[EM] Re: Arrow's axioms

[Ken Johnson]

>
> From: "Steve Eppley" <seppley at alumni.caltech.edu>
> Date: Wed, 03 Mar 2004 06:58:22 -0800 
> ...
>
> I consider Arrow's axioms justifiable.  In the decades leading up to 
> Arrow's theorem, economists and social scientists had struggled in 
> vain to find a good way to compare different individuals' utility 
> differences (known in the literature as the problem of "interpersonal 
> comparison of utilities") in order to be able to calculate which 
> outcome is most utilitarian.  That is, they were interested in being 
> able to sum for each alternative the utility of that alternative for 
> each voter, so they could elect the alternative with the greatest 
> sum.  By Arrow's time, they'd learned that, lacking mind-reading 
> technologies, they couldn't elicit cardinal utilities that could be 
> compared between individuals, for instance to compare the utility 
> difference between your "100" candidate and your "0" candidate to the 
> utility difference between my "100" candidate and my "0" candidate.  
> Simply summing our reported numbers, which don't have units (such as 
> dollars) attached, would not help them find which alternative had the 
> greatest utility.  If each voter is constrained to assign numbers 
> within a given range, such as 0 to 100, then the sum would not be the 
> utilitarian sum.  Maybe these sums aren't worthless, but they need 
> careful scrutiny. 

Arrow's axioms could well be justifiable, but his proof doesn't provide 
the justification. There may be good reasons why CR should be rejected 
as a viable election method, but Arrow's premises don't elucidate those 
reasons because if the theorem were generalized to encompass cardinal 
methods, its conclusion would be that rank methods cannot satisfy the 
axioms whereas CR can.

>
> Also, as you know, asking each voter to freely assign numbers within 
> some range would create a strong incentive for individual voters to 
> exaggerate, so that in the long run the information elicited from the 
> voters by a cardinal utility method would be no greater than the 
> information that can be elicited by Approval. 
> In the worst case, the socially responsible voters would fail to 
> exaggerate and the selfish voters would exaggerate. I consider this 
> case extremely disturbing. 

I agree, but even though CR may be an impractical election method, I 
think the economists' and social scientists' original idea of "maximal 
social utility" (i.e., "sincere CR") would provide a useful basis for 
comparing the merits of alternative voting methods. Even though you 
cannot know voters' sincere cardinal ratings of candidates, you could 
predict the outcome of any particular voting method given some presumed 
sincere CR profile and could quantify the outcome's "social utility" 
over a statistical ensemble of all possible CR profiles. Which voting 
method would result in greater social utility, on average? How would 
alternative methods compare in terms of their worst-case performance 
relative to sincere CR?

>
> Arrow also reasoned that the information about the voters' preferences 
> that can be elicited by Approval is far less than the information that 
> can be elicited by letting each voter express an ordering of the 
> alternatives. ...

Seems reasonable, but by the same rationale, the information that can be 
elicited from preferences is much less than that of CR (e.g. "A>B" just 
means my A rating exceeds my B rating by some number in the range 
1...100; there is no way that I can express strong preference distinctly 
from weak preference).

This brings up an interesting question. Suppose my sincere CR profile is 
the following,
A(100), B(90), C(0)
If I strategically increase my B rating, I make it less likely that C 
will win, but at the expense of also decreasing the chances for my 
favorite candidate A. If I decrease my B rating, this increases A's 
chances but increases the risk that C will win. There is a tradeoff, so 
what is my optimum strategy? In this case, with no prior information 
about how other voters will vote, my best strategy is to give B a rating 
of 100. My question is, does an analogous condition hold with ranked 
methods? Should I rank A and B equal (A = B > C)? Since my ranking 
options allow me to express indifference I can always provide a ranking 
that is equivalent to Approval. Would that be my best strategy, as it is 
with CR (assuming that all other voters follow a similar strategy)?

> ...That makes sense to me, and I further believe that the best methods 
> of tallying preference orders will lead to better outcomes for society 
> than if Approval is used, over the long run.  ...

You could be right, but can you formalize your notion of "better 
outcomes for society" and "over the long run" in such a way that the 
correctness of your belief could be demonstrated by way of a 
mathematical proof?

Ken Johnson