From: "Steve Eppley" <[EMAIL PROTECTED]>
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
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