Arrow's theorem 70 years on

Preferences are not Decisions. Beliefs are not Knowledge- they have no conceptual tie to action because they need not be related to a coherent Structural Causal Model and thus gain no benefit by displaying consistency or meeting any other criteria of rationality.

Language clearly distinguishes between what one prefers and what one chooses and what one believes and what one knows or uses to motivate one's actions. It is easy to generate intensional 'paradoxes' and counterintuitive 'impossibility results' by abolishing distinctions which extensional pragmatics, that is natural language, has long found discovered to be useful.

Consider the opening sentence in  Arrow's famous paper-

In a capitalist democracy there are essentially two methods by which social choices can be made: voting, typically used to make "political" decisions, and the market mechanism, typically used to make "economic" decisions.

In a Capitalist democracy, voters don't make social choices, they elect representatives who have their own agendas and constraints. If voters could make 'political decisions' directly, the expectation would be that periodic redistribution of Wealth would occur and so Capitalism would disappear.

Similarly, Market mechanisms don't make economic decisions. We may say, metaphorically, that people 'vote with their money'. But that is merely a figurative expression. It is not the case that markets 'aggregate preferences'. Opinion polls may do that. All that markets can do is coordinate decisions involving constraints and contingent events of various types.

Arrow's next sentence shows extraordinary ignorance and prejudice. He says Britain, whose Labor Government, at that time, had created the National Health System and which had Nationalized various industries, was an 'emerging democracy'. Jim Crow riddled America, however, was not.

In the emerging democracies with mixed economic systems, Great Britain, France, and Scandinavia, the same two modes of making social choices prevail, though more scope is given to the method of voting and to decisions based directly or indirectly on it and less to the rule of the price mechanism.

This is a wildly inaccurate picture. 'Social choices'- stuff like deciding whether to build a Hospital or a battleship- were made by those authorized to do so on the basis of economic considerations and involved considerable discussion and negotiation. Voting, more often than not, had a symbolic, legitimating role. It was not the basis of any decision.

Elsewhere in the world, and even in smaller social units within the democracies, the social decisions are sometimes made by single individuals or small groups and sometimes (more and more rarely in this modern world) by a widely encompassing set of traditional rules for making the socia1 choice in any given situation, e.g., a religious code.

This is nonsense. All social or communal or even family decisions are made on the basis of authority which may be guided by habit, or expediency, or consultation, or anything else it pleases.

The last two methods of social choice, dictatorship and convention, have in their formal structure a certain definiteness absent from voting or the market mechanism.

Dictators make decisions in the same way as everybody else. They may legitimate their decisions by holding a referendum. What makes them different from Democratic leaders is that they may suspend the Rule of Law and incarcerate or kill those who oppose them.

Social Choice in 'Democratic Capitalism' is just as likely to be based on habit, or convention. Indeed, if transactions are more frequent, this is more, not less, likely to be the case.

Arrow's take on Dictatorship is equally cartoonish.

In an ideal dictatorship, there is but one will involved in choice;

Nonsense! The dictator may know his own will but he does not know what resources are available and how they can be combined so as to yield what he wants. There is still an information aggregation problem involving what people need in order to work the dictator's will.

Furthermore, since Rousseau, there was the notion that the ideal leader- Philosopher King or 'Enlightenment' Despot- was legitimate if he or she incarnated the 'General Will'.  A Democratically elected leader may make the same claim- perhaps by invoking some personal askesis or spiritual kenosis such that his or her own ego had been pulverized and the 'Spirit of People' has replaced it.

Arrow understood this. He writes-

in an ideal society ruled by convention, there is but the divine will or perhaps, by assumption, a common will of all individuals concerning social decisions, so that in either case no conflict of individual wills is involved.

This would also be the case in a relatively homogeneous Capitalist Democracy in a steady state.
If the Dictator, or the Convention, aims at the same outcome as the Capitalist Democracy, the two would be indistinguishable.

The methods of voting and of the market, on the other hand, are methods of amalgamating the tastes of many individuals in the making of social choices.

Yet, Enterprises and Governments do market research and conduct opinion polls and organize focus groups and so forth. Votes and Market mechanisms may validate or reinforce decisions made by Enterprises or Governments. However, they may be disregarded for perfectly rational reasons.

The methods of dictatorship and convention are, or can be, rational in the sense that any individual can be rational in his choice.

Nonsense! No individual can be rational in his choice, unless the thing is trivial or it isn't a choice at all. That's why people agonize over the thing and seek the advise of others and pray to God and consult fortune tellers and so forth. Under uncertainty, utility maximization is not rational. Something based on hindsight- on backward induction- like 'regret minimization' may evidence superior rationality. But 'Hannan consistency' or 'Muth rationality' involve interdependence and represent correlated, not competitive, equilibria. In other words, Public signals are needed to improve individual choices so that Social outcomes are better for everybody.

Arrow, writing 70 years ago, may be pardoned for taking a different view-

Can such consistency be attributed to collective modes of choice, where the wills of many people are involved?

We now know that the notion of Hannan Consistency enables us to see why we can get to optimal correlated equilibria without formal Public signals. But this type of Consistency is very different from the crude mechanical sort Arrow is invoking.

It should be emphasized here that the present study is concerned with the formal aspects of the foregoing question. That is, we ask if it is formally possible to construct a procedure passing from a set of individual tastes to a pattern of social decision making, the procedure in question being required to meet certain natural conditions. 

It is not possible to determine even one individual's tastes and relate that determination to that individual's choices in a consistent manner under any natural conditions.

Consider 'unrestricted domain'- if there is no constraint on how the individual formulates his preferences, and thus the set of preferences that he might have, then there is no consistent mechanism which will deterministically provide the same choices, if the preferences are presented the same way.

If this were not the case, we wouldn't have two different words to represent what we prefer and what we actually choose. A person who said 'I preferred wine but I chose juice' would be considered a lunatic if there really was some deterministic connection between preferences and choices.

In practice, we have internally inconsistent preferences which is why we suffer cognitive dissonance. But without that dissonance our ability to flourish- even as a Robinson Crusoe, let alone as a member of a complex civilization- would be severely compromised.

No doubt, there are a lot of preferences which we have which consistently map to choices- however this only happens if the thing doesn't matter very much or else there is no real choice. I prefer wine to juice, but I order juice at breakfast because I have work to do. It would matter a great deal to me if I got fired for being under the influence.
By contrast, at dinner, I may finish off the juice rather than open a bottle of wine precisely because I'm going to go to sleep anyway and so my choice of beverage doesn't matter very much one way or the other.

It is wholly unnatural to think 'consistency' in translating preferences to choices is desirable save if the domain of the former is restricted to stuff which doesn't matter very much at all.

Moving on to another of Arrow's conditions, we find 'non Dictatorship'.
Surely I would resent a Dictator making my choices for me? The answer is- no. If the Dictator knows me better than I do myself and cares even more than me about my well-being, I would prefer that my choices reflect this 'meta-preference' of mine. The Dictator need not be another person. It could be my own super-ego.

Let us now look at the crucial assumption Arrow makes in his paper.

It is assumed that each individual in the community has a definite ordering of all conceivable social states in terms of their desirability to him. It need not be assumed here that an individual's attitude toward different social states is determined exclusively by the commodity bundles which accrue to his lot under each. The individual may order all social states by whatever standards he deems relevant.

Suppose Arrow's assumption were true in the real world. What would happen if Robinson Crusoe, alone on his island, could know his preferences with respect to all conceivable social states? The answer is his life would be much easier. He'd know when to wake up and which tree to stand under to be rewarded with a ripe fruit dropping into his hands and where to throw his fishing net and when to hide in the forest when cannibals appear and so on and so forth. He would also be able to build a teleportation device, using conceivable technology, to travel the Universe. He wouldn't need to put any thought into any of these activities. He'd just need to do whatever he preferred to do at any given moment.

Arrow's assumption destroys the need for Economics or Politics or Scientific Education or even Human Language. We would all just do what we preferred and everything would work out wonderfully- provided that's what we really wanted.

Amartya Sen has suggested that Arrow misses a trick by not allowing 'interpersonal comparisons of utility'. However, so long as people wanted a perfect Benthamite Utopia, they'd be able to achieve it just by doing what they preferred. This is because they are magically endowed with a complete set of preferences over all conceivable social states including the one which would have been computed by an omniscient Benthamite deity.

I suppose, Arrow could say 'by conceivable social states', I mean those conceivable by the agent given his mental endowments. However, in that case, the 'commodity bundle' is still a constraint. It isn't really true that the agent can have any sort of preferences. There is an element of determinism built in.

Alternatively, Arrow can stipulate for 'common knowledge. Thus

A member of Veblen's leisure class might order the states solely on the criterion of his relative income standing in each;

but can't steal a march on his rivals because he gains no extra information. But, in that case, his choice is still constrained by his commodity bundle.

However, Arrow adds-

a believer in the equality of Man might order them in accordance with some measure of income equality.

If such a person does this it must be common knowledge that the thing is feasible. But that means there must be a common knowledge mechanism to bring it about. This in turn means that Democratic Capitalism would be an oxymoron. The proletarian majority would expropriate the plutocratic minority through the ballot box.

Indeed, since, as mentioned above, some of the components of the social state, considered as a vector, are collective activities, purely individualistic assumptions are useless in analyzing such problems as the division of the national income between public and private expenditure. The present notation permits perfect generality in this respect.

In other words, it allows people to choose the ideal cooperative equilibrium where all actions are coordinated not by the invisible, yet greasy, hand of the market but the prompting of the Holy Spirit within our bosom.

Needless to say, this generality is not without its price. More information would be available for analysis if the generality were restricted by a prior knowledge of the nature of individual orderings of social states. This problem will be touched on again.

No it won't. The moment it is touched this whole availability cascade collapses.

In general, then, there will be a difference between the ordering of social states according to the direct consumption of the individual and the ordering when the individual adds his general standards of equity (or perhaps his standards of pecuniary emulation) .We may refer to the former ordering as reflecting the tastes of the individual and the latter as reflecting his values. The distinction between the two is by no means clear cut. An individual with aesthetic feelings certainly derives pleasure from his neighbor's having a well-tended lawn. Under the system of a free market, such feelings play no direct part in social choice; yet, psychologically, they differ only slightly from the pleasure in one's own lawn.

Nonsense! The aesthete buys a house in a gated community where there are rules for things of this sort. Alternatively, he could pay to get his neighbor's lawn tended to. Externalities can be internalised in various Coasian ways.

Intuitively, of course, we feel that not all the possible preferences which an individual might have ought to count;

Common sense tells us that preferences count for nothing. Only deeds matter.

his preferences for matters which are "none of his business" should be irrelevant without  challenging this view, I should like to emphasize that the decision as to which preferences are relevant or not is itself a value judgment and cannot be settled on an a priori basis.

These value judgments count for nothing. We know that on an a priori basis. Only deeds matter.

From a formal point of view, one cannot distinguish between an individual's dislike of having his grounds ruined by factory smoke and his extreme distaste for the existence of heathenism in Central Africa.

Formal point of views involve stabbing oneself in the eyes. This may be useful if you have a fundamentally useless job, not otherwise.

There are probably not a few individuals in this country who would regard the former feeling as irrelevant for social policy and the latter as relevant, though the majority would probably reverse the judgment.

What is important is that everybody agrees that this shite is irrelevant.

I merely wish to emphasize here that we must look at the entire system of values, including values about values, in seeking for a truly general theory of social welfare.

Why must we? Social welfare does not depend on what 'values' people may or may not have. It depends on the outcome of decisions made about how to allocate scarce resources.

My Mum did not ask us what we wanted for dinner. Whatever she cooked- we ate. This raised Social Welfare. A Mum who panders to the preferences or values of her children and sex slaves may reduce Social Welfare.

A general theory of Social Welfare does not contribute to Social Welfare. The thing is a waste of resources.

We do spend money compiling various types of Statistics. Nobody has ever bothered to try to compute a Social Welfare Function. The reason is not far to seek-

It is the ordering according to values which takes into account all the desires of the individual, including the highly important socializing desires,

thus, it is pie-in-the-sky such that everybody is declared to be the sexiest, smartest, coolest person alive

and which is primarily relevant for the achievement of a social maximum.

Nonsense! It is primarily relevant for the achievement of wanking.

The market mechanism, however, takes into account only the ordering according to tastes.

Rubbish! The market mechanism has no access to 'orderings according to taste'. It may have access to 'orderings according to values' in that philanthropy and thrift can be mediated through the market. However, the information aggregated by markets has to do with decisions made under constraints arising from scarcity of various types.

The market does not forbid actions of a Coasian type which internalize externalities. Similarly, charitable institutions as well as enterprises which seek to pool risk, do trade-off 'values' as opposed to tastes such that external effects are taken into account. Thus, both for charity or insurance purposes, certain courses of action may be avoided as repugnant or involving moral hazard precisely for this reason.

This distinction is the analogue, on the side of consumption, of the divergence between social and private costs in production which has been developed by Professor Pigou.

In other words, just as the economic theory of externalities dispels any foolish notion that we might have that deterministic market mechanisms have some magical property, so too does it destroy our faith in deterministic voting mechanisms.

That's why the following is shite-

Just as for a single individual, the choice made by society from any given set of alternatives should be independent of the very existence of alternatives outside the given set.

This is nonsense. No alternative is irrelevant for Robinson Crusoe. It is either useful to contemplate or not useful at any given time. Deciding it is useless, however, may be useful because it illumines a whole class of decision problems.

One might as well say that 'alternatives' of the sort used in thought experiments should always be ruled out. But then Arrow's own work would be ruled out. Yet, by examining it and declaring it to be shite we can save ourselves a lot of time & tummy trouble because we learn to recognize a particular type of turd in the punch- bowl.

For example, suppose an election system has been devised whereby each individual lists all the candidates in order of his preference, and then, by a preassigned procedure, the winning candidate is derived from these lists. Suppose an election is held, with a certain number of candidates in the field, each individual filing his list of preferences, and then one of the candidates dies. Surely, the social choice should be made by taking each of the individual's preference lists, blotting out completely the dead candidate's name. and considering only the orderings of the remaining names in going through the procedure of determining the winner.

Nonsense! The knowledge that Rajiv Gandhi had been killed after the first polling day affected the over all outcome because voters used the new information to change their vote in the next two polling dates. This was perfectly rational. The information set had changed.

Alternatives which arise out of the same Structural Causal Model are going to exhibit interdependence. Thus no alternative can a priori be considered 'irrelevant'. However if the thing has an almost perfect substitute and no 'income effect'- i.e. doesn't matter very much- then we can ignore it. But, if that were true in general, regarding the alternatives considered by Social Choice, then we should ignore Social Choice as wholly inconsequential.

If you have an election slate of bland interchangeable nonentities, it doesn't matter if you don't vote or if some of the candidates die because as far as Social Choice was concerned there was no real Choice.

That is, the choice to be made among the set of surviving candidates should be independent of the preferences of individuals for the nonsurviving candidates. To assume otherwise would be to make the result of the election dependent on the obviously accidental circumstance of whether a candidate died before or after the date of polling.

Election results do depend, and should depend, on stuff like whether a candidate is dead or not. Such deaths increase the information set and the only reason they shouldn't be taken into account is if the decision doesn't matter very much. But if this were true of all decisions, then Social Choice theory doesn't matter at all.

Therefore, we may require of our social welfare function that the choice made by society from a given set of alternatives depend only on the orderings of individuals among those alternatives.

Individuals don't want their own existing ordering to be used but rather the one they'd prefer to have with hindsight. This could be a regret minimizing ordering. Alternatively it may be the one they'd have at the end of an almost infinite process of negotiation and bargaining so that everybody agreed they had the best possible preference profile and that it yielded the best possible result.

It only makes sense to focus only on individual orderings if there are no externalities, no concurrency type problems, no uncertainty, no information asymmetry and so forth. But, in that case, there is no point having a Society. There is no 'social' dimension to anything. It's like a dinner party where everybody brings their own food and converses only on topics of interest to themselves. Would we really consider it a 'social' event? The thing is more akin to a bunch of overlapping transactions. Nobody is really being 'sociable'. They may as well have stayed home, ordered takeaway, and tweeted to like minded assholes. By contrast, at a well conducted dinner party, agape has a mellowing effect, people modify their opinions and see that  the true virtue of cooperative solutions is that they create an incentive for building trust and mutuality not merely to create more cooperation but as things good in themselves.