# [EM] Arrow's theorem and cardinal voting systems

Steve Eppley seppley at alumni.caltech.edu
Fri Jan 10 03:41:28 PST 2020

```I think the spirit of Rob's question is (or should be):  Can any plausibly democratic voting method satisfy this Independence criterion: "Assuming voters' preferences don't change, the winner must not change if another candidate chooses not to compete." (Let's call methods plausibly democratic if they don't privilege any candidates or voters.  In other words, the Neutrality and Anonymity criteria in the literature of social choice theory.)

None can satisfy that Independence.

The last time I checked, advocates of Range Voting acknowledge that when there are only two candidates, optimal voting strategy is to give the highest possible score to the voter's most preferred candidate and the lowest possible score to the voter's least preferred candidate. (Some of those advocates may even think that's sincere voting, bless their little hearts.)  Since that's such an obvious strategy, it's reasonable to assume at least some of the voters will eventually learn to use it (just as many voters have learned to vote for a compromise to help defeat a "greater evil" given Plurality Rule).  Now consider an example: Suppose that given a majoritarian voting method such as Plurality Rule, Rock would beat Scissors, Scissors would beat Paper, and Paper would beat Rock, by a narrow majority in each pairing.  Suppose also that Rock would be the winner if Rock, Paper and Scissors compete given Range Voting (or Approval).  Which one wins if only Rock and Paper compete
given Range Voting (or Approval)?  Obviously, Paper can win, since the majority who prefer Paper can elect Paper using their optimal voting strategy.  Thus the winner can change from Rock to Paper when Scissors doesn't compete, which violates Independence.

Given the fact that strategic voting is possible, many (most?) criteria definitions are "naive."  For instance, a "Condorcet winner given sincere voting" can lose given a voting method that satisfies the Condorcet criterion.  I hope we can all agree that the spirit of the Condorcet criterion is that the sincere Condorcet winner should win (when it exists).  Call that the Sincere Condorcet criterion.  No plausibly democratic voting method can satisfy Sincere Condorcet, but some voting methods can perform better on it than others, by making defensive strategies easier and/or more palatable, or offensive strategies riskier.

I presume the same is true regarding Independence: some plausibly democratic voting methods perform better than others on Independence, even though none satisfy Independence.

For the criterion that matters most to me, I don't have a rigorous definition.  Here's a non-rigorous definition:  The voting method should give candidates who want to win a strong incentive to take positions that the voters themselves would collectively choose given a well-functioning direct democracy... even on issues that most voters don't care strongly about.  Here's how I relate that to voting methods like Maximize Affirmed Majorities (MAM), which facilitate competition, count all pairwise majorities, and pay attention to the sizes of the majorities:  Suppose candidate Alice wants to win, and is considering taking position p on some issue.  Although she knows a majority of the voters prefer alternative q over p, her wealthy campaign donors favor p and most voters care more about other issues.  Given a voting method like MAM, the risk to Alice is that by advocating p, she would create an opportunity for another candidate Bob to enter the race, take position q and copy
Alice's positions on all other issues.  The larger the majority who prefer q over p, the larger the majority who would tend to rank Bob over Alice.  Defeating Alice.  A deterrent against taking unpopular positions to benefit donors.

--Steve

On 1/9/2020 6:17 PM, Rob Lanphier wrote:
> Hi folks,
>
> As some of you might have seen, Electowiki is a lot more active than
> it used to be.  I'm 99% convinced that's a good thing.  The 1% of me
> Arrow's theorem.  I'm hoping you all can do one of the following:
> a)  change my view about Arrow's theorem, -or-
> b)  offer me some help in better articulating my view about Arrow's theorem.
>
> Many Score voting[1] activists claim that cardinal methods somehow
> dodge Arrow's theorem.  It seems to me that *all* voting systems (not
> a mere subset) are subject to some form of impossibility problem.
> Arrow's impossibility theorem deserved great acclaim for subjecting
> all mainstream voting systems of the 1950s to mathematical rigor, and
> it's clear that his 1950 paper and 1951 book profoundly influenced
> economics and game theory for the better.  His 1972 Nobel prize was
> well deserved.  It seems that it has become fashionable to find
> loopholes in Arrow's original formulation and declare the loopholes
> important.  Even if the loopholes exist, talking up those loopholes
> doesn't seem compelling, given the subsequent work by other theorists
> broaden the scope beyond Arrow's version.
>
> But, what the heck, let's actually talk about Arrow's original
> formulation.  I believe Score voting fails unrestricted domain:
> <https://en.wikipedia.org/wiki/Unrestricted_domain>
>
> In particular, let's say that 90% of voters prefer candidate A over candidate B:
> 90:A>B
> 10:B>A
>
> Arrow posits that there should only be one way to express that, and
> Score fails it.  In Score, it's possible to sometimes pick A, and
> sometimes pick B, depending on the score values on the ballots.  If
> Score *always* chose either A or B, then it would pass Universality.
>
> Score advocates claim that this isn't a bug, it's a *feature*.  If
> (for example), voters for A only mildly prefer A over B, but voters
> for B strongly detest A, then the correct social choice is B.
> However, it doesn't seem practical to inflict this level of nuance on
> voters.  I suspect that the first election where the Condorcet winner
> is beaten by a minority-preferred candidate (e.g. like what happened
> in Burlington 2009 [2]) will result in a repeal (like what happened in
> Burlington).  Back to the A/B example above, It's hard to imagine
> voters would consider the selection of "B" to be fair in a large
> election.
>
> It's fine to hold the opinion that Universality is an uninteresting
> criterion, and that therefore, Arrow's set of criteria isn't very
> interesting.  For example, a few years ago, we went through a phase
> where Condorcet advocates promoted "Local IIAC" as a IIAC[3] as a more
> interesting criterion, and advocating for Condorcet variants that meet
> that criterion.  Regardless, just because we find one criterion less
> compelling than another, we should talk accurately about the failed
> criterion.
>
> My way of thinking about Arrow's theorem (and being thankful for it)
> is to think of it like the physics of voting systems.  For example, in
> real-world physics, a "perfect" vehicle is impossible, because it's
> impossible to meet these criteria:
> * Goes faster than the speed of light
> * Has infinite capacity
> * Has a luxurious and comfortable passenger cabin
> * Fits in a small coat pocket
> * Is easy to produce
> * Is cheap (or even free)
>
> Just because a perfect vehicle is not possible, I'm glad
> transportation innovation didn't stop with Ford's Model T.  Of course,
> automobile sellers compete on the tradeoffs between the criteria
> above, and much public policy debate is about mode-of-transport
> tradeoffs between planes, trains and automobiles (and bicycles, and
> scooters, and and and...).  We need public policy debates around
>
> I'm hoping we can try to stop trying to declare clever loopholes in
> Arrow's theorem, and just acknowledge the reality that *all* voting
> systems involve tradeoffs.  I hope we all can acknowledge that Arrow's
> central insight (there's no "perfect" system given perfectly
> reasonable criteria) is valid, and that it's only on the specifics of
> the exact criteria chosen for the 1951 proof that might be flawed.  I
> believe that election method activists should speak (and write) with
> gleefully declare that Arrow's theorem doesn't apply to their voting
> method (and imply perfection), the credibility of the writer drops
> *precipitously* in my mind.
>
> Am I wrong?
>
> Rob
>
> p.s. I've been meaning to write this email for a while.  What inspired
> me to finally write it has been reading the current state of
> Electowiki and Wikipedia articles on the topic, like the "Arrow's
> impossiblity theorem" article on Electowiki[4]
>
> [1]: https://electowiki.org/wiki/Score_voting
> [2]: https://en.wikipedia.org/wiki/2009_Burlington_mayoral_election
> [3]: https://electowiki.org/wiki/IIAC
> [4]: https://electowiki.org/wiki/Arrow%27s_impossibility_theorem
> ----
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