Which voting systems yield Condorcet Winners under reasonable assumptions about voter strategic behavior?

We had argued under "reasonable assumptions" about strategic voter behavior, that range voting with strategic voters would always elect Condorcet winners (based on honest voters) whenever a Condorcet winner (CW) exists. ( The "reasonability" of these assumptions can be questioned, but it at least [1] should be clear exactly what they are and [2] should be clear that they enjoy at least some approximate validity in practice. )

Counter-intuitively, with strategic voters Condorcet voting systems (which are designed to elect Condorcet winners) actually can (and in some simulation studies do) exhibit smaller probability of electing CWs than range voting.

The argument for why Range elects CWs seems very simple and quite general. Therefore you might expect/hope that, via slight variants of that argument, we could show that strategic voters under many other voting systems also elect CWs. We now examine that and find that the argument clearly works better when applied to range voting, than when applied to Plurality, Approval, and Instant Runoff (IRV).

Which voting systems elect CWs?

We go through systems one at a time.

Plurality voting. You might (as a first try) argue that plurality voters will elect Condorcet winners because

1. If they were to elect somebody else P, then they would not because
2. They'd view both the Condorcet winner C and P as top-2 competitors (?) in which case
3. They'd strategically vote for one of those top-two
4. causing C to win, not P.

However, the way voters learn who are the top-2 that they must strategically vote for (else they "waste their vote") is via e.g, pre-election polls. Such polls, assuming they too are conducted using plurality voting, will not necessarily indicate that a Condorcet winner is in the top two. (E.g. one who is everybody's second choice would get zero plurality votes, completely escaping notice. There are many other less-dramatic scenarios. This is very common when, for example, a third-party candidate is preferred by the voters from each of the two major parties over the major-opponent, and the two majors have approximately equal support. In that case the third-party candidate is the Condorcet "beats all" winner but gets a very tiny percentage of the plurality top-rank votes. To see how common this is, see the appendix.)

Hence plurality voting will not necessarily elect CWs under reasonable assumptions about strategic voter behavior.

Instant Runoff Voting (IRV): You might at first argue, IRV voters will elect Condorcet winners because

1. If they were to elect somebody else "I", then they would not because
2. They'd view both the Condorcet winner C and I as top-2 competitors (?) in which case
3. They'd strategically vote for one of those top-two
4. causing C to win, not I.

However, the pre-election polls (presumably now conducted using IRV) would again not necessarily indicate a CW (e.g. in the same third-party CW scenario we just explained). Again a CW who was everybody's 2nd choice would get eliminated instantly by IRV and would score pessimally in any poll. So such a CW would never be spotted as a strong contender by IRV pre-election polls. Indeed, any CW (or anybody) placing 3rd-or-worse as in Peru 2006, would then not be elected by (this kind of) strategic IRV voters. (To see how common it is for a Condorcet winner not to be one of the two top finishers in IRV, see the appendix. This seems to have happened 9 times in the 150 federal IRV elections in Australia 2007.)

So again – IRV voting will not necessarily elect CWs under reasonable assumptions about strategic voter behavior.

Range voting: In order for the argument to work to cause strategic range voters to elect an honest-voter Condorcet winner C, it suffices if the range voters get the idea that C is a strong contender. If they believe C is one of the top-2 most likely to win, then (the argument roughly proceeds)

1. If somebody else R is the winner of the pre-election range-voting polls, then after strategic range voting, C will win.
2. else (C won the pre-election polls and R was the 2nd-placer) then C will still win.

So if range voters get the idea that the Condorcet Winner C is one of the top 2 contenders, then when they vote strategically, C will win.

In short, if range were going to elect C as the winner or second-place finisher, then range would elect C with strategic voters. Strategic voting amplifies the chances range elects a Condorcet winner to be at least as large as the chances it elects the CW as top or second.

In practice, I suggest to you, those chances are pretty big. See the appendix to see how big – clearly there is a greater chance the Condorcet winner is among the top-two range finishers, than for any other voting method tried (namely IRV, plurality, approval) in that appendix, and as we'll explain next paragraph, those numbers are only a lower bound on the chance strategic range voters will elect an honest-CW (unlike for IRV and plurality where they are also an upper bound).

If, however, the range voters in honest-voting pre-election polls make C be the 3rd-place finisher, then what? Well (unlike IRV & plurality), strategic range voters then still might elect C. (It's pretty likely, but no longer a sure thing.) So we can see immediately that the argument works better for range than for Plurality or IRV.

Finally, what about Condorcet methods? Will they elect CWs?

Well... presumably C (the CW) wins the pre-election (assumed conducted with Condorcet) poll!

Now what happens in the subsequent real election? Now the anti-C voters strategically rank C bottom. As a result, it is possible that now no Condorcet winner exists, and hence C no longer wins. See this and this for exactly how that can happen plus some discussion of how often that happens in different Condorcet flavors.

Thus, in this sort of scenario, strategic Condorcet voters will, quite plausibly, fail to elect an (honest-voter) Condorcet winner, even assuming there are honest voters in the pre-election poll so that one is "elected" by that poll. (The difference is because the "strategy" worked.)

Observe that this will not happen with range. With range, if C wins the pre-election poll, C will still win the real election with strategic voters this time. Guaranteed. With Condorcet: not guaranteed.

That is an example of the strange fact that range voting can be more likely to elect Condorcet winners, than "official" Condorcet methods. Counterintutive, but it can be true (simulation experiments have clearly verified this).

Appendix: Numbers (to back this all up)

All numbers here are from IEVS 3.24 Monte Carlo runs. (29999 elections per datapoint. Each election using 31 honest voters. I have also done runs with 63 and 127 voters but the results change little, hence are not tabulated here.)

Given that a Condorcet Winner exists, it is one of the top two finishers in a plurality election this often [in the random elections model]:

Given that a Condorcet Winner exists, it is one of the top two finishers in an approval election this often:

Given that a Condorcet Winner exists, it is one of the top two finishers in an Instant Runoff election (and hence the IRV winner) this often:

Given that a Condorcet Winner exists, it is one of the top two finishers in a range election this often:

The reader may not love the random elections model. Those readers should check the 2D positional politics models: Range also usually elects Condorcet winners there. Each picture there contains 40,000 pixels, each one representing a different election result. If you check the pictures based on the set of 14 random candidates, you will observe that the range and Condorcet pictures very nearly coincide. However, in contrast, the Condorcet picture disagrees with the IRV picture in about 25% of the pixels (i.e. in about 25% of 2D-positional-model elections).

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