In article <email@example.com>, Dan Goodman <dog@fcbobDOTdemon.co.uk> wrote: >> What does "statistically unbiased" mean here? In statistics, >> an unbiased estimator is one whose expectation is the value >> of the parameter; in testing, an unbiased test is one for >> which the probability of rejection under any alternative is >> at least as great as under the null hypothesis.
>> If one adds randomization, the proof of the Arrow Paradox >> becomes extremely short; the problem is that the utility >> function has an arbitrary scale parameter.
>What I meant by it was that in a "random democracy", nobody's vote is more >important than anyone else's, which isn't true in general for nonrandom >democracies. In the recent American elections for example, a vote for Nader >(at least in terms of deciding who the president is, it has other purposes) >is essentially a wasted vote. The random democracy system has a few things >going for it, for instance strategic voting wouldn't happen, because there >really would be no point voting for anyone except who you really want to get >in. Of course it has disadvantages too, it's much more susceptible to >corruption for instance.
I still have no idea what you mean by "random democracy". Whatever voting scheme you propose, one can produce a situation in which the results will be paradoxical.
>How would you adapt Arrow's proof to deal with randomization? I'm not >entirely sure what this would mean.
Let me first state what the essence of Arrow's thesis is.
The object is to come up with a method for society to decide what action to take, based on the preferences of the individuals, and you want this to be consistent with itself. In addition, the comparison of two actions cannot depend on how other actions would be rated; it is this one which needs to be slightly modified for randomization, as if A is preferred to B, A with probability 90% and B with probability 10% is preferred to A with probability 40% and B with probability 60%.
It is well known that all of the voting schemes fail, including whatever yours may be, if we could figure out what it is.
I suggest you look at my paper on utility in _Statistics and Decisions_, 1987. It is self-contained.
-- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399 firstname.lastname@example.org Phone: (765)494-6054 FAX: (765)494-0558