"Later no harm" – an actually-silly (& massively overhyped by "FairVote") voting system criterion

In July 2011, FairVote posted a webpage that began

Approval voting is a method of voting to elect single winners that has adherents among some voting theorists, but is unworkable in contested elections in which voters have a stake in the outcome. Once aware of how approval voting works, strategic voters will always earn a significant advantage over less informed voters. This problem with strategic voting far outweighs any other factor when evaluating the potential use of approval voting in governmental elections.
    Other methods that should not be used in meaningfully contested elections include range voting, score voting, the Borda Count and Bucklin voting. They all share approval voting's practical flaw of not allowing voters to support a second choice without potentially causing the defeat of their first choice. Such voting methods have their potential value, but only in elections where voters have no particular stake in the outcome.
    The only voting methods that should be weighed seriously for governmental elections are methods that do not violate this "later-no-harm" criterion (plurality voting and forms of runoff elections and instant runoff voting) or only do so indirectly (such as Condorcet voting methods).

and then it got worse from there. Well, first of all, what is this "later-no-harm" criterion that they consider to be so crucial and incredibly pre-eminent among voting system criteria? Well, FairVote's page (and also their many-page-long so-called "analysis" it cited) did not actually formally define it (!!), but since we're trying to be better than them, we actually shall:

The Later-No-Harm criterion (LNH): In a voting method based on rank-order ballots (i.e. each ballot is a ranking of the candidates in order from most to least liked), a voter, by honestly stating her second-choice candidate, should never be able to "harm" the candidate she wrote as her top preference. Here "harm" means "cause to lose (or prevent from outright winning) the election."
[Warning: This wording by me has intentionally been based on FairVote's. But an actual published paper by Douglas R. Woodall (1994) Properties of Preferential Election Rules defining over 30 voting system criteria, many new, stated the somewhat stronger and different definition "Adding a later preference to a rank-order ballot should not harm any candidate already listed," which was repeated in Woodall's 1997 paper in Discrete Applied Maths. But Woodall felt it necessary to note that a referee had described LNH as "unpalatable" and that Michael Dummett had called it "quite unreasonable." LNH and "House monotonicity" were the only criteria Woodall felt it necessary to attach such a health and safety warning to. Also, Woodall 1997's theorem 2 proved no Condorcet method could obey LNH, which contradicts the assertion by FairVote that it does so "indirectly" – whatever that was supposed to mean.]
    But what about voting methods whose ballots are not rank-orderings? For example, with score voting, each ballot gives a numerical score on some fixed scale (say from 0 to 9) to each candidate (greater scores for better candidates). Presumably, LNH then would mean "by honestly scoring somebody with the second highest score used on that ballot, a voter should be unable to harm the candidate she gave her greatest score to." Unfortunately it then would remain unclear just what LNH should mean if, say, she scored more than one candidate co-equal top. Should doing so be unable to "harm" any of them? FairVote never said, and as we've said they did not even try to define LNH. And this equality issue gets rather relevant with approval voting, because it is just score voting where the only permitted scores are 0 and 1 ("disapprove" and "approve"). Obviously, an approval voter in an election with ≥3 candidates is forced to treat some as equal (which is a disadvantage of approval versus score voting).

And with plurality voting (unfortunately the most-used voting method in the world at present) your ballot is "name one candidate, then shut up" so it is impossible for a voter even to state her "second choice"... except actually it is possible, but if you try, then your ballot is discarded as "spoiled." FairVote therefore (quote above) considers Plurality to satisfy LNH, but obviously it really does not, because: if your plurality-vote for X caused X to win, but then you indicated a second choice Y, that would cause your ballot to be discarded, which would "harm" X by causing him no longer to be an outright winner.

So now that we have at least some idea what LNH is: which voting methods obey or disobey it? FairVote is correct that Instant Runoff (IRV) obeys LNH and that score and Borda disobey LNH.

What about approval voting? That's debatable. A voter by approving several candidates, thus regarding them as "equal," can only "harm" approved-candidate X, by "helping" (i.e. electing) also-approved candidate Y – which is fine with that voter if X and Y were truly equal in her view! Of course in reality, most voters do not really regard candidates X and Y as equal; they simply were forced by the rules of approval voting to pretend they were equal. And a voter who honestly prefers X over Y, can by approving both X and Y, harm X! She could avoid that harm by disapproving Y, but in that case she could still be in trouble, because then she might be (say) disapproving both Y and Z (and she hates Z), and she by the foolish act of disapproving Y, might cause Z to win! So with approval voting in a ≥3-candidate election, every ballot must take such risks. They are unavoidable. If you approve too many candidates you risk electing one of the worse among them. If, on the other hand, you disapprove too many candidates, you risk electing one of the worse among them.

A good compromise, which tends to minimize such risks, is "approve all the candidates you consider better than the expected value of the winner." You of course do not know who the winner will be, ahead of time – you can only make probabilistic guesses about everybody's winning chances – which is why I am using the word "expected." Any approval-voter who follows this strategy will never help anybody to win whose quality is below expectation; and never will harm anybody whose quality (in her judgment) lies above expectation. Therefore, by voting in this way, her vote will always increase the expected quality (as reckoned by her) of the winner (versus if she had not voted at all).

Similarly, with score voting, a voter by using the approval-voting strategy we just described (i.e. give the maximum-allowed score to each candidate you consider above expectation, and the minimum-allowed score to each candidate you consider below expectation) can guarantee that her vote will increase the expected value of the winner. Further, with score voting, by casting an honestly-ordered vote, you never can cause a worse election result than you would have gotten by not voting at all.

Those desirable properties, sadly, are not enjoyed by IRV. More precisely, a voter, by voting her honest rank-ordering in IRV, can cause the election result to worsen by her reckoning. She in such cases would have been better off casting a dishonest rank-ordering, or not voting at all.

But FairVote ignores all that, plus also pretends that the most strategic way to approval-vote is to approve exactly one candidate, and they further pretend that once approval voters are "aware" of this brilliant strategy that will give them a "significant advantage over less informed voters." That all was usually wrong advice! Usually, single-approval votes are not the best strategy, so FairVote is giving its readers exactly the wrong strategic advice, then pretending that advice was a hidden advantageous secret only the FairVote geniuses knew.

(Comparison of various score & approval voting strategies indicating the "mean based thresholding" strategy was the best one tried in all experiments with ≥30 other voters; and "scaled honesty" appears to be at least 2/3 as good as the best score-voting strategy in every experiment tried so far.)

Why is LNH a silly criterion?

1. Suppose Joe wins an election. Don't you agree that, if Mary has enough support, then Mary should have won? Don't you agree that, if Mary gets "Mary is my 2nd choice" votes from a lot of voters, that ought to mean Mary is more-worthy of thus-winning? In other words, it is desirable for a voter to be allowed to indicate that she also likes Mary, even if she likes Joe more. And it is desirable that this should sometimes cause the election of Mary, not Joe, because that extra Mary-support tips the balance. So, objectively better election results are obtained, at least with honest voting, in LNH-disobeying voting systems than in LNH-obeying voting systems. I.e. objectively, LNH is not even a desirable property with honest voters unlike (say) monotonicity.

2. Now, one might argue that LNH is desirable with strategic=dishonest voters. Because LNH hopefully will encourage the strategists to provide honest second choices. Without LNH, they'd lie more.

That's a sensible idea. But if that is your view, and your goal is to motivate voters to provide honest 2nd choices, then it is better instead to use as your criterion, the straightforward "with this voting method, providing an honest 2nd choice, is never a strategic mistake, i.e. never forces the election result to worsen in that voter's perspective." That would be exactly the criterion that truly would motivate honest 2nd choices. Call it the "HSC criterion" (motivates Honest Second Choices). But since instant runoff voting (IRV) disobeys that superior criterion, the IRV propagandists at FairVote have always hidden this issue. Instead, they very carefully word their LNH criterion, which we now see is the wrong criterion for their very own declared goal, so that IRV obeys it. Then they act as though LNH=HSC, i.e. act as though LNH were the correct criterion, hoping nobody notices this sleight of hand. And then they take the deception even further – in the FairVote web page on this (local copy) they actually declared that this wrong LNH criterion was actually the pre-eminent criterion which should "far outweigh" all others in importance for judging voting systems – and if you disobey it, your voting method is doomed and should never even be considered.

This is absurd.

3. And as long as we are worried about motivating honest 2nd choices, why aren't we worried about honest first choices even more? The "HF criterion" would be, say, "with this voting method, honestly scoring your 1st choice candidate with the maximum allowed score (or top ranking), is never a strategic mistake, i.e. never forces the election result to worsen from that voter's perspective." Well, IRV disobeys HF. (So do Borda, plurality, and every Condorcet method.) Range and approval both obey HF. It seems intuitively clear HF is more important than HSC, which in turn is more important than LNH. So the very chain of reasoning FairVote is using, when redone correctly, actually leads us to support approval & score voting, and not support IRV – exactly the opposite of the conclusion FairVote drew!

4. All that was theory. What about practice? Well, in practice about 85% of Australian voters rank one major party top, other bottom or 2nd to bottom, which is clear massive strategic exaggeration voting behavior, and it causes major harm to Australia, making it essentially impossible for a 3rd party to win an IRV seat. In 3 consecutive house elections (150 IRV seats each) in 2001, 2004, 2007, their third parties won zero seats. In the 4th (2010), they finally won one seat – the Green MP Adam Bandt.

Aside: How did this miracle occur? Chip LeGrand: "Greens celebrate historic lower house victory,", The Australian 21 Aug. 2010 discusses this election in the "Melbourne inner city" district. Bandt won with 56.1% of the final-round votes versus Cath Bowtell (Labor party) who had 43.9%. In the first IRV round, the top contenders were Bandt(Green) 36.2%, Bowtell(Labour) 38.1%, and Simon Olsen (Liberal) 21.0%, plus 4 candidates from other parties ("Family First", "Sex", "Democrats", and "Secular") who in aggregate got 4.5%. Australia's top two parties are the Liberal/Nationals and Labour. The Green party is a "third" party. This was the first time a Green had ever won a House seat in a general election in Australian history – but they had won one once before in a special election in which one of the major parties did not run anybody – and it happened because Olsen's voters almost unanimously preferred Bandt over Bowtell. So in this case IRV seems to have worked exactly as its proponents would hope.

So strategic voting really happens in IRV, at a massive rate of occurrence, and it causes huge damaging effects.

But why? Wasn't FairVote claiming those Australian voters, motivated by the wonderful wonderful essential pre-eminent LNH property of their voting system, would vote honestly, not strategically? Whatever is the matter with them?

I mean, if Australians supported the Greens, why not rank Greens top, and then honestly as their 2nd choice, name one of the two major parties, and do this in the full confidence this 2nd choice will in no way hurt their Green 1st choice?

Well, experimentally, they don't. Experimental fact is, that about 85% of Australians vote the exaggerated way I described with the 2 major parties artificially shoved to top & bottom. So any hope the LNH criterion will prevent strategic voting, is just garbage in the real world. And it is a theorem that with IRV, if over 75% of voters vote for one of the two major parties top, then it is mathematically impossible for any third party to win a seat. Ever.

And this all is not some mere quibble. This is approximately the most massive refutation one could possibly have hoped for.

So: All this LNH nonsense spewed by FairVote is both theoretically and experimentally bankrupt.

Going further

5. But let us not completely waste our time. Let's try to turn this investigation into something useful, not merely yet another demonstration of the perpetual lies "FairVote" continually spews as their standard modus operandi.

First of all, we just said HF is a good criterion. And score and approval voting both obey it, while such other voting methods as IRV, Borda, Plurality, and Condorcet do not.

Second, we showed that if you like the LNH criterion, you obviously must like the HSC criterion even more, although presumably not as much as you must like HF. So, Natural question: Can we invent a voting method obeying both HF and HSC?

Answer. No deterministic voting system based on rank-order ballots is possible that obeys both HF and HSC, except for a "dictatorship" (i.e. one special voter, the 'dictator,' decides all election results all by himself). Proof: this is a restatement of the famous Gibbard-Satterthwaite theorem in the case of 3-candidate elections.

Jan Kok described this as: "(For ranked-order-ballot deterministic non-dictator votng methods) Later-no-harm implies earlier-yes-harm"!
    In the 2009 Burlington, VT mayoral IRV election, if you voted for the Republican as your favorite and you also preferred the Democrat over the Progressive, you shot yourself in the foot. You should have dishonestly voted for the Democrat as your first choice. That would have allowed the Democrat to win, instead of the Progressive who actually won.
    In other words, voting for your favorite as your first choice can actually harm you.

(Woodall also proved some statements of the form "later no harm forces various kinds of non-monotonicity.")

6. But the key phrase in that theorem statement was "based on rank-order ballots." Score voting is not based on rank-ordering the candidates as ballots. It is based on numerically scoring the candidates, as your ballot. Could range voting thus manage to evade the Gibbard-Satterthwaite theorem to achieve the holy grail of satisfying both HF and HSC in 3-candidate elections? Even though neither IRV, nor any other rank-order ballot voting system, can?

The answer to that follow-up question is: "partially." With score voting, honestly giving your true favorite candidate, the maximum score, is always strategic. It can never worsen the election result from your perspective. And also: With score voting, honestly giving the candidate you truly consider the worst, the minimum score, also is always strategic and can never worsen the election result from your perspective. So once you as a voter in a 3-candidate election have done those two things, there is only one task remaining: to determine your score for the candidate you honestly view as the middle one (i.e, your honest second choice). Choose whatever the strategically best score is (and if more than one such score exists, then, e.g, choose the "most honest" within this strategically-best set). So what have we now done? We now have voted, using score voting, in a maximally strategic way, and at the same time, in a "semi-honest" way – meaning for every candidate pair (A,B) such that we honestly think A>B, we have stated scores on our ballot obeying A≥B and we've never dishonestly scored as though A<B. [Also, for every (A,B) such that you honestly think A=B, you can strategically score-vote fully honestly, i.e. such that your scores for A and B are equal.]

So with score voting (and this also works for approval voting) in 3-choice elections, we've just proven it always is possible to vote "semi-honestly." Indeed, with score voting, but not approval voting, we actually can go a bit further: with probability=1 if the other score-voters include some continuum randomness (e.g. if at least one voter someplace scores randomly-uniformly within the allowed-scores interval), then it always is possible to score-vote fully honestly, in the sense your scores obey the same <, =, or > relations as your honest opinion.

But the Gibbard-Satterthwaite theorem shows that no rank-order ballot voting method – not IRV, not Borda, not Condorcet, not any of the infinite number of rank-order voting methods that nobody has even invented yet – can say that, because for all of them, that feat is impossible. With IRV or any such voting method, there are elections in which the only way for you to get a good election result, is to flat-out lie by pretending A>B when your honest belief about that candidate pair is A<B.

Think such "forced to lie" scenarios are uncommon? Sorry, the Friedgut-Kalai-Nisan 2008 quantitative version of the Gibbard-Satterthwaite theorem would appear to kill that hope... and if we are only taking about IRV, then our computer simulations definitely kill it.

So, things are really looking quite bad for FairVote's so-called argument "against" score voting and "for" IRV at this point.

And it's about to get even worse. Now, let's forget about 3-candidate elections; let's go to C-candidate elections for any C≥3. In that case, I claim this theorem: with score or approval voting (either work) if you knew everybody else's vote, you always could select a vote that would simultaneously be:

  1. maximally strategic (assuring the best possible election result in your view), and
  2. semi-honest, i.e. for any candidate-pair (A,B), if you honestly consider A>B your vote features scores A≥B; while if you honestly consider A=B then your vote's scores obey A=B.
  3. With score (but now not with approval) voting we can further claim that with probability=1 if the other score-voters include some continuum randomness, then it always is possible to score-vote fully honestly, in the sense your scores obey the same <, =, or > relations as your honest opinion.
  4. No non-dictatorial deterministic rank-order-ballot voting method (including, of course, IRV) can achieve (i) and (ii) simultaneously even in the simplest case C=3 of 3-candidate elections, and even for voters who knew the votes from everybody else.

Proof: (iv) is just the Gibbard-Satterthwaite impossibility theorem. And (i) & (ii) are accomplished thus: find the best candidate (in your reckoning – call him B) whom your score-vote could elect. Give B the maximum score. Give all better candidates that same max score, and give all worse candidates the minimum score. Now, to show (iii), perturb all C scores by amounts smaller than ε in such a way that the ordering becomes fully honest and all candidates in the upper, and all in the lower group, are each equi-spaced. With probability=1 there will exist some ε small enough that this vote will still elect B. For example, any ε less than (2C)-1 times the minimum spacing between the candidates' summed-scores from all the other voters, will work.

Bottom line

So approval voting, and even better score voting, both accomplish precisely the dishonesty-discouraging feats FairVote claimed to want (once their idiocy and deceptions are corrected), and do so not only to a greater extent than IRV does, but also than any rank-order-ballot voting method can. It is mind-boggling that FairVote claims to use all this as an argument against Approval and Score voting and for IRV.

More generally, FairVote, over and over, tries to take the strategy of Appealing to the Idiot. They come up with arguments that sound naively appealing to people who've never thought about voting systems before, then push them whether or not they have any actual validity, and no matter how absurd they really are, and apparently never correcting themselves even after years of actual experts pointing their nonsense out.

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