nothing to them. The modern élan vital which is to distinguish, not the living, but the thinking being, is intentionality (Searle, 1980, 1984).
One of Searle’s keenest critics has been computer scientist and psychologist Douglas Hofstadter, who, together with the philosopher Daniel Dennett, has gone to considerable lengths to defend the computational theory of mind. Hofstadter’s response
“is basically the ‘Systems Reply’: that it is a mistake to try to impute the understanding to the (incidentally) animate simulator [Searle]; rather it belongs to the system as a whole ... The weakness of Searle’s position is that ... he merely insists that some systems [humans] have intentionality by virtue of their ‘causal powers’ and that some [machines] don’t.” (Hofstadter and Dennett, 1982: 374-5)
It would be out of place to go into more detail here on the original Chinese Room. The interested reader should consult: Searle (1980), Searle (1984) Ch 2 ‘Can computers think?’, Hofstadter and Dennett (1982) Ch 22, Poundstone (1988) Ch 11 ‘Mind: Searle’s Chinese Room’, and Dennett (1993).
Little’s thought experiment can be re-cast in the Chinese Room format as follows. It is Little, now, who is imprisoned in the room with a book of instructions and who has to process the characters on slips of paper which are pushed under the door. Unknown to him the slips of paper are individual orderings, the book constitutes the SWF and the output which he pushes back under the door is the social ordering. Now, just as Searle thought that no understanding was taking place as he personally did not understand any Chinese, Little can claim that no preferring is going on in the SWF Room, as he cannot perceive it. Just as the answers Searle provides to the questions were not his answers (he doesn’t even know that they are answers, let alone what they say) so the social preferences provided by Little are not his preferences (he, likewise, doesn’t even know they are preferences, let alone what they say). All Little experiences is (a) his own preference which has been submitted for processing along with everyone else’s, and (b) the utter tedium of carrying out the mechanical processes dictated by the book of rules. The answer is the same: it is not to be expected that Little will experience himself preferring anything since the preference formulated in the SWF Room is not his preference. The preference attaches to the whole system: all the individuals who are asked in some way to code their preferences, the formula for aggregating them, the room and its furnishings, Little (or the computer we would normally expect to do his job), and so on. What is the special ingredient – corresponding to Searle’s notion of intentionality – whose absence prevents the output of the SWF Room from being a genuine preference? Little does not tell us.
In conclusion, therefore, we have seen that Little adopts an illicit dualism which privileges the level of the individual person. His approach implies that there are two fundamentally diverse kinds of thing in the world: individual humans, which can prefer, and everything else, which cannot. Systemists such as Hofstadter and Dennett have argued strongly that all sorts of systems – including genes and memes, and complexes of genes and memes, individual organisms and collectives of individual organisms, species and populations – can be sensibly thought of as having interests and hence as preferring one state of affairs to another. For Little, however, the privileged status of the individual is simply an assumption without explanation.
3.8 How serious a problem is Arrow’s impossibility theorem?
The above account has shown how the libertarian right has responded to Arrow’s impossibility theorem by rejecting the concepts of social rationality and of a social welfare function. I have also suggested that this rejection is ultimately untenable, as it illegitimately privileges the individual and adopts a dualistic standpoint. We need now to say something about the seriousness of the challenge the impossibility theorem poses for the invisible hand theorem.
In my opinion the challenge posed by Arrow’s theorem is not so serious as that posed by the prisoners’ dilemma. The latter shows something profound: namely, how agents can be locked into a situation where their individually rational behaviour leads to socially sub-optimal outcomes. Further it shows that this occurs when there is interdependence but no reciprocity, when decision making is social in content but privatised in form. What does Arrow’s theorem show? It shows that there is no aggregating procedure which is guaranteed to produce an acceptable SWF, based only on ordinal individual preferences, for every conceivable constellation of preferences. Is this so serious? That depends on (a) whether the constellations of preferences encountered in reality are anything like the constellations that cause the Arrow problem, and (b) whether the restriction to ordinal preferences is a legitimate one.
On (a), I have already explained, above, that the problem lies in non-single-peaked preferences, and that such preferences do occur widely in reality. Non-single-peaked (or bimodal) preferences occur in the presence of indivisibilities: when one wants ‘all or nothing’, for example. Half a loaf is better than no bread, but half a baby is worse than none. Babies are indivisible, a fact that Solomon was able to use in judging a difficult case. Indivisibilities are ubiquitous in a heterogeneous world. Non-single-peaked preferences form the basis for the ‘paradox of voting’. Suppose three voters having the preferences illustrated in Table 3 and Figure 1, above or in the example below. Think of them as parties or candidates standing for election. Voting would produce a two-to-one majority for x over y, for y over z and for z over x. Majority voting cannot produce a consistent result. Now, although this is known as the ‘paradox’ of voting, there is not much paradoxical about it. All it says is that if people have different preferences, they may not be able to agree. Indeed, one can imagine a simpler case: two agents, A and B have preferences over two alternatives regarding a £20 note, (i) that A has the £20, and (ii) that B has it. Again, they will not agree. Nothing very profound seems to be going on here. The paradox of voting, and hence the Arrow theorem, which incorporates it, only illustrates the problems that arise when agents have conflicting interests; the bite in the prisoners’ dilemma arises from its combination of conflicting and converging interests.
On (b), the seriousness of the challenge posed depends on how strictly neoclassical one is. The neoclassical paradigm depends upon the assumption of ordinal and interpersonally non-comparable preferences. Whether it is right to do so is debatable and has inspired a huge literature, and it would be inappropriate to take up this very controversial issue here. I will simply note that we do actually make interpersonal comparisons so frequently and unselfconsciously as to invite the speculation that the brain contains special organs for that very purpose. If I bang my finger with a hammer, and you lose your leg in a road traffic accident, few would hesitate to say who was likely to be worse off, or who had suffered the larger decrement in happiness. But for those neoclassicals also committed to the ordinalist paradigm, as in fact most are, the Arrow result is indeed a problem.
There is a further point to be made about the significance of Arrow’s result. This starts by recognising that it is a special case of Gödel’s incompleteness theorem. The latter says that in any consistent formal system (of at least the level of complexity of arithmetic) there must be true statements whose truth cannot be proven within the system, that is, it cannot be both complete and consistent. The SWF is a formal system and must at the very least impound arithmetic – how else is it to aggregate preferences? – and Arrow’s general possibility theorem shows that it cannot be complete (ie satisfy condition U) and consistent (condition O) while at the same time satisfying the other three conditions (P, I, D).
Gödel proceeds by formulating a statement in a formal language, say, PC (for predicate calculus), which says ‘this statement cannot be proved in PC’. It would not be possible to prove the statement without proving a contradiction – for if one proved that it was not provable, by proving it one would have proved that it was provable. Hence a statement and its negation would both be true and PC would be inconsistent. Hence it is the case that it is unprovable; but that is just what it asserts, so it is also true.
So Gödel works by setting up a formal system and then importing a paradox, namely, a version of the Liar Paradox, ‘this statement is false’. The version actually used says, not ‘this statement is false’, but ‘this statement is unprovable in PC’. Arrow also works by setting up a formal system and importing a paradox: the paradox of voting. The paradox of voting, as we have seen, says that if three voters hold the preference orderings
A: x > y > z
B: y > z > x
C: z > x > y
then majority voting on each pair yields the binary social preference rankings
S: x > y, y > z, and z > x, in each case by a majority of 2 to 1.
Hence, for society, every option is preferred to every other option: the social ranking is intransitive (and hence not an ordering).
In the Arrow proof, A, B and C are, respectively, one member of the smallest decisive set, the complement of the smallest decisive set, and the smallest decisive set minus one person, A. The decisive set referred to is decisive for x against y. The pattern of choices set out above is then deployed to show that wherever z is placed on the social preference ranking, this must imply that either A or C is decisive, contradicting the assumption that they were both smaller than the smallest decisive set. This result is then used to show that if all five conditions are satisfied simultaneously, the system is inconsistent.
Completeness in the Gödel context means that all and only the true statements are provable. Completeness in the Arrow context means that for every possible pattern of preferences there is a unique social ordering. Incompleteness is shown in Gödel by importing a ‘paradoxical’ or self-defeating statement: if it can be proved (ie, if the system is complete) then the system must be inconsistent. Incompleteness is shown in Arrow by importing a ‘paradoxical’ or self-defeating pattern of preferences: if an SWF is derived from it the system is inconsistent.
It has been suggested that the Arrow impossibility theorem presents as much a problem for proponents of planning as for those of laissez-faire[7]. This is false, even apart from the cardinal-ordinal issue mentioned above[8]. In the case of Gödel’s theorem, a human intellect standing outside the formal system in question can detect the truth status of the statement ‘this statement is unprovable in PC’. We can see that the statement is true and that the formal system is incomplete. Further, we can see how to make it more complete by incorporating true but unprovable statements into the system as axioms, thereby expanding the system. Similarly, as we encounter paradoxical preference constellations, we can expand the preference aggregating procedure to encompass them by identifying cyclical preferences, taking their transitive closure and impounding the result as a new axiom of the aggregating procedure. This, to be sure, will still leave the aggregation procedure vulnerable to further paradoxical preference constellations, but as soon as one is encountered it can be treated in the same way. If the number of policy options is finite then eventually all comparisons will have been made and we will have a complete social welfare ordering. Otherwise, the result is an infinite regress with each obstacle being overcome, only to give way – potentially, at least – to a new one. It is, perhaps, in the nature of things that we can get there, not all in one go, but only as an unending series of approximations.
3.9 Conclusion
In this chapter I have argued that Arrovian impossibility, like the prisoners’ dilemma, has been perceived as presenting a fundamental challenge to the Smithian ‘invisible hand theorem’. This challenge has prompted a number of conservative thinkers in the social sciences to argue that we should abandon the concept of collective rationality itself. Micro-level rationality is all that we can ask for. The policy prescription, therefore consists of maximum freedom for individuals to maximise their individual utilities, coupled with passive acceptance of whatever emerges at the macro level. As Barry and Hardin say, sympathetically summarising Little’s view, ‘The outcome is the outcome, and it may have nothing apart from that to be said for it’ (268). Pursuing a libertarian critique of the idea of a SWF, these writers accuse Arrow, and hence, implicitly, mainstream neoclassical economics, of standing in a tradition leading from Rousseau and Hegel to modern totalitarianism.
In this chapter I have also argued that Little’s argument against the possibility in principle of a social welfare function closely parallels Searle’s Chinese Room thought experiment designed to show the impossibility in principle of artificial intelligence. Both arguments fail for the same reason: they involve a dualistic vision of the world in which individual humans are set apart from the rest of nature by some innate quality – such as ‘intentionality’ – the absence of which is supposed to prevent systems other than individual humans from being conscious or forming purposes.
Finally, I noted the links between Arrow’s and Gödel’s results, and argued that, although still an important challenge for the neoclassical paradigm, the Arrovian theorem is of less theoretical significance than the prisoners’ dilemma. While the essence of the Arrow theorem was a conflict of interest between agents, that of the prisoners’ dilemma was a combination of conflict and convergence of agent interests.
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Revised: Saturday, 13 September 2003