Range voting with mixtures of honest and strategic voters

(Executive summary)

It was conjectured by Range-doubters that range, while it works well with 100% honest voters, and also well with 100% strategic voters, might not work so well with a mixture of the two. The problem would be if (say) more Bush than Gore voters were strategic, leading to a huge advantage for Bush.

In the computer simulation experiments below we employed a 50-50 honest-strategic voter mix (a composition that seems approximately realistic based on our polling studies) in which each voter chose whether to be honest or strategic by flipping a coin. Note: it then is entirely possible in each election for Bush (or Gore) to get more strategic voters, and since the number of voters in each election below was only 61 (not huge) such an imbalance was in fact highly probable and typical and must have hurtfully impacted range's Bayesian regret numbers.

As Kevin Venzke put it:

If I don't want to assume that voters will courteously vote sincerely (even when this limits their power to affect the results), then I wouldn't use Range, as the result will be rather randomly skewed based on who chose to exaggerate and who didn't.

So – what happens? Answer: despite these circumstances, Range still kicks the butt of all the other ≈50 voting methods available in our computer sim package IEVS 2.53, getting the lowest Bayesian regret scores. In particular, Range is superior to Approval voting (in which all voters are "forced" by the rules to be strategic, even if they want to be honest), i.e. what Venzke called "random skewing" actually helped, not hurt.

Here is a typical run trying 29999 elections:

Honfrac=0.50, NumVoters=61, NumCands=5, NumElections=29999, IgnoranceAmplitude=0.001000
Voting Method Regret #Agreements with (true-utility-based) Condorcet Winner (when CW exists)
SociallyBest 0 13612
Range2Runoff 0.15947 17187
Range 0.16379 13279
Approval2Runoff 0.17343 16770
Range3 0.17353 12901
TopMedianRating 0.2012 12472
MCA 0.20512 12574
Approval 0.20512 12574
HeitzigDFC 0.20903 14123
Sinkhorn 0.29385 12088
Borda 0.29652 12066
KeenerEig 0.301 12027
CondorcetApproval 0.30741 11615
MDDA 0.30741 11615
BramsSanverPrAV 0.31307 11764
Black 0.33593 11615
CondorcetLR 0.36118 11615
HeitzigRiver 0.36169 11615
SchulzeBeatpaths 0.36242 11615
... ... ...
AntiPlurality 0.66083 6278
RandomBallot 0.86788 4789
Hay 0.94342 4068
RandomWinner 1 3655
SociallyWorst 1.9943 31

and now, just to make things ultra-clear, let's do a run with 13-voter elections just to make range really suffer heavily from this effect:

Honfrac=0.50 (or 1.00), NumVoters=13, NumCands=6, NumElections=29999, IgnoranceAmplitude=0.001000
ErrorBar for RandomWinner's regret=±0.0048; This (experimentally always?) upperbounds the error bar for every other regret.
Voting Method Regret (50-50) Regret (100% honest) #Agreements with (true-utility-based)
Condorcet Winner (when CW exists)
SociallyBest 0 0 12653
Range2Runoff 0.14785 0.1206 15574
Range 0.16329 0.04802 11796
Approval2Runoff 0.16795 0.15796 15054
Range3 0.17322 0.098206 11439
HeitzigDFC 0.19874 0.18676 12857
MCA 0.20012 0.17836 11491
Approval 0.215 0.18983 10997
TopMedianRating 0.23625 0.15327 11012
ContinCumul 0.26115 0.071422 10327
Sinkhorn 0.27859 0.093701 10806
Borda 0.28234 0.093768 10821
KeenerEig 0.28989 0.094234 10790
CondorcetApproval 0.30501 0.13742 10342
MDDA 0.30501 10342
BramsSanverPrAV 0.31801 0.14148 10410
Black 0.32771 0.11277 10342
UncAAO 0.13306 10342
CondorcetLR 0.35602 0.13781 10342
Copeland 0.3589 0.15425 10342
IterCopeland 0.35969 0.15421 10342
ArrowRaynaud 0.36433 0.15143 10318
HeitzigRiver 0.36691 0.15187 10342
TidemanRankedPairs 0.36774 0.1521 10342
Bucklin 0.3683 0.20312 9744
SimmonsCond 0.36861 0.15447 10342
Coombs 0.3738 0.18881 9792
SimpsonKramer 0.37449 0.15981 10342
SchulzeBeatpaths 0.37452 0.15792 10342
SmithIRV 0.3806 0.17825 10342
NansonBaldwin 0.38295 0.17252 10342
RaynaudElim 0.38709 0.1821 10342
UncoveredSet 0.39004 0.19653 10342
BTRIRV 0.39549 0.16867 10342
SmithSet 0.41488 0.23681 10342
SchwartzSet 0.41508 0.23808 10342
VtForAgainst 0.41726 0.20101 8940
ArmytagePCSchulze 0.42209 0.12331 8926
DMC 0.42409 0.18731 10342
LoMedianRank 0.43065 0.2893 8850
HeismanTrophy 0.43908 0.14424 8931
Top2Runoff 0.49909 0.23604 8823
IRV 0.50115 0.21684 8387
BaseballMVP 0.50532 0.14569 8094
Top3IRV 0.53124 0.23683 7842
PlurIR 0.53726 7717
Dabagh 0.55227 7497
Nauru 0.55258 7519
RandomPair 0.62973 0.62733 4932
VenzkeDisqPlur 0.63942 6385
Plurality 0.64371 0.33137 6357
AntiPlurality 0.67998 0.45963 4953
RandomBallot 0.72014 0.72305 5185
Hay 0.89489 0.91057 3484
RandomWinner 1 1 2930
SociallyWorst 2.0024 2.0098 6

So... the range-doubters look to be just wrong.

Note that Range Voting is at the top of the heap (lowest Bayesian regret) in both these tables, except for

  1. the "socially best" fake method (magically choose optimum winner) which has zero regret,
  2. the Range+Top2-Runoff method, which actually does better than plain range voting in these two simulations (albeit in the first one its advantage is statistically insignificant). That is because range voting can indeed be distorted by strategic voters; adding a top2-runoff after the range voting election, can partially correct for that because in a 2-candidate runoff, even strategic voters will always be honest. Range2Runoff tends to improve slightly over plain Range (up to 10-30% regret reduction) when 75% or more of the voters are strategic. But when 75% or more of the voters are honest, plain range is better than Range2Runoff by a lot (up to ≈3 times smaller regret). In view of that, plain Range still appears to be the best method overall.
No other method of those listed here (aside from Range2Runoff and trivial Range variants) is capable of bettering Range's Bayesian regret by a statistically significant margin under any simulation parameters so far tried.

Range is more Condorcet than Condorcet methods!

Also note that Range and its variants (Range2Runoff, Approval2Runoff) in these simulations with 50% honest voters actually yield the true-utility-based Condorcet winner more often than any other method, including "Condorcet methods" shown colored. That counterintuitive conclusion was forecast in a different model of strategic voting than the one simulated here. (The one here involves voters who believe a priori that candidate k+1 is far less likely to win than candidate k, and act accordingly to maximize their vote's impact.) This is a very strong reason not to prefer Condorcet voting methods over range – with a 50-50 mix of strategic and honest voters, range actually does their own job better than they do!

Caveats: IEVS 2.53 is still an early version and does not contain a lot of voter strategies and utility generators planned for later IEVS versions and which had been in my old (1999-2000) simulator. In particular, all rank-order voting methods shown here are strict rank-order, i.e. with rank-equalities forbidden in ballots. It has been suggested that variant-methods in which rank-equalities are permitted – especially Condorcet methods based on the "winning votes" concept – would handle strategic voters better. That is probably true. If wv-condorcet strategy were the same as approval-voting (and it definitely is not, but that might be a semi-decent approximate view) then in fact range and wv-Condorcet methods would all exhibit identical Bayesian Regrets with 100% strategic voters, and presumably considerably closer-than-here-shown BRs for a 50-50 mix, although I would expect Range still would have better (i.e. smaller) BRs. Also, in that case, range might no longer be better than Condorcet methods for the purpose of generating honest-voter Condorcet winners in the presence of strategic voters. All this, however, has to be regarded as speculation until more advanced IEVS versions appear that are capable of handling equalities in rank-order votes.

What if IRV & Condorcet voters are honest more often than range voters?

This objection was raised by one of our critics. We respond

  1. Where is any evidence for that?
  2. Suppose it is so. Suppose Condorcet and IRV voters really are little paragons of total honesty – angels – while exactly 50% of the range voters are nasty strategic exaggerators out to hurt the poor little innocent honest ones (other 50%). Fine.
    To cast light on this we reran the simulation in the green table with IEVS 3.24 this time for 100% honest voters. (See the second regret column.) What happened? As you can see, the Condorcet methods yield regrets between 0.1127 and 0.1331 (for Black's Condorcet method and UncAAO respectively, which are the two best Condorcet methods in this run) and 0.1783 and 0.1821 (for SmithIRV and Raynaud respectively, which are the two worst Condorcet methods in this run). And IRV delivered regret=0.2168. (Error bars are ±0.005 or less.) Meanwhile range with a mix of 50% honest and 50% strategic voters yielded regret 0.1633, while range+top2runoff yielded 0.1479.
    In other words, even then, and even in the situation in the green table designed to maximally make range voting look bad due to Venzke's "random skewing" effect, range and range+top2runoff still outperform IRV and perform comparably to Condorcet methods even when the latter two have angels as voters! (But we suggest to you that, sadly, Condorcet and IRV voters will not actually be angels who vote 100% honestly.)

Any questions?

Why were the range-doubters wrong?

Two possible ways to look at it:

  1. Range indeed is hurt when honest voters are replaced by strategic ones, but not enough to kick it out of first place (and/or the other methods are hurt by strategic voting too, enough to keep them out of first).
  2. The idea that the horrible Bush voters will gain a huge advantage over Gore voters if there are a larger number of Bush than Gore voters who are strategic, perhaps is the wrong way to look at it. Perhaps a better way to look at it is: there are two voter populations: strategic and honest. The strategic ones use approval voting. (Fine point: strategic and honest approval voting are not exactly the same thing. They differ slightly.) The honest ones use range voting. Then you add up all the votes. Sure, with only 61 voters (or especially with only 13 voters!), the strategic voter set will generally be imbalanced and include more Gore than Bush supporters (or the reverse) – but so what? Approval voting is a pretty good voting system (and it is even better if some honest range votes are added on). So the results will be pretty good. Meanwhile non-range systems like Condorcet, IRV, etc, all will have to deal with the same typical imbalances. The net result is range ends up doing better than them in terms of Bayesian Regret, which is exactly what these computer simulations of thousands of elections show. (We also have many, many, more such simulations, far too many to put on this page. This is just one typical results-table example, got from the random-normal elections model. They all behave this way. And you can download IEVS and run it yourself in different models.) The fact is, fear of the strategic imbalance "bugaboo" simply is not justified by the evidence once we go beyond intuitive worries about "fairness" and actually try the experiments.

Another set of computer sims

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