Serious-looking Range Voting problems caused by strategic voters

By Warren D. Smith inspired by important contributions by Chris Benham.

Chris Benham rejects the idea that voting honestly is a "fault."

The voting system should try to minimise the advantage of strategists over sincere voters, and of informed strategists over less well informed and zero-information strategists. It should give the voter a clear way of voting sincerely, and if there is a zero-info strategy it should be straight-forward and similar to sincere voting.

A minimum standard is that the voting method should give good results in the zero-info case with strategic voters.

Note: by "zero information" Benham means he is considering a voter who has to decide how to vote without having any information about how the other voters are voting. This voter has information about the candidates, but not about the other voters.

Benham's attack

Here is an example where 0-9 range voting fails that minimum standard.

#voters their vote
45 A9 > B4 > C0
36 B9 > C5 > A0
19 C9 > B4 > A0

(Here the first line "A9 > B4 > C0" means these 45 voters each rate A with honest score 9, rate B with 4, and C with 0.)

B is both the Condorcet and big sincere ratings winner. But if these voters all use the asymptotically best zero-info Range/Approval strategy (mean-utility as approval-threshold) then the voting proceeds

#voters their vote
45 A approved
36 B & C approved
19 C approved

whereupon C, the sincere ratings loser, wins, A is second, and the best candidate B finishes last!

Further, with more-than-zero info, such as, the voters just heard of the above and figured A & C were the top threats to win, and hence placed their threshold midway between them, the above bad election would merely recur.

In contrast, with many rank-order-based voting schemes (such as many Condorcet schemes) we expect that the best voter zero-info strategy is honest voting. (I.e. best strategy is honesty under the assumption all other voters are independently random and cast all votes with equal likelihood. We do not actually have a proof of that, but it sounds very plausible.) So in this sense, those other schemes and in particular Condorcet voting, behave in a way superior to range voting in this example assuming strategic zero-info voters.

Our defenses against this attack

The election example above (based on one by Benham) certainly carries some punch! We have two ways to try to defend range voting from the onslaught.

Preliminary Remark

Benham's attack takes advantage of the fact that, in Range Voting, the best "zero-info voting strategy" is generally not the same thing as casting an "honest range vote." However, it is always a "semi-honest" range vote, i.e. never states A>B if your honest belief about candidates A and B is B>A.

First Defense

Usually voters do not have zero information about the other votes. We claim that if candidate X is regarded as most likely to win, candidate Y is second-most-likely, and all other candidates are regarded as having comparatively negligible winning chances, then your best approval-voting threshhold-based strategy is to place the threshhold between X and Y, and nearer to X (in the limit where X is far more likely to win than Y, place it extremely near to X). To explain why, realize that (a) if the threshold is not between X and Y, you entirely sacrifice your chances to decide the election; (b) if it is placed very near to X, then in the small chance that it comes down to an X vs Z tie, your vote decides the issue in the maximum number of possible cases, at the cost of sacrificing ability to decide YZ ties (but these are presumably far less likely than XZ ties for first).

So if in the election example above, the voters believe A and C (top 2 finishers) are most likely to win with C (top) considerably more likely, they should place their thresholds between A and C but considerably closer to C. In that case the bad election would not recur, but instead would proceed

#voters their vote
45 A & B approved
36 B & C approved
19 C approved

whereupon B wins and all is well. Furthermore, if the voters instead reckoned that the top threats were B & A – or if they reckoned them to be B & C – then again in either case B would win. So in this example, the behavior is always good if the strategic voters have better-than-zero information to act on.

Although that ("more than zero information") defense is valid against Benham's example, it is possible to devise anti-range-voting examples in which it does not work. Here is one with 1001 voters:

#voters their vote
73 Z12 > A8 > B7 > C6 > D5 > E1
73 Z12 > E8 > A7 > B6 > C5 > D1
73 Z12 > D8 > E7 > A6 > B5 > C1
72 Z12 > C8 > D7 > E6 > A5 > B1
73 Z12 > B8 > C7 > D6 > E5 > A1
71 A8 > B7 > C6 > D5 > E1 > Z0
70 E8 > A7 > B6 > C5 > D1 > Z0
70 D8 > E7 > A6 > B5 > C1 > Z0
70 C8 > D7 > E6 > A5 > B1 > Z0
70 B8 > C7 > D6 > E5 > A1 > Z0

Zero-info voters would threshold at 6.5 or 4.5 (for the two types of voters here) causing A and B to finish with the top and second-top number of approvals.

Voters now armed with the information that A & B were the two most likely to win, would place their threshold between them, now causing A and E to finish with the top and second-top number of approvals (80% and 60% respectively).

Voters now armed with the revised information that A & E were the two most likely to win, would place their threshold between them, now causing E and D to finish with the top and second-top number of approvals (80% and 60% respectively).

And so on: each new election the top-2 pair just shifts one step left round and round the ABCDE cycle. Z never wins. In fact Z always gets 51% approval, causing it to place fourth out of six every time after the first couple of iterations. But Z is the Condorcet winner (beats all others individually by a 51-49 vote ratio) and also the sincere range voting winner.

To sum up, in this counterattack-example, strategic range voters, even acting on information updated from an infinite number of previous elections, still refuse to elect the Condorcet and sincere-range winner Z, provided the threshold is placed not too far from the utility-midpoint of the guessed-top-2. (If it is placed much nearer to the topmost previous finisher, then Z comes third, not fourth, which actually would be good enough to win the union of all the elections combined, despite not winning any individually. But if fourth, then Z is the last-place loser of the union of all elections combined.)

Second Defense

Our second defense is that, while admitting these strategic effects do cause damage, we claim that such examples either are rare, and/or that when they do occur, the damage they cause often is not large. The basis for that claim is that our Bayesian regret experimental computer-simulation studies, must have found many instances of this kind of scenario, but still Range Voting ended up with lower Bayesian Regret than everything else tried...

Still, the attacker could argue that those studies tended to pit strategic range versus strategic Condorcet, and honest versus honest – and the attackers disparage that attempt to compare apples versus apples because the attackers believe voters are more likely to be strategic with range voting than they are with Condorcet voting.

So: let us compare zero-info (mean-based threshold strategic) approval voting with honest Condorcet voting! Here are some numerical Bayesian Regret numbers with a variety of different "knob settings" extracted from our dataset (see paper #56 here) where C is the number of candidates and lower regret values are superior:

Bayesian regrets
Cond[hon] 0.053570.077430.088020.09275
AV[strat-zero] 0.053570.076450.099850.11654
Cond[hon] 0.030760.044570.050730.05347
AV[strat-zero] 0.030760.046540.062950.07531
Cond[hon] 0.023080.033540.038150.04023
AV[strat-zero] 0.023080.035710.048860.05904
Cond[hon] 0.019140.027790.031680.03343
AV[strat-zero] 0.019140.029750.040950.04978
Cond[hon] 0.016660.024230.027590.02919
AV[strat-zero] 0.016660.025970.035800.04360
Cond[hon] 0.203780.294630.335590.35606
AV[strat-zero] 0.203780.308090.413900.50062
Cond[hon] 0.105950.144150.161570.17096
AV[strat-zero] 0.105950.102570.131060.15176
Cond[hon] 0.061080.082960.093200.09855
AV[strat-zero] 0.061080.062970.083450.09913
Cond[hon] 0.045730.062330.070030.07406
AV[strat-zero] 0.045730.048490.065050.07796
Cond[hon] 0.037760.051560.058040.06147
AV[strat-zero] 0.037760.040550.054680.06596
Cond[hon] 0.032840.044910.050590.05356
AV[strat-zero] 0.032840.035510.047900.05787
Cond[hon] 0.395810.541970.612590.65092
AV[strat-zero] 0.395810.424760.560390.66834
Cond[hon] 0.142030.189890.211790.22247
AV[strat-zero] 0.142030.140410.178830.20575
Cond[hon] 0.081510.109980.122420.12860
AV[strat-zero] 0.081510.086770.114480.13559
Cond[hon] 0.061300.082700.092380.09720
AV[strat-zero] 0.061300.066920.089510.10664
Cond[hon] 0.050680.068420.076670.08087
AV[strat-zero] 0.050680.056130.075270.09052
Cond[hon] 0.044240.059630.066830.07056
AV[strat-zero] 0.044240.049120.066020.07944
Cond[hon] 0.534690.725680.814800.86252
AV[strat-zero] 0.534690.591360.774500.91993
Cond[hon] 0.217170.287980.318860.33252
AV[strat-zero] 0.217170.218540.276540.31754
Cond[hon] 0.124960.166760.185010.19362
AV[strat-zero] 0.124960.135230.177510.20922
Cond[hon] 0.093740.125340.139670.14640
AV[strat-zero] 0.093740.104190.138880.16549
Cond[hon] 0.077460.104150.116120.12172
AV[strat-zero] 0.077460.087530.117100.14005
Cond[hon] 0.067680.090640.101030.10623
AV[strat-zero] 0.067680.076800.102800.12306
Cond[hon] 0.821571.108061.241721.30925
AV[strat-zero] 0.821570.926371.210631.42940
Cond[hon] 0.301930.401500.443450.46143
AV[strat-zero] 0.301930.308140.388280.44416
Cond[hon] 0.174800.232240.256810.26826
AV[strat-zero] 0.174800.190140.248840.29327
Cond[hon] 0.130340.175680.193950.20318
AV[strat-zero] 0.130340.147390.194870.23286
Cond[hon] 0.108560.145320.161660.16935
AV[strat-zero] 0.108560.123630.164630.19754
Cond[hon] 0.094800.126760.141070.14832
AV[strat-zero] 0.094800.108270.144820.17290
Cond[hon] 1.152501.550281.732641.82449
AV[strat-zero] 1.152501.306821.706482.00369
Cond[hon] 0.424480.562280.621010.64618
AV[strat-zero] 0.424480.431510.548090.62466
Cond[hon] 0.245690.326570.359570.37765
AV[strat-zero] 0.245690.267250.352070.41392
Cond[hon] 0.184540.245310.273270.28530
AV[strat-zero] 0.184540.206970.276560.32875
Cond[hon] 0.152150.204510.227020.23778
AV[strat-zero] 0.152150.174290.232430.27795
Cond[hon] 0.133020.177170.197930.20795
AV[strat-zero] 0.133020.152620.204030.24350
Cond[hon] 1.616312.183962.438472.57293
AV[strat-zero] 1.616311.852112.401812.83800
Cond[hon] 1.782462.655833.219103.62039
AV[strat-zero] 1.782462.635393.214843.63070
Cond[hon] 1.143531.705822.064222.31975
AV[strat-zero] 1.143531.698852.063262.33086
Cond[hon] 0.880381.314871.600031.79878
AV[strat-zero] 0.880381.308931.594501.80465
Cond[hon] 0.738481.101691.338441.50411
AV[strat-zero] 0.738481.097821.338931.50959
Cond[hon] 0.644690.965441.169391.32120
AV[strat-zero] 0.644690.963351.171691.32409
Cond[hon] 1.620432.180462.440472.55936
AV[strat-zero] 1.620431.845312.389682.83686
Cond[hon] 1.341511.995952.397792.67292
AV[strat-zero] 1.341511.944572.383542.67942
Cond[hon] 0.970861.454001.752341.96743
AV[strat-zero] 0.970861.428571.753381.97274
Cond[hon] 0.781201.173421.417131.58658
AV[strat-zero] 0.781201.166651.412231.59399
Cond[hon] 0.671750.998561.201081.36525
AV[strat-zero] 0.671750.991091.201251.37501
Cond[hon] 0.594070.892911.076271.21086
AV[strat-zero] 0.594070.887131.070001.21093
Cond[hon] 1.606162.182212.437392.57289
AV[strat-zero] 1.606161.858852.392312.82517
Cond[hon] 1.346181.985212.410992.73145
AV[strat-zero] 1.346181.927962.338482.63158
Cond[hon] 0.957021.417701.726341.95916
AV[strat-zero] 0.957021.406261.701281.91378
Cond[hon] 0.766201.136221.391981.57910
AV[strat-zero] 0.766201.136041.380361.55270
Cond[hon] 0.655560.980811.194011.34977
AV[strat-zero] 0.655560.976651.188811.33113
Cond[hon] 0.582840.866831.059411.19832
AV[strat-zero] 0.582840.864731.054461.18366
Cond[hon] 0.903521.291481.538771.71467
AV[strat-zero] 0.903521.196811.449201.64860
Cond[hon] 0.719281.048031.270931.43756
AV[strat-zero] 0.719281.016821.235401.40059
Cond[hon] 0.614570.902081.096571.24695
AV[strat-zero] 0.614570.888911.073461.20784
Cond[hon] 0.544460.803700.975471.11024
AV[strat-zero] 0.544460.792130.961381.08519
Cond[hon] 0.496130.733590.891361.01554
AV[strat-zero] 0.496130.724210.883020.99384
Cond[hon] 1.603892.162442.424792.57387
AV[strat-zero] 1.603891.848722.395992.84532

Summary: AV[strategic zero-info voters] usually does better than Cond[honest voters] with 3 candidates but Cond[honest] usually does better than AV[strat-zero] with 5 candidates (although not always in either case). The differences are not tremendous though; I don't think any difference ever was as bad as a factor of 2 (although I might have missed an occurrence; this was a quick skim by eye) and usually the ratio is ≤1.4.

So in conclusion, Benham's nasty example does bother me but it appears to occur rarely enough, and/or with low enough severity, that in practice it should not hit too hard.

If you believe, though, that Condorcet voters will always be honest and approval/range voters will always employ the zero-info mean-based threshold strategy, and if further you believe there will always be 5 or more candidates, then Condorcet is a better voting method than AV/Range.

I personally believe that a lot of Condorcet voters will be strategic and employ exaggeration of the 2 frontrunners to max & min ranking; and also a lot of range voters will be honest, and/or when they do choose to strategize, they'll do it based on considerably more than zero info. I'm pretty confident these things will happen to a great enough extent that Range Voting will be superior to Condorcet (plus it is simpler).

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