It all begins with Kenneth Arrow

 

I have been nibbling at the edge of Amartya Sen's book, COLLECTIVE CHOICE AND SOCIAL WELFARE in recent posts. 

Today I'd like to dive into the heart of it. But to do that, I have to start with Kenneth Arrow. Arrow is the author of the "impossibility theorem," the logical argument that there is no optimal way to make collective choices -- no voting system, in particular, that will not be open to devastating objections. 

Arrow's argument begins with premises about what is an acceptable, or unobjectionable. We want a system in which each of the five will be true:

1) The system can handle any level of pluralism in its inputs (universal domain).

2) The system does not produce paradoxical circles (in which a beats b, b beats c, and c beats a). This demand is known as ordering. 

3) If all individuals in the society prefer x to y, then so does "society." The "weak Pareto principle."

4) The independence of irrelevant alternatives. The choice between x and y depends on what voters think of x and y, not a or b.

5) There  be no ‘dictator’, who always determines the social preference, regardless of other individuals' preferences. 

Simple pairwise majority voting satisfies all of these conditions except ordering.

Now: Arrow offered a logical proof that no system can ever satisfy all of these requirements. This is the "impossibility theorem." I won't try to replicate it now -- I have frankly no confidence that I understand it. But I will say this: 

(A) Arrow's argument has generally been thought to be valid given his premises as he understood them 

(B) Sen generally supported this inference -- in part of the book he sets out a proof that he says is better than Arrow's own, with some of its kinks ironed out,

(C) much of the debate about this theorem since Arrow first set it out  in 1950 turns on which principle it would be best to modify -- that is, if we can lessen our expectations in one of these five respects can we achieve something that is good, and doable? 

(D) it seems to me that part of the proof is this -- many systems fail because they create a dictator, in much the sense that Manchin is now the dictator for the US Senate, and so fall afoul of the fifth premise. 

(E) Sen seems to believe that there are sinister implications hidden in the third premise, that of Pareto. So he attacks the problem largely by attacking those implications.