Monday, 2 September 2019

Maskin on Arrow's Impossible Social Choice

E.S Maskin writes-
Arrow's first paper in this field was “A Difficulty in the Concept of Social Welfare” (Arrow 1950), which he then expanded into the celebrated monograph Social Choice and Individual Values (Arrow 1951a, 1963a, 2012). His formulation starts with two things: (a) a society, which is a group of individuals, and (b) a set of social alternatives from which society must choose. 
The interpretation of this setup depends on the context. For example, imagine a town that is considering whether to build a bridge across the local river. In this case, society comprises the citizens of the town, and the social alternatives are simply the options to build the bridge and not to build it. We can also think of a situation involving pure distribution. Suppose that there is a jug of milk and a plate of cookies to be divided among a group of children. In this case, the children are society and the different ways to allocate the milk and cookies among them are the alternatives. As a third example, think of a committee that must elect a chairperson. In this case, society is the committee and the social alternatives are the various candidates for chair.
Is this a reasonable interpretation of how Societies make choices? Bridges cost money to build. Why should hobos or small children have a voice in whether or not a bridge should be built? Jugs of milk and plates of cookies don't appear out of thin air. Kids can't be fed by any old child molester. A committee is not Society. Its existence may represent a social pathology.

A Society is not a memoryless collection of social atoms. It has its own folkways and shibboleths and rules and hierarchies.  The sort of choice Arrow is speaking of is merely an abstract decision space which has no properly social features whatsoever.
Those are just a few interpretations of the Arrow setup, and there is clearly an unlimited number of other possibilities. An important feature of the formulation is its generality.
Presumably, each member of society has preferences over the social alternatives. That means that the individual can rank the alternatives from best to worst. Thus, in the bridge example, a citizen might prefer building the bridge to not building it. A social welfare function (SWF), according to Arrow, is a rule for going from the citizens’ rankings to social preferences (i.e., a social ranking). Thus, social preferences are a function of citizens’ preferences.5
My preferences change depending on the Society I expect to keep. I wear a suit and tie to work but jeans to the pub and pajamas at home in bed. When I move to another country, I adjust to the new society. In India, I'd wear pajamas to work because I'd look foolish in jeans and like a pimp in a suit.

Individuals accommodate themselves to Society. Only if being a Social animal conveyed no survival value to our species would Arrow's approach not be utterly foolish.
In the bridge setting, one possible SWF is majority rule, meaning that, if a majority of citizens prefer building the bridge to not building it, then building is socially preferred—the town should build the bridge.
This is foolish. The majority may want a bridge to the moon. There may be a SWF which describes what Society chooses and which everybody, with hindsight, thinks socially correct but, unless that Society already had a shibboleth w.r.t majority voting, it would have no use for the thing. Suppose, a voting rule were imposed by a conqueror. Then such social pressure would be brought to bear, or penalties would be so extreme, that the outcome would be the same as before.
Although highly permissive in some respects, this way of formulating a SWF still excludes some important possibilities. First, it rules out making use of the intensities of individuals’ preferences (or other cardinal information).
But it still has a preference revelation problem. Some may tactically oppose or endorse the bridge for some reason which has nothing to do with their preferences.
For example, it disallows a procedure in which each individual assigns a numerical utility (or grade) to every alternative (say, on a scale from 1 to 5), and alternatives are then ordered according to the median of utilities (for a recent approach along these lines, see Balinski & Laraki 2010). Arrow's rationale for excluding cardinality—following Robbins (1932)—is that such information cannot be reliably obtained empirically unless individuals trade off alternatives against some other good like money, in which case the set of alternatives that we started with does not fully describe the possibilities.
Information about preferences can never be 'reliably obtained'.  They have a very complex relationship with our 'social being'. We are seldom aware of the entire inwardness of our signalling- one aspect of which is the preferences we reveal.
A second (and closely related) omission is that the formulation does not allow for interpersonal comparisons (for formulations that do permit such comparisons, see Sen 2017, chapter A3*). For example, there is no way of expressing the possibility that individual 1 might gain more than individual 2 loses if alternative a is replaced by alternative b.
People may make interpersonal comparisons and order their preferences accordingly. It may be that people in some Societies do this by default or some people in our Society can do this in a manner we find prescriptive.
Thus, Arrow's setup excludes classical utilitarianism à la Bentham, according to which a is socially preferred to b if the sum of individuals’ utilities for a is greater than that for b.
The Benthamite legislator would have so arranged things that preferences would be eusocial. People would feel utility for doing utile things. This is the theological side of the invisible hand. Synderesis synchronized by Charisma or Providence or some guy who thinks he's real smart causing everybody to fulfill of their own volition, that Divine Plan, which can make a Heaven on Earth.

It was this side of Bentham which appealed to Raja Ramohun Roy and Dwarkanath Tagore and which, even in the unlikely shape of 'Harbhat Pendse', became a but strand worn loose from the entropy reversing yantra of Umaswati or the empathic tantra of Shantideva or even the Vasudeva mantra of the humblest Vaishnav.
The formulation also rules out a comparison such as “Individual 1 is worse off with alternative a than individual 2 is with alternative b.” Thus, Rawls's (1971) maximin criterion (in which a is socially preferred to b if the worst-off individual with alternative a is better off than the worst-off individual with alternative b) is also off the table.
One 'maximin' condition which everybody would require to be fulfilled before submitting preferences to a social choice rule is that there in no probabilistic fatal- or even somewhat undignified- outcome. But this stipulation can be extended to maximin criteria. In other words, preferences could forbid the application of Social Choice mechanisms which don't have a 'safety net'. Indeed, the great utility of Social Choice is it allows risk pooling which reduces uncertainty and thus increases cooperation- i.e. creates a positive sum game.
Arrow avoids interpersonal comparisons because, again, he argues that they lack an empirical basis (he doubts that there are experiments that we could perform to test claims such as “This hurts me more than it hurts you” or “My welfare is lower than yours.”)
Our history as a species is nothing but examples of such experiments. It's the reason we've gotten rid of corporal punishment. Contrary to expectations, our society hasn't lapsed into illiteracy just because we are not battering our kids adequately.

Finally, the requirement that social preferences constitute a complete ranking6 may seem to attribute a degree of rationality to society that is questionable (see, in particular, Buchanan 1954). Arrow's reason for positing a social ranking, however, is purely pragmatic: The social ranking specifies what society ought to choose when the feasibility of the various alternatives is not known in advance. Specifically, society should choose the top-ranked alternative, a, if a is feasible; should choose the next best alternative, b, if a is infeasible, and so on.
Knightian Uncertainty means we don't know all the alternatives- feasible or otherwise. It may make sense to pursue a 'regret minimizing' course with some resources being used up in potlatches or devoted to the upkeep of white elephants or, more senselessly yet, in whatever it was that was the forerunner of what we now recognize as Science & Technology.
Yet requiring a social ranking is a potential problem for the best-known way of determining social preferences, majority rule. I note above that majority rule works fine in the bridge example, where there are only two possible choices. Imagine, however, that there are three alternatives: building a bridge (B), building a tunnel (T), and doing nothing (N). Suppose, for example, that 35% of the citizens in the town prefer B to T and T to N, 33% prefer T to N and N to B, and 32% prefer Nto B and B to T (these preferences are summarized in Table 1).

Citizens’ rankings of three alternatives

In this case, under majority rule, N is socially preferred to B because a majority (33% + 32%) prefer N. Furthermore, T is socially preferred to Nbecause a majority (35% + 33%) prefer T. However, B is socially preferred to T because a majority (35% + 32%) prefer B. Clearly, majority rule does not give rise to a well-defined social ranking in this case.
Nothing does. There are too many unknowns. Who ever heard of a tunnel coming in under budget? As for this so called bridge- you can't tell me the Mayor isn't getting paid off.
As far as we know, it was Condorcet who first noted the possibility of a Condorcet cycle, in which majorities prefer N to B, T to N, and B to T(although he was himself a strong proponent of majority rule).
Everybody had already noticed it which is why election rules, were they survived- e.g. in Venice- were so darned complicated. The fact is, it makes sense to oppose things proposed by a guy you don't like as well as to support things you don't like because someone else will have to stop them from happening.
Condorcet cycles were Arrow's starting point in his thinking about social choice (see Arrow 2014). Interestingly, he was, at that time, unaware of Condorcet's work but rediscovered the above problem with majority rule for himself. It led him to wonder whether there is some other reasonable way of determining social preferences that does succeed as a SWF.
But this wondering didn't get him very far because he couldn't see that the prevalence of Knightian Uncertainty means that Social Choice should have mimetic features and feature dramatic saltations in individual profiles. Ultimately, for incentive compatibility to obtain, individual 'hedonic' utility needed to align with what was socially utile. Thus, the 'reverse game theory' of mechanism design- not the imbecility of deterministic voting rules- was 'economic'- i.e could pay for itself- whereas 'impossibility results' were a puerile exercise in Akrebia.
By “reasonable,” Arrow first required that the SWF should always work. That is, it should determine the social ranking no matter what preferences individuals happen to have. This is called the Unrestricted Domain (UD) condition. It is the UD condition that majority rule violates.
So if everyone says 'take your voting scheme and shove it up your arse' then either 'majority rule' is violated or else domains have to be very restricted indeed.
Nobody wants everything to be a matter which is voted on. The Domain of Social Choice must be very small relative to that of Individual Choice and Family Choice and Corporate Choice and Voluntary Association Choice and may feature quite diverse 'Tiebout Models' in different Local Authorities.
Second, he insisted that, if all individuals prefer alternative a to alternative b, then society should rank a above b.
Utterly crazy! All individuals may be wrong. Some outsider may be right. Society should prefer the right thing not the wrong thing- even if everybody would prefer to live in a Universe where that wrong thing yielded the best result.
After all, it would be quite perverse for society to choose b when everyone thinks that a is better.
Nonsense! We may say 'of course, we all prefer to burn witches- as the Good Book tells us to- but, as a Society, let us express no preference in this matter coz otherwise we come across like crazy yokels with pitchforks who are constantly burning our grandmothers.' It is wise to exit a Society which starts expressing preferences on all sorts of things simply on the basis of preference falsification or virtue signalling or vulgar prejudice.
This is called the Pareto (P) condition.
Pareto's work had a sociological component. He was the last man to recommend applying this condition in situations where 'residues' and 'derivations' were such that everybody preferred crazy shit over whatever it was that Science would approve and which might prove more socially utile.
Third, Arrow required that the social preference between alternatives a and b should depend only individuals’ preferences between a and b, and not on their views about some third alternative c.
This was foolish. The third alternative is to tell Arrowvian Social Choice to fuck itself. It is always better than expressing a preference between any two things. In practice, this is what happens, when some shithead starts burbling about Democracy and pretending we have to choose sides on all manner of wedge issues. We tell the fuckwit to go fuck himself. He says 'for evil to triumph, it is sufficient for good men to do nothing. Don't you see, there is a slippery slope! Even now we risk slipping into Nazism and Fascism and like the Spanish Inquisition!' We suddenly realize we'll have to vote for Trump and keep voting for Trump till these fuckers fuck the fuck off.
He argued that c is irrelevant to society's choice between a and b, and so that choice should be independent of c. This is called the Independence of Irrelevant Alternatives (IIA) condition.7
It is also called 'forcing a card' or 'the gaslighting strategy'. Your choice is to admit that you are a repressed homosexual who wants to suck me off or else that you are a repressed homosexual who wants to suck me off but is too fucking repressed to do it. The third choice, which is that you are not homosexual at all is irrelevant. Anyway, it is homophobic.  What should you do? Report the guy for sexual harassment and get him fired. Me-too the stupid fuck. Don't kick his head in. The slimy goo inside his cranium may stick to your boots rendering them malodorous.
These three conditions—UD, P, and IIA—all seem quite natural and, on the face of it, not terribly demanding.
They are as stupid as shit. No Society has ever featured anything like them. They are utterly unnatural.
However, remarkably, Arrow showed that the only SWF satisfying all three conditions is a dictatorship, a highly extreme sort of SWF in which there is a single individual—the dictator—who always gets his way: If he prefers alternative a to alternative b, then society prefers a to b, regardless of other individuals’ preferences. Thus, if we introduce an additional requirement—a Nondictatorship (ND) condition, which demands that the SWF should not have a dictator—then we obtain Arrow's Impossibility Theorem: With three or more possible social alternatives, there is no SWF that satisfies UD, P, IIA, and ND.
Arrow's discovery amounts to saying 'if there is no Dictator then there is a Dictator'. Accept this lie and you can prove cats are dogs and dogs are turtles.

How does Arrow work the trick? He uses this definition-
Nondictatorship (ND): There exists no individual who always gets his way in the sense that, if he prefers a to b, then a must be socially preferred to b, regardless of others’ preferences. Formally, there does not exist i* such that, for all and all a, b , if , then , where .
This means that if we choose a particular person who has most knowledge and most public spirit to choose for us then that person is a Dictator. However, this is not true. Dictators are guys like Hitler and Stalin. They kill people who disagree with them. The guy we choose may be a Saint. No doubt, the Commies would say 'General Eisenhower is a Dictator enforcing the will of the gangster plutocrats' or 'Mahatma Gandhi is a Dictator enforcing the will of the gangster bania class' - but the Commies were habitual liars. Gandhi wasn't a dictator. Neither was Eisenhower.

It is perfectly possible for an SWF which we all agree, after the fact, to have been, if not optimal, then 'regret minimizing' and therefore eusocial, to be implemented on the basis of the protocol bound decisions of a group as expressed by its leader. This would look 'non-deterministic'. Individuals, in Arrow's framework, can have non-deterministic preferences but, for some reason, the voting algorithm can't. Thus he can only prove his impossibility result by saying an omniscient Benthamite planner is a Dictator even if he isn't a Dictator at all. But, by ex falso quodlibet, this also implies that all cats are dogs.

How did this egregious nonsense end up exercising such a hold over the minds of academicians? The answer, I think, is that they were stupid donkeys. They sought for a mathematical 'Akrebia' in a field where such precision was misapplied and bound to lead to outright cretinism. Society is bound together by a Ciceronian 're-ligio'- a scrupulous re-examination of choices- and this yields a pious eusebia which is economic, not akrebic. Its scope is limited and discretionary and based on mutuality and good faith. It isn't some monstrous pseudo-mathematical machinery whose gears bite at the empty air. Rather, its propositions feature Structural Causal Models such that people can help each other incrementally and in a piece meal manner. Doing good is not different from theorizing about the good because the theory stands or falls by its immediate, normative, tie to action. Mimetics are important. They permit runaway effects. But pathos is the path to Mathos. Mathematical Economics can go somewhere else to wank.

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