"Social choice and threshold phenomena: bad news and good news"
Abstract: Condorcet's ``paradox'' demonstrates that if we use the majority rule we may reach a situation where the society will prefer A over B , B over C and C over A. Arrow's impossibility theorem asserts that under some natural conditions, if there are at least three alternatives then every non-dictatorial social choice lead to such ``irrational'' choices for the society. A class of social choice functions is called genuine if there is no small set of individuals with decisive power.
The bad news:
1. As the society grows, (under mild conditions) every genuine social choice mechanism must lead to chaotic (not merely irrational) outcomes for the society. 2. When the individual preferences are restricted if the majority may lead to a non-rational outcome, then so will every genuine rule. (Related to a recent theorem by Dasgupta and Maskin.)
Some good news: Under some probabilistic assumptions on the preferences of the individuals (bias, positive correlation, etc.), every genuine social choice function leads typically to rational choices for the society.
The tool: These results are consequences of recent insights concerning the case of TWO alternatives, mainly threshold behavior and also sensitivity to noise.
More bad news: 3. (with Saharon Shelah) Arrow's theorem applies for general classes of choice functions.