Note on authorship: This result was discovered by Warren D. Smith & published on these web pages in August 2006. We in 2011 found out that essentially the same discovery had been made independently by Jean-Francois Laslier in Paris and published in an online economics-paper-archive, in December 2006. Laslier's analysis also is interesting and quite different from ours. Then in 2021 Adrien Faure pointed out to me that it already had been stated in Laurent Mann's PhD thesis at Ecole Polytechnique in Palaiseau 1995.
A "Condorcet Winner" is a candidate who would beat any opponent in a simple majority vote (2-man race). Unfortunately there are cyclic election examples where no Condorcet Winner exists. But when one does exist, it seems difficult (but possible!) to dispute that he ought to be the election winner. And so, much fear and gnashing of teeth arose from the realization that the Approval Voting system (and hence the Range Voting system) could fail to elect a Condorcet Winner.
But we shall now see that, in practice under reasonable assumptions about the strategic behavior of approval voters, Approval elections will choose Condorcet winners whenever they exist, and in fact (counterintuitively!), plausibly will do so better in practice than "official" Condorcet voting methods! (See also puzzle #61 for another reason that might be so.)
And because strategic range voters generally vote approval-style, the same would be true of range voting elections with strategic voters. In other words, to the extent range voters are strategic they will elect Condorcet winners; whereas to the extent they are honest, range voting should perform better than Condorcet. Even if you don't quite buy all that, we think you still will agree that in practice, one should expect no great advantage for Condorcet methods over the simpler range voting system.
The reasonable assumption: Approval voters act according to the following strategy: they order the candidates from best-to-worst, then select a "threshold" T, and they approve the candidates above T. They choose T to cause their vote to have the most impact.
The claim: Let N≥2. There does not exist an N-candidate tie-free election in which the Approval (A) and Condorcet (C) winners differ, provided that, if they were going to differ (A≠C) that all the approval voters would place their threshold T strategically under the assumption the winner was going to be either A or C, i.e. would place T somewhere between them.
Proof: Assume for a contradiction that A≠C. Then the approval voters will strategically place their thresholds between C and A. But that will cause C to be approved more times than A is approved (since C, being the Condorcet winner, is preferred over A by a majority). Hence the claim A was the Approval winner and A≠C, leads to a contradiction. Hence either A=C or there is no Condorcet winner C. Q.E.D.
Remark: Actually, essentially the same argument might seem to show that Condorcet winners (when they exist), will be elected by strategic voters not only under Range and Approval voting, but in fact under a large variety of voting systems. When we investigate that, though, we find that the argument works better for range than for the three most-common alternative voting systems. That analysis subpage tries to gain more understanding of exactly which voting systems work and under exactly which assumptions about strategic voter behavior – and exactly how "reasonable" are those assumptions? – and exactly what are the experimental numbers?
Anyhow, definitely Range and Approval with our "strategic thresholding" voter-behavior, work.
In view of this fact, it seems that much of the so-called "conflict" between the Approval and Condorcet "philosophies," is illusory.
We now see that approval voting always will elect the Condorcet Winner C (if C exists) provided merely that the voters consider it likely that both C and whoever (A) the putative non-C approval winner might be, might be elected – and hence vote strategically about C versus A.
So we see that for practical purposes, Approval is a Condorcet method!
Indeed – counterintuitively – it might actually be that Approval Voting is more likely to elect the Condorcet Winner in practice, than Condorcet methods! (Indeed, experiments indicate that happens.)
Why? Because in approval voting it is quite rare that strategically voting dishonestly, is wise. (And when it is wise, it is even rarer that people will actually realize it and do it.) In other words, with Approval, people will tend to honestly order the candidates, and the only strategic decision they'll make is where to locate their approval "threshold."
In contrast, with Condorcet methods with rank-order ballots, it seems a lot more common to see a way to be usefully strategic-dishonest in your vote. (See also puzzle #62.) In other words, I suspect it will be comparatively common that voters will dishonestly misorder their preferences to, e.g. try to elect lesser evils.
In that case, it might well be that Condorcet rank-order ballot systems will do something silly (like not electing whoever should have been the Condorcet Winner with honest votes, such as in this example) more commonly than Approval will in practice fail to elect the Condorcet Winner.
In that case, for practical purposes Approval will be a better Condorcet method than actual "official" Condorcet methods!
I really think this is actually quite likely, it is not just some unlikely theoretical speculation. And if so, it is quite a remarkable counterintuitive conclusion.
A different discussion of the same phenomenon