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<div dir="ltr" data-setdir="false">Arrow published a mathematical theorem so presumably everything was rigorously defined and not open to interpretation, so that would include unrestricted domain. On the Wikipedia it says "<span>In social choice theory, unrestricted domain, or universality, is a property of social welfare functions in which all preferences of all voters (but no other considerations) are allowed.</span>" And while it might be defined differently and more precisely in Arrow's paper, I wouldn't say score fails by that definition. But it doesn't really matter anyway. You can define it in a way that score fails or define it in a way that it doesn't apply to score. Similarly you could define a condition where degrees of liking must be allowed, and ranked methods would fail that. And as has been said, it's not as if Arrow's Theorem is the be all and end all. All methods have their own problems, and whether they happen to be covered by one particular theorem is neither here nor there.</div><div dir="ltr" data-setdir="false"><br></div><div dir="ltr" data-setdir="false">But what I would say is that I consider Arrow's Theorem to be possibly the most overrated and overstated theorem of all time. If you look at the criteria that ranked methods must fail one of according to Arrow's Theorem, most of them are just criteria that any remotely reasonable method would pass. The theorem, stated more informally, is basically that with a few reasonable background assumptions, all ranked-ballot methods fail independence of irrelevant alternatives. Which is interesting enough itself, except that this was known for centuries anyway from the Condorcet Paradox. If head to head A beats B, B beats C, and C beats A, then any winner in the three-way election has to overturn one of the head to head results as a result of an irrelevant alternative being added.</div><div dir="ltr" data-setdir="false"><br></div><div dir="ltr" data-setdir="false">Toby</div><div><br></div>
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On Thursday, 9 January 2020, 23:17:56 GMT, Rob Lanphier <robla@robla.net> wrote:
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<div><div dir="ltr">Hi folks,<br></div><div dir="ltr"><br></div><div dir="ltr">As some of you might have seen, Electowiki is a lot more active than<br></div><div dir="ltr">it used to be. I'm 99% convinced that's a good thing. The 1% of me<br></div><div dir="ltr">that has reservations is regarding how some advocates talk about<br></div><div dir="ltr">Arrow's theorem. I'm hoping you all can do one of the following:<br></div><div dir="ltr">a) change my view about Arrow's theorem, -or-<br></div><div dir="ltr">b) offer me some help in better articulating my view about Arrow's theorem.<br></div><div dir="ltr"><br></div><div dir="ltr">Many Score voting[1] activists claim that cardinal methods somehow<br></div><div dir="ltr">dodge Arrow's theorem. It seems to me that *all* voting systems (not<br></div><div dir="ltr">a mere subset) are subject to some form of impossibility problem.<br></div><div dir="ltr">Arrow's impossibility theorem deserved great acclaim for subjecting<br></div><div dir="ltr">all mainstream voting systems of the 1950s to mathematical rigor, and<br></div><div dir="ltr">it's clear that his 1950 paper and 1951 book profoundly influenced<br></div><div dir="ltr">economics and game theory for the better. His 1972 Nobel prize was<br></div><div dir="ltr">well deserved. It seems that it has become fashionable to find<br></div><div dir="ltr">loopholes in Arrow's original formulation and declare the loopholes<br></div><div dir="ltr">important. Even if the loopholes exist, talking up those loopholes<br></div><div dir="ltr">doesn't seem compelling, given the subsequent work by other theorists<br></div><div dir="ltr">broaden the scope beyond Arrow's version.<br></div><div dir="ltr"><br></div><div dir="ltr">But, what the heck, let's actually talk about Arrow's original<br></div><div dir="ltr">formulation. I believe Score voting fails unrestricted domain:<br></div><div dir="ltr"><<a href="https://en.wikipedia.org/wiki/Unrestricted_domain" rel="nofollow" target="_blank">https://en.wikipedia.org/wiki/Unrestricted_domain</a>><br></div><div dir="ltr"><br></div><div dir="ltr">In particular, let's say that 90% of voters prefer candidate A over candidate B:<br></div><div dir="ltr">90:A>B<br></div><div dir="ltr">10:B>A<br></div><div dir="ltr"><br></div><div dir="ltr">Arrow posits that there should only be one way to express that, and<br></div><div dir="ltr">Score fails it. In Score, it's possible to sometimes pick A, and<br></div><div dir="ltr">sometimes pick B, depending on the score values on the ballots. If<br></div><div dir="ltr">Score *always* chose either A or B, then it would pass Universality.<br></div><div dir="ltr"><br></div><div dir="ltr">Score advocates claim that this isn't a bug, it's a *feature*. If<br></div><div dir="ltr">(for example), voters for A only mildly prefer A over B, but voters<br></div><div dir="ltr">for B strongly detest A, then the correct social choice is B.<br></div><div dir="ltr">However, it doesn't seem practical to inflict this level of nuance on<br></div><div dir="ltr">voters. I suspect that the first election where the Condorcet winner<br></div><div dir="ltr">is beaten by a minority-preferred candidate (e.g. like what happened<br></div><div dir="ltr">in Burlington 2009 [2]) will result in a repeal (like what happened in<br></div><div dir="ltr">Burlington). Back to the A/B example above, It's hard to imagine<br></div><div dir="ltr">voters would consider the selection of "B" to be fair in a large<br></div><div dir="ltr">election.<br></div><div dir="ltr"><br></div><div dir="ltr">It's fine to hold the opinion that Universality is an uninteresting<br></div><div dir="ltr">criterion, and that therefore, Arrow's set of criteria isn't very<br></div><div dir="ltr">interesting. For example, a few years ago, we went through a phase<br></div><div dir="ltr">where Condorcet advocates promoted "Local IIAC" as a IIAC[3] as a more<br></div><div dir="ltr">interesting criterion, and advocating for Condorcet variants that meet<br></div><div dir="ltr">that criterion. Regardless, just because we find one criterion less<br></div><div dir="ltr">compelling than another, we should talk accurately about the failed<br></div><div dir="ltr">criterion.<br></div><div dir="ltr"><br></div><div dir="ltr">My way of thinking about Arrow's theorem (and being thankful for it)<br></div><div dir="ltr">is to think of it like the physics of voting systems. For example, in<br></div><div dir="ltr">real-world physics, a "perfect" vehicle is impossible, because it's<br></div><div dir="ltr">impossible to meet these criteria:<br></div><div dir="ltr">* Goes faster than the speed of light<br></div><div dir="ltr">* Has infinite capacity<br></div><div dir="ltr">* Has a luxurious and comfortable passenger cabin<br></div><div dir="ltr">* Fits in a small coat pocket<br></div><div dir="ltr">* Is easy to produce<br></div><div dir="ltr">* Is cheap (or even free)<br></div><div dir="ltr"><br></div><div dir="ltr">Just because a perfect vehicle is not possible, I'm glad<br></div><div dir="ltr">transportation innovation didn't stop with Ford's Model T. Of course,<br></div><div dir="ltr">automobile sellers compete on the tradeoffs between the criteria<br></div><div dir="ltr">above, and much public policy debate is about mode-of-transport<br></div><div dir="ltr">tradeoffs between planes, trains and automobiles (and bicycles, and<br></div><div dir="ltr">scooters, and and and...). We need public policy debates around<br></div><div dir="ltr">election method tradeoffs, too.<br></div><div dir="ltr"><br></div><div dir="ltr">I'm hoping we can try to stop trying to declare clever loopholes in<br></div><div dir="ltr">Arrow's theorem, and just acknowledge the reality that *all* voting<br></div><div dir="ltr">systems involve tradeoffs. I hope we all can acknowledge that Arrow's<br></div><div dir="ltr">central insight (there's no "perfect" system given perfectly<br></div><div dir="ltr">reasonable criteria) is valid, and that it's only on the specifics of<br></div><div dir="ltr">the exact criteria chosen for the 1951 proof that might be flawed. I<br></div><div dir="ltr">believe that election method activists should speak (and write) with<br></div><div dir="ltr">clarity about the tradeoffs involved. Whenever I see someone<br></div><div dir="ltr">gleefully declare that Arrow's theorem doesn't apply to their voting<br></div><div dir="ltr">method (and imply perfection), the credibility of the writer drops<br></div><div dir="ltr">*precipitously* in my mind.<br></div><div dir="ltr"><br></div><div dir="ltr">Am I wrong?<br></div><div dir="ltr"><br></div><div dir="ltr">Rob<br></div><div dir="ltr"><br></div><div dir="ltr">p.s. I've been meaning to write this email for a while. What inspired<br></div><div dir="ltr">me to finally write it has been reading the current state of<br></div><div dir="ltr">Electowiki and Wikipedia articles on the topic, like the "Arrow's<br></div><div dir="ltr">impossiblity theorem" article on Electowiki[4]<br></div><div dir="ltr"><br></div><div dir="ltr">[1]: <a href="https://electowiki.org/wiki/Score_voting" rel="nofollow" target="_blank">https://electowiki.org/wiki/Score_voting</a><br></div><div dir="ltr">[2]: <a href="https://en.wikipedia.org/wiki/2009_Burlington_mayoral_election" rel="nofollow" target="_blank">https://en.wikipedia.org/wiki/2009_Burlington_mayoral_election</a><br></div><div dir="ltr">[3]: <a href="https://electowiki.org/wiki/IIAC" rel="nofollow" target="_blank">https://electowiki.org/wiki/IIAC</a><br></div><div dir="ltr">[4]: <a href="https://electowiki.org/wiki/Arrow%27s_impossibility_theorem" rel="nofollow" target="_blank">https://electowiki.org/wiki/Arrow%27s_impossibility_theorem</a><br></div><div dir="ltr">----<br></div><div dir="ltr">Election-Methods mailing list - see <a href="https://electorama.com/em " rel="nofollow" target="_blank">https://electorama.com/em </a>for list info<br></div></div>
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