[gjm11]

A few remarks:

1. Arrow's theorem concerns the situation where your election procedure needs to deliver (not just a single winner, but) a ranking of all the candidates. You might hope that relaxing this condition will help, but ...

2. There's a closely related theorem with the magnificent name of Gibbard-Satterthwaite, which says that if you have more than two candidates, any procedure that takes in ranked preferences and spits out a single winner must (1) give all the power to one voter, or (2) leave at least one candidate unable to win whatever the voters' preferences, or (3) be susceptible to tactical voting, meaning that in some situations a voter does best to rank the candidates in an order that doesn't match his or her actual preferences.

3. However, there is a loophole "at the other end". For instance, if the input consists not of rankings but of scores (e.g., from 0 to 100), then the conditions of Arrow and Gibbard-Satterthwaite don't apply. And, in fact:

4. If there are only three candidates then "range voting" or "score voting" (each voter scores every candidate and the candidate with best average or total score wins) has the desirable properties Gibbard & Satterthwaite forbid for ranking-based voting systems. (Almost: sometimes optimal voting strategy might require you to give two candidates the same score even though you have a definite preference between them.) But, alas,

5. With more than three candidates no score-based system has those properties either.

(An interesting simplification of range voting is "approval voting", where the only possible scores are 0 and 1.)

[MichaelDickens]

For #2, what if you relax the "no tactical voting" requirement to say that voters cannot predict how to vote tactically unless they have an impractically large quantity of information about other voters?

[YokoZar]

This is the insight. Proving "tactical voting is always theoretically possible" sounds bad, until you realize that there exist systems where tactical voting requires not only nearly perfect information about every other voters exact preferences ahead of time, but also solving cryptographically hard problems

[dllthomas]

I think I'd be satisfied with that. Or ideally, "given attainable information, an honest vote is the most likely to make the output most agree with me."

[logicchop]

I don't follow this "loophole" you mention. Afaik there are proofs of Gibbard-Satterthwaite that allow indifference in the rankings, and this "distinction" between rankings and scores is methodologically dubious: Arrow himself was sensitive about the meaningfulness of quantitative reports of preference, and I don't see any reason to believe that a "score" issued by a voter is any better or more meaningful than a simple ranking (including rankings of indifference). This approach has always struck me as an unmotivated anti-empirical gimmick to get around this-or-that condition in the theorem(s).

[reagency]

Scoring lets you indicate ties, and strong preferences.

[dragonwriter]

> Scoring lets you indicate ties, and strong preferences.

Non-forced rankings allow you to indicate ties, and quantitative indications of strength of preference are unlikely to have consistent meanings among voters, so treating them has having the same meaning (as, presumably, any voting system using them must) is inherently problematic.

[logicchop]

indifferent rankings already indicate ties, and what does a 'strong preference' value amount to here? I'd suggest that it only makes sense if you try to understand the "score" as being a relative ranking of N imaginary alternative candidates in addition to the actual candidates.. So a 1 to A and 99 to B only indicates 'strength' in that it suggests if there were 96 other candidates you'd put them above A and below B.. but now we are back to orderings over candidates. At any rate, you ought to be suspicious of quantitative assignments of preference..

[ClayShentrup]

No. Utility is not about ordering. E.g. suppose you prefer X over Y over Z, and can have a guarantee of Y, or a 50/50 probability of X or Z. If your preference for Y is greater than the average of X and Z, then you would want to take Y. If Y is less than that average, then you want the lottery.

You can arbitrarily change the 50/50 probability to e.g. 40/60 or what have you, to make it equally preferable to any guaranteed item.

[logicchop]

Yes, we know that preference involves more than ordering and that "strengths" are relevant to decision making. We know this because it explains an obvious empirical matter, that there are scenarios like the one you cite where we prefer X over Y, but end up choosing Y over X. Not a puzzle; not a paradox; how it is. And of course the way we tease out the "strength" of your preference involves, e.g., looking at how your choice varies between counterfactual scenarios involving different probabilities.

But the question is whether a "score" given by a voter on a ballot indicates anything psychologically interesting, and, more importantly, whether it indicates anything relevant to the "judicious" selection of a candidate. It seems to me that, empirically, the only meaning we can _seriously_ assign to a "score" on a ballot is a relative-scoring-as-indicative-of-relative-preference. I don't know how "10 to A, 20 to B" could indicate anything beyond the simple fact that the voter prefers B to A. And surely my "10 to A, 20 to B" needn't have the same meaning as your "10 to A, 20 to B" -- I think all we can really say is that we both prefer B to A. What else?

Now, you might want to try to interpret these "scores" as something like a hypothetical question put to the voter about how much they might pay to have this-or-that person elected; but if you know anything about self-reports of this kind about hypothetical scenarios, you know that people are horribly inaccurate (either intentionally or not), and that the setup is contrived. Serious measures of preferences involve actual stakes, markets, etc. Good luck with that here.

[ClayShentrup]

> I don't know how "10 to A, 20 to B" could indicate anything beyond the simple fact that the voter prefers B to A.

Scores are actual utilities with added loss, from:

  - ignorance    
  - normalization    
  - rounding error    
  - tactical voting

The way you measure the combined impact of that loss is with Bayesian regret. http://ScoreVoting.net/BayRegsFig.html

So those scores HAVE meaning. That meaning is muddied by the aforementioned loss, but it is lunacy to say it only "could indicate anything beyond the simple fact that the voter prefers B to A".

[_dps]

You are implicitly presuming an expected utility framework (more generally, that people rank outcome distributions based solely on their long run central tendencies).

This isn't the case (or even uniquely formalizable) in base (non-e.u.) utility theory which requires choice-order preservation under arbitrary monotonic transformations of the scoring function. Expectation ordering is not necessarily preserved under such transformations.

E.U. is a common paradigm because it is a useful approximation and allows numerical calculations but it is not all (or even, the core) of utility theory except in very limited circumstances (e.g. purely monetary payoffs and linear utility of money).

[ClayShentrup]

> You are implicitly presuming an expected utility framework

This isn't an assumption. Organisms have specifically evolved to maximize the expected number of copies of their genes they make. Or rather, _genes_ have specifically survived in proportion to their expected impact on the number of copies of themselves they make, and therefore we are to a very close approximation utility maximizers.

> ..linear utility of money

Utility is approximately log(money).

A caveat here is that certain decisions themselves are costly (particularly in terms of the most precious resource: time) and hence we can sometimes make clearly "irrational" choices because the expected utility of spending more time choosing was lower than the expected utility of making the more optimal choice. This leads to apparent paradoxes like the Allais Paradox.

Here's some background on utility theory. http://scorevoting.net/UtilFoundns.html

[ClayShentrup]

> Arrow's theorem concerns the situation where your election procedure needs to deliver (not just a single winner, but) a ranking of all the candidates.

Well, any social welfare function that can pick a single winner can also be used to form a complete ordering, by repeating the process with the previous winner eliminated. So any social welfare function is a social ordering function.

[ClayShentrup]

Some useful links.

http://ScoreVoting.net/ArrowThm.html http://ScoreVoting.net/GibbSat.html

Last year, I got some books signed by Arrow at his Palo Alto retirement home. It was pretty cool.