Why Social Welfare functions can't exist

Our welfare- that is happiness- may go up or down without any change in how much money we have or the state of our health or in other metrics regarding things we value.  Finding out if someone in like circumstances to us feels happier can motivate us to improve our lives.

Similarly, two areas may be alike according to earnings, health outcomes, leisure etc, etc, yet people from one area may keep emigrating to the other. It might be useful for us to have a way of measuring both individual and Social welfare because this could provide us with ways of making our life better.

 

If Tardean mimetic effects exist- i.e. people and Societies imitate those who are in a superior position- then there must be some way to rank individual and social outcomes at least for any specific purpose corresponding to a type of mimetic action. In this sense there are elements pertaining to individual and social welfare functions 'out there'. However, they may merely refer to 'local arbitrage opportunities' or a bunch of disparate cases of cognitive dissonance which don't cohere sufficiently such that we may speak of utility or welfare as having changed in aggregate- in other words, the elements of such functions may not link up in a substantive and comprehensive manner. 

There is a saying 'comparisons are odious'. It comes from a fifteenth century poem by John Lydgate in which the horse, the goose and the sheep contend as to which is most useful to man. Judgments about utility, it seems, were always known in England to be vainglorious for all forms of life, regardless of nobility or grace or even usefulness, are glorious to God. Lydgate, writing during the hundred years war, gives the palm to the peaceful sheep for the 'lamb of Grace' is none other than Jesus who vanquished Satan. 

However, the horse's military role and the goose feathers needed for the arrows of the English archers remind us that 'Welfare' is intimately related to who prevails in 'Warfare'. Ultimately, the measurement of an individual's and a Society's resources is linked to the 'threat point' they can establish to preserve themselves or prey upon others. In other words, measuring welfare arises out of the odious necessity to preserve welfare by warfare. 

Lydgate, however, gives a twist to this argument. The horse rebukes the sheep- the source of England's wealth as is still represented by the Lord Chancellor seated upon the woolsack- for creating the wealth that attracted and sustained aggression. Utilitarianism, as a political philosophy and economic doctrine, was associated with Britain's increasing affluence and security in the hundred years between Waterloo and Sarajevo. But as the two World Wars taught us, envious emulation of what increases Social Welfare can destroy it with increasing thoroughness as the natural Sciences advance under the spur of mutual assurance of destruction.

Thus, it seems, a concern to reduce Social Welfare to a mathematical function might imperil the very elements upon which it is based.

Thankfully, by the Thirties- with the work of Godel, Turing and Tarski- mathematical logic itself repudiated this odious misology. Sadly, Sen-tentious cretins didn't get the memo. But much of the blame must fall on Tarski's student- Ken Arrow- for this outcome. He of all people should have known that what he was doing was foolish.

Consider the following, fairly typical, sentence from a Welfare Econ paper-

A social welfare functional assigns a social ordering defined on the set of social states X to each possible profile of individual utility functions in its domain.

 Either individuals are interested on Social Welfare or they are not. If they are not, they may have utility/disutility functions re. some things such that there is some trade or robbery. But there is no Society in the full sense of the world. On the other hand, if people are tightly bound to each other and act as a cohesive Social whole then there are no 'individual utility functions' in the domain of the Social welfare functional. There are only individual social welfare functions representing a ranking of social states X under different Social Choice rules by different people. For any given Social Choice rule, it is possible that some or all of these individuals rankings are undecidable by reason of impredicativity- e.g tactical voting, 'income effects', antagonomia or its reverse- the wish to go with the majority, etc, etc.

 Furthermore the Rule is itself a member of X- i.e. a social state. If the implementation of the Rule uses up resources, it is always inferior to the same outcome arising without any Rule whatsoever. Thus, a social welfare function never assigns a social ordering because nobody wants it to. 

Is there really a 'set' of Social States X? Naïve set theory defines a set as any well-defined collection of distinct elements. An expression is well defined if its definition assigns it a unique interpretation or value.  Is there a unique vector of quantities x assignable to K individuals corresponding to Social State X? Sure. Provided there is no Social connection between those K individuals- i.e. no linguistic, emotional, or other type of communication or exchange between them affects or could affect their welfare. It is sufficient to describe their commodity bundles to give a complete account of their meaningful interactions. Inter alia, this means this is a situation where there is no sexual reproduction, no Price equation type kin selective altruism and no Dawkins type extended phenotype.  

If there is linguistic or other communication or non-commodity based exchange or interaction between agents, then it can't be the case that the Set of Social States is a well-defined bunch of physicalist vectors. Rather, by Tarski's undefinability theorem- No sufficiently powerful language is strongly-semantically-self-representational- it must be, if it is anything, 'second order' with respect to a commodity space- i.e. there is nothing in the first order language that corresponds to semantic operations- e.g. establishing preference, transitivity relations, or ordering relations. But is there a well-defined commodity space? Not if agents are 'Social'- i.e. who consumes what has no unique representation in the configuration space. The fact is, I may eat cake today while you go hungry. But this may be because you gave me your cake because I am an abject beggar while you went hungry because you are having a very expensive surgical procedure done tomorrow which will completely rejuvenate you. This is a very different situation from one where I got cake because I was rich and you didn't because you were poor. In the former case, I ate cake with a feeling of gratitude- or, it may be, secret resentment- while in the latter, cake eating was purely hedonic.

 Language- natural language- is 'being for others'. It is essentially social. It corresponds to a wide range of welfare outcomes corresponding to a given physicalist assignment. To be fed by charity feels different from feeding others for charitable reasons. Everybody may eat the same but how they feel about it depends on things expressible only in a second order language of obligation, duty, benevolence etc which are not captured in the first order, physicalist, language specifying the configuration space. Arrow was trained by Tarski. Yet this is the mistake he makes in his possibility theorem paper- though his error is less egregious because he is not using set theory- there is no naive set of Social States- but there may be 'univalent foundations' for his relational algebra. But those foundations could only be known after the human race- or 'Social' relationships- become extinct.

Consider the following

Once I had mustered up enough courage, I would talk with Arrow with complete freedom, and before long I had told him that I had once spent an entire day with Piero Sraffa at Cambridge, and that he had given me a signed copy of his book ...I told [Arrow] that I admired the book for its apparent demonstration that the distribution of income across labor income and profit could not be fully pinned down by economics — that some reference to the political system was needed. Arrow looked at me with a mix of irritation and pity: “I’ll get you out of that soon enough.” And of course he did convince me that Production of Commodities By Means of Commodities had an extra degree of freedom in it that generated a fake indeterminacy, though I still harbor a sneaking suspicion that Sraffa was on to something.--Debraj Ray "Kenneth Arrow, 1921-2017"

Presumably, Arrow showed that Sraffa was committed to constant returns- because 'Sraffian prices' magically maintain a 'steady state'- i.e. things remain as they are because actually there are constant returns though Sraffa does not specify this and thus, effectively, hides the truth. 

The truth, however, is that there are infinite degrees of freedom unless there is perfect information- in which case there would be no need for either language or markets and thus there would be no 'Social' realm. 

It is a fact about the world that there aren't just 'contingent contracts'- e.g. something observable stochastically triggering an outcome under a futures contract-  but that there are 'counterfactual contracts' where nothing observable has this effect. Thus, on block chains, a contract is counterfactually instantiated, if all parties in the channel act as though it has been deployed, even though it has not. This occurs if one party can unilaterally make the counterfactual happen on chain. However, if this condition is weakened, there still is quasi counterfactual instantiation whose strength depends on cost and probability of unilateral deployment. What of impossible, or non compossible, contracts? As a matter of fact, not theory, all sorts of 'ontologically dysphoric' hedges and 'positional goods' and signaling and screening devices exist. Organized Religion and Elite Paideia and State Ideologies and Juristic concepts of uttermost good faith and the bonus paterfamilias etc, etc, all invoke things which are useful or valuable but 'not at home in this world'. 

Under Knightian Uncertainty self-interested agents will form relationships to pool risk such that physical outcomes are associated with a range of welfare distributions. But even under Perfect Information, Arrow Debreu models will have a range of ex ante social orderings associated with any given ex post physicalist vector because 'futures markets' have hedging and income effects rendering them- by the Sonnenschien-Mantel-Debreu theorem, 'anything goes'. How could it be otherwise? 'Futures' are epistemic and depend on shared mathematical assumptions. But one could have 'future contracts conditional upon 'Bayesian' criteria- e.g. if P=NP then, under such and such physicalist vector, one assignment is implemented. If, as we believe, P is not equal to NP, another assignment obtains. 

Of course, what I have said applies only to naive set theory. If human beings have a finite life time in this Universe, it must be the case that there is some 'reverse mathematics' such that there is a canonically parsimonious axiomatic set theory which yields a 'set of Social States'. But it won't be constructible, or, indeed, knowable till long after everybody is dead. The owl of Minerva can only take wing after none can hear it hoot. 

Can we have an ordering over a set we don't know? Only in the sense that there is a 'black Sun' which shines on the back of all things revealing their true purpose in that Cosmic economy we may term the Mind of God. 

In Physics we could speak of a teleological function which assigns a trajectory to every object in the Universe. Knowledge of this function would save us from having to work out the result of interactions between objects. Our grasping after God's gift would have us dispense with Yoneda's Lemma and thus Physics and Economics and Language. We may as well become trees or bushes and live by photosynthesis. 

Knowledge of 'teleological functions' mean we wouldn't need to know about Gravity or the other fundamental forces. Furthermore we could make normative judgments- e.g. that's a bad apple because it will fall on the head of Newton and cause him to invent a useless branch of Physics. On the other hand, that banana is a good banana because the monkey who eats it will discard its skin at exactly the right spot such that our enemy slips upon it and falls down and fractures his stink bone. 

Why do Physicists not do research into the 'teleological function'? The answer is because the thing is useless. We don't know what objects exist in the universe. We don't know their trajectory. Talk of this kind represents mere magical thinking.

Why do some economists, those afflicted with 'Physics envy', still talk of a Social Welfare Function on a physicalist configuration space when this is equivalent to a teleological function? The answer is because they are as stupid as shit. This is not to say that each of us is incapable of conceiving of how Society could be made better or worse. But equally, we could say that some physical objects may have good or bad trajectories. If astronomers see an asteroid hurtling towards the earth they may say 'that's a bad asteroid. Let us nuke the fuck out of it, or else our species will go the way of the dinosaurs.'

However, to nuke the fuck out of asteroids we need to have been doing Newtonian, not teleological physics. Similarly, talk of Social Welfare Functions can't make Society better. The thing is actively mischievous. Consider the following- 

A social ordering R is a reflexive, complete and transitive binary relation that orders social states.

An ordering orders. But 'Social States' can't be ordered because they are not distinct, uniquely specifiable, semantically, extensionally, or exhaustively ostensively, definable, or otherwise constructible as a set. 

In particular, no physicalist vector assignable to a 'Social state', is not equally assignable, by the same method, absent arbitrary 'buck-stopping', to a non denumerable infinity of 'tensed' interpretations, Montague grammars or other witless Presentist Research Programs of pure Time wasting shite.

The set R represents the set of all orderings defined on X.

No. That's exactly what it can't represent because no 'Reflection principle', no Universal Set, can be associated with it. It's just random shite is all. X is a Tarskian primitive arising out of other Tarskian primitives from one point of view till X dissolves that point of view into its constitutive blind spots- or doesn't. Meaningless statements have no entropic arrow. Of such is 'Presentism'. Regret alone constitutes our stasis as a vanishing present, our concurrency deadlock as Kurukshetra's Krishna- apportioning oblivion to Truth's warriors. That's why that charioteer's 'Cosmic form' is his own self-slaying and our lives but its hysteresis.