Wednesday, 25 November 2015

Lecture 20

I have thought that existing proofs of the Gibbert-Satterthwaite theorem are necessarily complicated and hard to explain. I was delighted to find a recent paper Yet Another Very Short Proof of the Gibbard-Satterthwaite Theorem, by Ville Korpela, October 2015, which I have adapted to provide the proof in section 20.3 and which I presented in today's lecture. I have added to pages 96-98 of the notes the proof that was given to students last year. You can compare and see which you think is easier to understand.

I did not prove Arrow's impossibility theorem. You can see at the link that its proof is quite long-winded.

It may at first seem a bit strange in 20.1 to call $f$ a social welfare function. Perhaps "social preference function" sounds more natural. However, in general, a social welfare function is a function which allows us to compare any two alternatives and say which is better, or that the are equal. In other contexts this can be done because the social welfare function associates to alternative $a$ a real number, say $f(a)$, and then alternative $a$ is preferred to $b$ iff $f(a)>f(b)$. Arrow's set up is more general,because $f$ outputs a linear order on the set of alternatives, $A$. So $a$ is preferred to $b$ iff $a\succ b$, where $\succ=f(\succ_1,\dotsc,\succ_n)$.

I have pretty much finished writing the lecture notes up to the end of the course, Lecture 23. The notes now include a table of contents and index. Since this year I have made a new reworking of the material I would appreciate hearing from you about any typos that you find in the notes, or things which you think could be better explained, while maintaining conciseness. I am deleting this year the topic of stable matchings which was originally listed in the draft schedules.