Sen's mathsy masturbation

The Stanford Encylopedia of Philosophy has this to say on- 

 Sen’s extension of Arrow’s framework

An assumption built into Arrow’s framework is that

Preferences are well defined and thus, by Zorn's lemma, well-orderable or Noetherian- i.e. they satisfy certain ascending or descending chain conditions. The problem is that Zorn's lemma and the well ordering principle are equivalent to the Axiom of choice which can give rise to the Banach Tarski paradox such that an object can, by purely mathematical means, be turned into an infinity of identical objects. Perhaps, this gives rise to Sen's belief that we have as many 'identities' as we have predicates even though we can never be sure a particular predicate is properly applied to us.

Knightian Uncertainty- the fact that we know little about the future- means  preferences can never be well defined, they are 'intensional' and have an unknown 'extension'- at least for beings who evolved by natural selection and who participate in co-evolved phenomena like Society.  Arrow's theorem can only apply to agents who have no need for language or participation in any Social process.

The fact is, our preferences are conditional and mimetic. We only want what we want under certain conditions and provided other smart people are making the same choice. Arrow's framework is wholly useless for the analysis of any actual choosing within a society. All that it says is that when people vote for who should win the Oscars, or essentially meaningless stuff of that sort, the results could well be crap no matter how complicated the voting rule used. 

preferences are ordinal and not interpersonally comparable:

though we frequently say things like 'I know you like waffles more than me- so you have the waffles'  

preference orderings contain no information about each individual’s strength of preference

but no preference orderings actually exist. We neither have knowledge of all the options available to us nor will commit to ranking them. In any case, we can put money values to our preferences just as easily as we can rank them. Money is what greatly facilitates trade. It also explains why we buy things we don't want because they are cheap and we think we can sell them later at a better price. This violates Arrow's useless framework.  

or about how to compare different individuals’ preferences with one another. Statements such as ‘Individual 1 prefers alternative $x$ more than Individual 2 prefers alternative $y$’ or ‘Individual l prefers a switch from $x$ to $y$ more than Individual 2 prefers a switch from $x^{\ast }$ to $y^{\ast }$’ are considered meaningless.

This is because individuals aren't allowed to trade or bribe or beat each other. Preference intensity is revealed by paying money or screaming loudly and shitting yourself or knifing the other guy repeatedly. 

In voting contexts, this assumption may be plausible, as we often may not be able to elicit more information from voters than their ordinal rankings of the options.

But we can see some guys spending a lot of time and money on persuading people to vote one way or another. Also there are lots of people who scream and shit themselves or threaten to run amok if their side loses.  

But in welfare-evaluation contexts—when a social planner seeks to rank different social alternatives in an order of social welfare—the use of richer information may be justified.

Indeed. If the social planner knows he will get lots of money or else avoid being knifed repeatedly, his decision may indeed be affected. More generally, a social planner who wants to continue being a social planner has to make the type of decisions which will keep him in his job rather than occupying a nice slab in the mortuary.  

 

Sen (1970b)

who came from a shithole country where 'social planners' did whatever Sanjay Gandhi told them to do for fear of being beaten or losing their job 

generalized Arrow’s framework to incorporate such richer information.

As before, consider a set $N=\{1,2,\dots ,n\}$ of individuals $(n\ge 2)$ and a set $X=\{x,y,z,\dots \}$ of social alternatives.

Is X well defined? No. We don't really know if a scenario is genuinely a 'social alternative'. Indeed, in some respects it is likely to be 'incompossible'. Moreover, because of Knightian Uncertainty, there are some genuine alternatives which are wholly beyond our experience or imagination. Thus X isn't really a set at all. I am not saying the thing is non well-founded. It just isn't a set of any description. You may say, 'it is a set in the mind of God'. But that is a theological mystery. It isn't philosophy. It isn't economics. It is mystical nonsense.  

Now each individual $i\in N$ has a welfare function $W_i$ over these alternatives, which assigns a real number $W_i\left(x\right)$ to each alternative $x\in X$, interpreted as a measure of $i$’s welfare under alternative $\mathrm{x.}$

An element of a set must have an unambiguous description. This is unlikely if it represents a multiply realizable state of affairs. Moreover, if approximations are used there is an always an arbitrary and ambiguous aspect to the thing. 

Any welfare function on $X$ induces an ordering on $X$,

No. Since X is not well defined, there can be no function on it. An arbitrary ordering may be conjured up but we can't say it represents any sort of mathematical function.  

but the converse is not true:

Why not? There could be an 'upeksha' welfare function which is indifferent to circumstances. I suppose some stoic type of mysticism might approve of all that is- provided it is shitty enough.

welfare functions encode more information.

They are nonsense because there is no well defined set for them to operate on.  

A combination of welfare functions across the individuals, $⟨W_1,W_2,\dots ,W_n⟩$, is called a profile.

Why not call it a unicorn? Like unicorns, the thing does not exist.  

A social welfare functional (SWFL), also denoted $F$, is a function that assigns to each profile $⟨W_1,W_2,\dots ,W_n⟩$ (in some domain of admissible profiles) a social preference relation $R=F(W_1,W_2,\dots ,W_n)$ on $X$, with the familiar interpretation.

Relation algebras only work with finitely many identities. But if there is a Preference there may also be a meta-Preference and so on to infinity. In any case, RA is still incomplete, undecidable etc. 

Again, when $F$ is clear from the context, we write $R$ for the social preference relation corresponding to $⟨W_1,W_2,\dots ,W_n⟩$. The output of a SWFL is similar to that of a preference aggregation rule (again, we do not build the completeness or transitivity of $R$ into the definition[9]), but its input is richer.

But it is arbitrary and ultimately nonsensical because nothing is well defined or unambiguous.  

What we gain from this depends on how much of the enriched informational input we allow ourselves to use in determining society’s preferences: technically, it depends on our assumption about measurability and interpersonal comparability of welfare.

In other words, the whole procedure is arbitrary and the mathematical window-dressing is ignorant and foolish.  

4.2 Measurability and interpersonal comparability of welfare

By assigning real numbers to alternatives, welfare profiles contain a lot of information over and above the profiles of orderings on $X$ they induce.

By assigning money, beatings, and the nuisance posed by people shouting at you and shitting themselves, you can do whatever arbitrary shite you need to do to keep your job and maybe buy your mistress a mink coat. This is because this shite only had currency back when men drank Martinis and women put out in return for costly presents.  

Many different assignments of numbers to alternatives can give rise to the same orderings. But we may not consider all this information meaningful. Some of it could be an artifact of the numerical representation. For example, the difference between the profile $⟨W_1,W_2,\dots ,W_n⟩$ and its scaled-up version $⟨10\cdot W_1,10\cdot W_2,\dots ,10\cdot W_n⟩$, where everything is the same in proportional terms, could be like the difference between length measurements in centimeters and in inches. The two profiles might be seen as alternative representations of the exact same information, just on different scales.

So, just normalize them to yield a unit vector. What's so difficult about that? You are still stuck with arbitrary shite.  

To express different assumptions about which information is truly encoded

arbitrary shite doesn't encode information though it may give us an idea of who is paying off the planner or who is threatening her with a beating.  

by a profile of welfare functions and which information is not (and should thus be seen, at best, as an artifact of the numerical representation), it is helpful to introduce the notion of meaningful statement

Nothing meaningful can be said about functions on non well defined sets. One may as well talk about how to make unicorns less gay.  

Some examples of statements about individual welfare that are candidates for meaningful statements are the following (the present formulations come from List 2003a; for earlier analyses, see Bossert 1991 and Bossert and Weymark 1996: Section 5):

A level comparison: Individual i’s welfare under alternative $x$ is at least as great as individual $j$’s welfare under alternative $y$, formally $W_i\left(x\right)\ge W_j\left(y\right)$. (The comparison is intrapersonal if $i=j$, and interpersonal if $i\ne j$.)

This is not meaningful. It is arbitrary. You may motivate this arbitrary statement by saying 'look, both guys are pretty similar and, what's more could swap places. The fact that they don't tells us something'. The problem is that it is easy to knock down this 'antidosis' based argument.

A unit comparison: The ratio of [individual $i$’s welfare gain or loss if we switch from alternative $y_1$ to alternative $x_1$] to [individual $j$’s welfare gain or loss if we switch from alternative $y_2$ to alternative $x_2$] is $\lambda$, where $\lambda$ is some real number, formally $\left(\right(W_i\left(x_1\right)-W_i\left(y_1\right)\left)/\right(W_j\left(x_2\right)-W_j\left(y_2\right))=\lambda$. (Again, the comparison is intrapersonal if $i=j$, and interpersonal if $i\ne j$.)

Again, this is a wholly arbitrary procedure. Some people may be paid a little money to do shite like this but they will be bribed or beaten or shat on if they get in the way of anybody who counts.  

A zero comparison: Individual $i$’s welfare under alternative $x$ is greater than / equal to / less than zero, formally $\text{sign}\left(W_i\right(x\left)\right)=\lambda$, where $\lambda \in \{-1,0,1\}$ and $\text{sign}$ is a real-valued function that maps strictly negative numbers to $-1$, zero to 0, and strictly positive numbers to $+1$.

We don't know our own welfare under different alternatives. Getting drunk at the Christmas party seems like a swell idea. Buy you regret punching your boss after she asked you to put your clothes back on. 

Arrow’s view, as noted, is that only intrapersonal level comparisons are meaningful, while all other kinds of comparisons are not.

There is an arbitrary element to both. The difference is that the law assumes that any adult not certified as insane will make both intrapersonal and interpersonal comparisons of a reasonable kind. Thus, while the Employment tribunal might endorse your view that getting drunk is welfare enhancing for you- coz you are a drunken scumbag- they may not agree that your boss would have been delighted to be punched in the face. 

Sen (1970b) formalized various assumptions about measurability and interpersonal comparability of welfare by (i) defining an equivalence relation on welfare profiles that specifies when two profiles count as ‘containing the same information’,

The thing is done by arbitrary stipulation. The math is just window-dressing of an ignorant and foolish sort.  

and (ii) requiring any profiles in the same equivalence class to generate the same social preference ordering.

Arbitrary actions may be consistently arbitrary. This does not make them mathematical in any sense.  In this case the three conditions for an equivalence class are all violated. Because of ambiguity, reflexivity is violated (i.e. the thing does not always present as itself) and because the set is not well defined symmetry and transitivity too are out of the window. 

Of the three kinds of comparison statements introduced above, the meaningful ones are those that are invariant in each equivalence class. Arrow’s ordinalist assumption can be expressed as follows:

Ordinal measurability with no interpersonal comparability (ONC): Two profiles $⟨W_1,W_2,\dots ,W_n⟩$ and $⟨W_1^{\ast },W_2^{\ast },\dots ,W_n^{\ast }⟩$ contain the same information whenever, for each $i\in N$, $W_i^{\ast }={\varphi }_i\left(W_i\right)$, where ${\varphi }_i$ is some positive monotonic transformation, possibly different for different individual

Another way to say this is that nothing about how the ordering was arrived at matters. Yet, we know cognitive ability and affect driven behaviour have a strong influence on our preferences. We frequently say things like  'I was hungry and tired when I made that decision which I now regret.' 

Thus the individual welfare functions in any profile can be arbitrarily monotonically transformed (‘stretched or squeezed’) without informational loss, thereby ruling out any interpersonal comparisons or even intrapersonal unit comparisons.

This is foolish. Just have lots of unit vectors and assign each a weighting. People with money, or threat potential or nuisance value get more as do those who can exit the jurisdiction if fucked with.  Better still, don't bother with unit vectors because the underlying set is not well defined. Just pretend to have done a lot of mathsy research before coming up with a plan which will keep you employed or out of the mortuary. 

If welfare is cardinally measurable

by arbitrary stipulation- sure 

but still interpersonally non-comparable,

by arbitrary stipulation once again 

we have:

Cardinal measurability with no interpersonal comparability (CNC): Two profiles $⟨W_1,W_2,\dots ,W_n⟩$ and $⟨W_1^{\ast },W_2^{\ast },\dots ,W_n^{\ast }⟩$ contain the same information whenever, for each $i\in N$, $W_i^{\ast }=a_i{W}_i+b_i$, where the $a_i$s and $b_i$s are real numbers (with $a_i>0)$, possibly different for different individuals.

Again, the assumption here is that how a decision was made- stuff about the state of mind and the information available to the agent- is irrelevant. All relevant information is captured in the ordering. The problem is that there is no ordering. State of mind and information availability and other such ideographic matters have no 'natural' representation. They are arbitrary simply. 

The fact is, we learn to make better choices by choosing under different circumstances. One painful lesson, is don't make investment decisions when coked out of your skull. 

Here, each individual’s welfare function is unique up to positive affine transformations (‘scaling and shifting’), but there is still no common scale across individuals. This renders intrapersonal level and unit comparisons meaningful, but rules out interpersonal comparisons and zero comparisons.

Would it actually did so! The problem is that the minute you impute a preference ordering to a particular agent, you are already implicitly making 'interpersonal comparisons' such that the guy is held to do better by doing foolish shite you personally happen to think good and rational.  

Interpersonal level comparability is achieved

arbitrarily 

under the following enriched variant of ordinal measurability:

Ordinal measurability with interpersonal level comparability (OLC): Two profiles $⟨W_1,W_2,\dots ,W_n⟩$ and $⟨W_1^{\ast },W_2^{\ast },\dots ,W_n^{\ast }⟩$ contain the same information whenever, for each $i\in N$, $W_i^{\ast }=\varphi \left(W_i\right)$, where $\varphi$ is the same positive monotonic transformation for all individuals.

This just means multiplying unit vectors by the same scalar. It does not 'enrich' any fucking information base. On the other hand, to say 'people didn't know the effect of smoking or using leaded petrol or being a nasty racist or homophobic cunt' is quite useful. It explains why older people like me fucked up the globe so badly. Information is not in things, it is in the minds that choose between things. Those minds are socialized. Social Choice is about Society not some pseudo-mathsy shite indulged in by Professors conning their students into thinking that they will be able to reorder and reshape society just as soon as they get their sheepskin in Social Choice Theory. 

Here, a profile of individual welfare functions can be arbitrarily monotonically transformed (‘stretched or squeezed’) without informational loss, but the same transformation must be used for all individuals, thereby rendering interpersonal level comparisons meaningful.

How? It is not meaningful to be told that X has 100 utiles and Y has 100 utiles because that's how we rigged the game. If X is a smart guy who makes great decisions and is loved by all who know him and Y is a rapist rotting in jail, we have gained no information whatsoever. Our time has been wasted.  

Interpersonal unit comparability is achieved under the following enriched variant of cardinal measurability:

Cardinal measurability with interpersonal unit comparability (CUC): Two profiles $⟨W_1,W_2,\dots ,W_n⟩$ and $⟨W_1^{\ast },W_2^{\ast },\dots ,W_n^{\ast }⟩$ contain the same information whenever, for each $i\in N$, $W_i^{\ast }=a{W}_i+b_i$, where $a$ is the same real number for all individuals $(a>0)$ and the $b_i$s are real numbers.

Here, the welfare functions in each profile can be re-scaled and shifted without informational loss, but the same scalar multiple (though not necessarily the same shifting constant) must be used for all individuals, thereby rendering interpersonal unit comparisons meaningful.

If this is meaningful, what is nonsense?  

Zero comparisons, finally, become meaningful under the following enriched variant of ordinal measurability (List 2001):

Ordinal measurability with zero comparability (ONC$+0)$: Two profiles $⟨W_1,W_2,\dots ,W_n⟩$ and $⟨W_1^{\ast },W_2^{\ast },\dots ,W_n^{\ast }⟩$ contain the same information whenever, for each $i\in N$, $W_i^{\ast }={\varphi }_i\left(W_i\right)$, where ${\varphi }_i$ is some positive monotonic and zero-preserving transformation, possibly different for different individuals. (Here zero-preserving means that ${\varphi }_i\left(0\right)=0$.)

This allows arbitrary stretching and squeezing of individual welfare functions without informational loss, provided the welfare level of zero remains fixed, thereby ensuring zero comparability.

Wonderful! Everybody is zero. That's very helpful if you are trying to find yourself a spouse! 

After a little more mathsy masturbation, the article concludes-

Which assumption is warranted depends on how welfare is interpreted. If welfare is hedonic utility, which can be experienced only from a first-person perspective, interpersonal comparisons are harder to justify

fuck off! We've all seen 'when Harry met Sally'. The fact is, you choose the same dish as the lady who is experiencing orgasmic satisfaction by eating it.  

than if welfare is the objective satisfaction of subjective preferences or desires (the desire-satisfaction view)

this militates against interpersonal comparisons. People may be more different from each other than is convenient for suppliers of private or public goods.  

or an objective good or state (an objective-list view).

stuff like how many calories people are consuming. Sadly, affluence tends to mean people need less calories. A friend of mine is a billionaire. I noticed that all his furniture is second hand. I patted him on the back and promised to buy him something nice from Ikea. He started babbling about 'Chippendales'. Clearly the fellow has to prance around naked to earn a little money. I thanked my lucky stars I stuck to Socioproctology rather than specializing in Econophysics. 

The desire-satisfaction view may render interpersonal comparisons empirically meaningful (by relating the interpersonally significant maximal and minimal levels of welfare for each individual to the attainment of his or her most and least preferred alternatives), but arguably not in a normatively attractive way (Hausman 1995).

The thing is arbitrary. But so is normativity. The Taliban have very strong views on Social Choice. It is safer not to argue with them about this in Kabul.  

Different individuals’ most preferred alternatives may differ significantly with respect to how costly they are, for instance due to some individuals’ expensive tastes or adaptive preferences, and it is not obvious whether it is fair to treat a modest individual’s welfare under, say, a diet of cheap food as being equal to the welfare of someone who finds only caviar satisfactory.

What an amazing discovery! I now see that I was wrong to dismiss Sen's mathsy masturbation. It can lead to great insights- e.g. a guy who subsists on bread and water isn't as well off as a guy scoffing caviar and quaffing champagne on his super yacht.  

Resource-based, functioning-based, or primary-goods-based currencies of welfare, by contrast, may allow interpersonal comparisons in a way that is less morally problematic.

But such comparisons can be done better and more cheaply by uneducated and unemployable shitheads. They can improve on Sen's analysis by discovering that a guy earning a million dollars a year who has a hot g.f and drives a Porsche is better off than a guy who earns 50,000 a year and who keeps getting beaten up by his obese wife.

Social Choice, au fond, is biological. It is about who gets access to reproductive resources. One type of affluence leads to the rich having more progeny. Another type leads to demographic replacement. This can have important political effects but was little guessed at by academics working in this wholly useless field.