peteg's blog - noise - books - 2016 09 19 TheArrowImpossibilityTheorem

Maskin, Sen, Arrow: The Arrow Impossibility Theorem (Kenneth J. Arrow Lecture Series).

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Kindle. I discovered this and the lecture series via a review by Athan (see also Dick Burkhart's). It is OK. Amartya Sen's proof is quite slick, and while I didn't think about it too deeply, it seemed quite close to the proof I mechanized from his 1970 classic Collective Choice and Social Welfare. More modern proofs try to juice the theorem for other insights; Saari gives a geometric analysis, for instance, that I never get around to comprehending.

The book consists of two lectures by Amartya Sen and Eric Maskin, a response by Kenneth Arrow, three papers about related issues and an introduction. As usual Sen is mostly concerned about social welfare implications, and gets a bit obscure. In many ways he is doing philosophy here, and as a result where he ends up is not very satisfying. Maskin focuses on implications for voting, and with Dasgupta claims to show that the majority rule is the most robust one on offer, in a precise sense, in a sort-of generalization of May's Theorem. Unfortunately they require a continuum of voters, which seems nuts; unbounded, possibly countably infinite, well, maybe, but a continuum? (They claim things work just as well with a large but finite number (p108), and I would have kept reading if their main development had in fact used that. Also see the coment at the bottom of Athan's review about measurability.) Arrow is politely skeptical in his commentary:

I do not yet quite understand how Eric's results can help us in the case where his conditions fail. [...] When you are dealing with infinite dimensional elements, can you really compute the results? Some things are simply quite extremely difficult to compute. They’re not constructible in the sense that there is no finite process that will enable an individual to carry out the calculation. This applies to a lot of problems, not just those that are social in nature, such as climate change, but also to individual as well as social choice problems. To put it more simply, you could say, "You choose the best of that heap." But then how one exactly does that can be quite complicated if not impossible in a finite length of time.

Arrow also endorses the comparison of personal utilities ala the behavioural economists, if only because people find these questions meaningful (and despite "hard boiled" economists having difficulty in modelling them). He provides some cutesy anecdotes about this work of his, of more than sixty years ago.