Alex, I think you have it right. (In my experience, the only way to be sure
of 'not missing something' is to write out everything, including full
definitions, in one place, not partly in a message and partly on a web page
or textbook.)
Yes, CR can violate the conclusion of Arrow's theorem because CR's inputs
violate one or more premises of the theorem. Put positively, CR inputs
incorporate info of value to voters which Arrow's hypotheses disregard,
namely: not only the EXISTENCE of pairwise preferences (i.e., A v. B
evaluated as one of the three values +, 0, -) but moreover the quantitative
(potentially real-valued) STRENGTHS (i.e., importances to the voter) of
those preferences.
The only weighty objection raised to CR seems to be that CR is
'strategically' equivalent to Approval, so that a CR ballot is, allegedly,
needlessly complex.
Well, the same issues vis-a-vis Approval (arguably the simplest rational EM)
exist for other methods too.
The problem of describing and then correcting for 'strategy' exists for all
methods - and is rather grosteque for non-monotonic methods. If you choose
to let it be so, strategy can be an endless source of worry, not least
because of the infinite variety possible in the degrees of info completeness
or lack thereof - as you consider various possible elections, and within
each election the various factions.
Note that in a usual human mass election, prospects will anyhow be de facto
next to nil that your vote will make any instrumental difference. So, at
least with a monotonic method like CR - including Approval as the simplest
case - you can do rather well by voting reasonably 'close' to 'sincerely':
you thereby at least have the satisfaction of your vote expressing something
like your actual sentiments.
Joe Weinstein
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- [EM] Arrow and CR Alex Small
- Joe Weinstein
