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.

Benham's attack

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

(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

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

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.

Counterattack
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:

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:

Summary: AV usually does better than Cond with 3 candidates but Cond usually does better than AV 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. 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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