Friday, 21 April 2017

An Arrowvian Social Welfare Function is ab ovo Dictatorial.

Suppose there are only two feasible social states for the world identical in every way except that in one an Arrowvian Social Welfare function (ASWF) has been implemented.

Suppose there is only one agent. We don't know in advance which state of the world she would pick. It would appear that she has two choices because there are only 2 possible states of the world. However, there is a third possibility. She may want a world in which the ASWF is implemented by her choice.

If she wants this third possibility, and we don't rule it out before hand by an arbitrary or dictatorial action, she also may want to know this has been done. But, this means there is a new State of the World- viz. one in which the agent knows that the ASWF has been implemented.

Why stop here? Why not an infinite number of possible States of the World such that she has x per cent likelihood of discovering that an ASWF has been implemented with x being a real number between 0 and 100?

Clearly this means there is an uncountably infinite number of feasible social states if the existence or implementation of the A.S.W.F can itself be a subject of agents' preferences. Moreover, there is no way to verify if the A.S.W.F is doing its job. What if the agent stipulated that her likelihood of discovering that an ASWF has been implemented take an uncomputable value?

It is possible that the agent actually only wants or does not want an ASWF to be implemented. However, a priori, we can't rule out the possibility that she might have the following preference 'If it is going to rain on my picnic on Sunday, then I want to have certain knowledge that the ASWF has been implemented on Saturday.' It may be that there is enough evidence in the State of the World on Saturday to compute, with certainty, whether or not it will rain on her picnic on Sunday.

Clearly, if an ASWF can do complex calculations beyond our reach, a rational agent in a single person economy should want it to be implemented- unless it uses up scarce resources in its operations thus itself altering the State of the World.

I suppose we could bar an ASWF from giving agents this sort of information, or, indeed, from possessing it. We can restrict admissible preference profiles- and this is what Arrow does. However, this is a dictatorial act.  Moreover, if the functioning of an ASWF uses up scarce resources, it would be irrational to prefer its existence unless it itself generates Welfare. But if it is possible for it to generate Welfare simply by existing, then it is a proper input for agents' Preference profiles. To exclude it is arbitrary and dictatorial.

Returning to our one agent economy, what do we find? The agent, if rational, would not want a ASWF to be implemented because it is silly. The fact that a particular agent may be irrational or derive Utility from the knowledge that an ASWF has been implemented does not change the fact that it is possible for the agent to have preferred otherwise. Unlike a Bergsonian SWF, possible preferences matter to an ASWF. Thus, in a single agent economy, no ASWF would be implemented unless it were either dictatorial (in the matter of admissibility of preference profiles) or else non-deterministic and thus not an ASWF at all.

Suppose there are n agents in an economy. They think implementing an ASWF stupid and so it isn't implemented because that would be Dictatorial. Add one more agent. The ASWF still won't be implemented unless the new guy is a Dictator.

Thus an ASWF is dictatorial ab ovo.
It is a silly idea.

Is there some technical sense it which it isn't silly?
Notice in the following the crucial importance of the notion of 'weak ordering'.


The problem here is that the domain of f can be extended by adding 'prefer to have an ASWF implemented iff it makes no difference whatsoever' for every agent. Assuming people don't get negative or positive utility from an ASWF being implemented, then the conditions of U and SO are met by this newly extended domain since agents are indifferent to the new alternative. Whatever was a weak ordering of X is also a weak ordering of X extended to include the implementation of a completely neutral ASWF. 

By the conditions of 'Weak Pareto' and 'Independence of Irrelevant Alternatives'- this must be the case. Thus, an ASWF can itself feature in its own domain thus validating itself Democratically at the price of impredicativity. However, it will never be validated because even if people are irrational, it is possible that they might be rational and an ASWF can't dismiss that possibility in advance without being dictatorial.

To quote, once again, from the Stanford Encylopedia's article on Arrow's theorem, re. 'Independence of Irrelevant Alternatives'- 


Restricting Preference Profiles to (implement ASWF/ don't implement ASWF) the condition stated above requires that a third possibility viz.'conditionally implement ASWF'  is irrelevant. But this violates
because every rational person would prefer 'conditionally implement ASWF' to 'implement ASWF'. There may be an irrational person who chooses otherwise and Unrestricted Domain means we can't rule out this possibility. However, the ASWF must be dictatorial if it implements itself on the basis of such a person's possible existence.

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