Sunday 10 September 2023

Broughel wrong on Arrow's theorem

In Economics, one frequently has to look at how the choices you are faced with could result in different outcomes depending on what other people are doing. They too are making such decisions. This 'strategic' element in decision making, which arises out of 'interdependence', means that you sometimes falsify your preferences or do the opposite of what you might otherwise be expected to do. Under radical interdependence, which is what obtains in the political sphere, Preferences and 'the pay-out matrix' are not well-defined. Mathematics has no way to represent the situation. Still it can model it well enough by making some simplifying assumptions- e.g. that agents act more simply or the situation is more simple than it actually is. This is a rough and ready procedure of a purely pragmatic or utilitarian type. It can't produce any great or fundamental insight into anything. It merely takes some well known 'bug' of the system and gives the reductio ad absurdum of it as if it were actually a feature. 

One example is what happens when you try to make a spread sheet of alternative scenarios. You have to assume that interdependence is too small to bother with otherwise all you have is formulae with inputs from other cells  in the table in every cell of the table. There is no way to evaluate the result because of 'impredicativity'- i.e. no cell can be computed because every other cell has to be computed using the computed value of each cell- which hasn't been computed! To get around this, we can stipulate that cells don't really depend on each other. Alternative scenarios are irrelevant for the computation of a particular cell. Indeed, to say a person has a mathematically well defined preference function depends on just such an arbitrary stipulation. This is the 'Independence of irrelevant alternatives' axiom of Arrow's theorem.

Econlib has an article titled-  

Kenneth Arrow’s Overrated “Impossibility Theorem”

By: James Broughel

Kenneth Arrow‘s impossibility theorem, a pillar of neoclassical economic theory, argues that under certain “seemingly reasonable” conditions, no social welfare function can adequately represent the preferences of society’s members.

An individual's preferences are different from what he would choose acting as a member of a family, an enterprise, a particular religion, or as the patriotic citizen of a State.

None of us have any knowledge of or any 'individual' preference over the vast majority of Social States. That is why we delegate decision making. I let my wife manage the family budget and choose what food we will have for dinner even though I know I sometimes might not like the choices she makes.  I can choose which Church to attend and which Political Party to vote for. I can't get any Bishop or Cabinet Minister to follow my own, generally foolish, preferences. 

This means that there will always be 'local dictators'. Moreover, if preferences are epistemic- i.e. they change as information set changes- then two separate mathematical problems arise such that there can be no 'non-arbitrary' SWF. The first is 'impredicativity'. We may prefer to 'choose what the smart guys are choosing'. But the smart guys may want to choose what the majority chooses. This is like the problem of the 'Keynesian Beauty Contest'. 

The second problem is known as the 'intensional; or 'masked man' paradox'. Where the extension of a set has an intensional element- there can be no non-arbitrary mathematical function from it to anything else. 

A social welfare function is a mathematical formula that combines individual preferences in some manner to reflect societal preferences.

Sadly, for any general purpose, there can be no such mathematical formula save of a trivial or arbitrary type. But we can't prove this. I could say 'the mathematical formula will be found on page 13 of the first correct proof of the Reimann hypothesis'.  We can't be sure such a thing will ever exist. But we can't be sure it won't. 

The impossibility theorem is sometimes colloquially understood as implying that aggregating individual preferences to form a social preference ordering is impossible.

It is possible, but arbitrary.  

However, the restrictions Arrow placed for his theory to hold are far from reasonable, undermining the validity of his theorem.

They are crazy. He defines somebody who is clearly not a Dictator as a Dictator. This is ex falso quodlibet.  

Let’s take a closer look. One of the constraints Arrow includes in his theorem is the “independence of irrelevant alternatives” (IIA). This supposes that a person’s preference between two options shouldn’t be influenced by other choices.

Other wise the thing is not well ordered or its extension is unknown. It can't be the subject of any mathematical operation.  

Essentially, it posits that preferences shouldn’t be dependent on any other alternatives becoming available.

Which may be the case 'at the end of mathematical Time'- but not before that.  

Some economists have likened the IIA assumption to choosing amongst flavors of ice cream. Whether I prefer chocolate to vanilla ice cream shouldn’t depend on whether strawberry is available, or so it is argued.

Otherwise, preferences are 'epistemic'. Knowing strawberry is available tells you something about the type of vendor selling ice-cream. Only stupid kids eat strawberry ice cream. Posh vendors won't serve it. In this case you are safer with vanilla because the chocolate is bound to be crap.  

The ice cream metaphor is a poor one, however, given the trivial nature of the alternatives. So let’s break down the IIA assumption with another, more appropriate, example.

Suppose you’re torn between going out to party tonight and staying home to study for an exam. At face value, partying appears to be the more enjoyable option. However, a third option, like “getting into a good college,” may depend on your decision. Clearly, this third choice—the future consequence—ranks higher among your preferred alternatives than the immediate option to party. This is despite the fact that partying, when taken independently of future consequences, seems to be the preferred choice over studying.

Again, this is epistemic. I didn't know 'getting into a good college' was linked to studying boring shite. I thought it was what happened to nerdy peeps who are sent off to a place where they can bump uglies or become Communists. 

Herein lies the problem with Arrow’s impossibility theorem: it overlooks long-term consequences

which are 'epistemic' or 'intensional' 

and focuses too much on immediate gratification.

If all kids party instead of studying, 'good colleges' have to drop their entrance requirements. This is 'impredicativity'.  

This is troubling because much of today’s welfare analysis, including cost-benefit analysis, is based, at least indirectly, on Arrow’s theorem,

No. It is based on cardinal utility- or the transferable utility which is money- which thanks to Szpilrajn's extension theorem- all ordinal utility can be made to yield.  

leading economic theory broadly and public policy specifically to take a short-sighted approach.

If the Government is doing CBA it has a simple criteria- viz. spend on stuff which boosts future tax receipts or cuts future spending. If you go off a fiscal cliff your country may be governed by somebody else.  

Rejecting Arrow’s theorem shouldn’t be controversial. Nor should it be a politically divisive issue.

Rejecting Arrow-Debreu- which assumes no Knightian Uncertainty- shouldn't be controversial. In an Arrow-Debreu world humans would have no need for language or communication or education of any type.  

Many from the political left already contest Arrow’s theorem and its rigid IIA assumption.

Arrow's theorem merely says there is no non-arbitrary way to do Social Choice. The Left thinks that means they will get to do whatever crazy shit they like. They forget that what is sauce for the goose etc.

The fact is individuals and Societies and Enterprises have to do stuff which enables them to survive. Choosing to do stupid shit undercuts your future ability to choose to do stupid shit on the same scale. 

However, those on the political right have been slower to disavow it, perhaps because of a natural skepticism toward a social planner dictating societal values through a social welfare function.

Screw that. Social Choices, like a Family's choices or an individual's choices, depend on budgeting and being prudent and following a 'regret minimizing' strategy. On the one hand this means you do some crazy shit For Fear Of Missing Out but, otherwise, you mainly settle for 'Tardean Mimetics'- i.e. imitating what smart peeps be doing.  

This is a mistake. Ironically, rejecting the use of any social welfare function at all, as some libertarians do, also implies rejecting the market process—which itself is guided by a social welfare function of sorts.

Nonsense! Market processes are 'co-evolved' methods of taming complexity and exploring a fitness landscape. The aren't about 'optimization' or Category theoretical 'naturality'.  

Arrow himself acknowledges as much in his book that presents the impossibility theorem, Social Choice and Individual Values, when he concludes that “the market mechanism does not create a rational social choice.”

But Societies don't actually use markets to decide who gets to be a surgeon or a bishop or a General. I can't buy the right to operate on you- much as I would like to do so. I also can't pay King Charles to become his official concubine. Sad.  

Strangely, most libertarians have failed to heed the lesson.

A proper Libertarian will shoot me in the head if I turn up on their doorstep with a piece of paper saying I have purchased the right to perform surgery upon them.  

Without the IIA condition, the impossibility theorem falls apart.

It was nonsense to start off with. Social Choices are made on the basis of Tardean mimetics- us guys imitate what the smart countries in our neighbourhood are doing- and feature 'transferable utility'- i.e. paying off vested interest groups.  

The key takeaway here is that economists need to revisit some of their foundational theories.

No. The key takeaway is that 'foundational theories' which ignore Knightian Uncertainty are stupid shit.  

Indeed, much of modern welfare economics needs a fresh look and reevaluation.

No. Just concentrate on Budgeting. The Government should spend on stuff which boosts future Tax receipts otherwise there is a Fiscal Cliff or Ricardian fucking Equivalence and a massive crowding out effect.  

The mid-20th century, sometimes regarded as a golden age of economic theory,

Bretton Woods kept politicians on the straight and narrow. Guys who understood Public Finance were bureaucrats or businessmen, not fucking academics who were misapplying mathematics.  

may well have been a breeding ground for economic errors. We need to rectify these, and a good starting point would be to revisit Arrow’s impossibility theorem

No. We need to do whatever it is the Chinese have done so as to get ahead of us in some utterly vital technological fields. Maybe Vivek Ramaswamy understands that sort of stuff. What is certain is that Arrow and Sen and Chichilnisky are wholly irrelevant.  


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