## With strategic voters, Condorcet voting can fail to elect Condorcet Winner

A myth about Condorcet voting: Although Condorcet is vulnerable to strategic voting, it supposedly is only vulnerable in cycle situations with no Condorcet winner.

The bust: In fact it is vulnerable in situations with an honest Condorcet winner too: Let there be 3 candidates A,B,C (or there could be more D,E,...; it'll still work). Suppose C is the honest-voter Condorcet winner, with slight pairwise victories over A,B,...; and A has large pairwise victories over everybody but C. The A-voters instead of honestly voting A>C>B>... strategically vote A>B>...>C. This causes C no longer to be the Condorcet winner. In fact C now loses pairwise (by moderate margins) to every candidate except A. And A now wins pairwise over every candidate except C, who beats A by slight margin. Now A is the winner under all these popular Condorcet methods (plus more too, if you arrange it right): Schulze beatpaths, Tideman ranked pairs, basic Condorcet, Simpson-Kramer minmax.

Conclusion: Strategic Condorcet voters can fail to elect a Condorcet winner. Strategic range or approval voters will, however, elect an honest-voter Condorcet winner under certain (in practice highly reasonable) assumptions, as is proven by this theorem.

## But... with counterstrategy...

If the other voters "catch on" to the strategic action and realize it is coming down to an A vs. C battle, then they can also alter their votes to try to maximally impact that battle by always ranking A and C top and bottom (or the reverse) regardless of their honest rankings of A, C, and everybody else. If all voters do that, then C (if it truly is an honest-voter Condorcet winner) again will win. If C is not an honest-voter Condorcet winner, then still either A or C will then always win. However, observe that the voters had to pay a heavy price in dishonesty to do that. With range voting, voters who wish to rank A and C top and bottom are not thereby prevented from ranking their truly favorite and truly most-hated candidates top and bottom; range voting thus permits somebody other than A or C to win.

## Details

Some readers have insisted this was not obvious enough, they could not conceive of this kind of phenomenon, etc, hence they require a fully-worked out random-looking numerical example. (I personally thought it was obvious, but "the reader is always right.") So here is a set of 21 ballots for 4 candidates:

#votersTheir vote
6A>C>B>D
2B>C>A>D
3B>A>C>D
2C>D>A>B
2C>B>A>D
5D>C>A>B
1D>A>C>B

The defeats-margins matrix is:

 Option A B C D A 0 14 10 13 B 7 0 5 13 C 11 16 0 15 D 8 8 6 0

C wins (with every Condorcet voting system) because C has a victory pairwise versus each rival, indicated by the all-green row.

Now A's 6 supporters naturally are not happy about that. What can they do to make their man win? There is one, and basically only one, change to their "A>C>B>D" vote they can make which causes A to win. That is to "bury" C with "A>B>D>C." (Well, the alternate burials A>D>B>C and A>D>C>B also work in this example, but they both are at-least-as-dishonest votes and anyway still count as burying C.) If they do this, we get

#votersTheir vote
6A>B>D>C
2B>C>A>D
3B>A>C>D
2C>D>A>B
2C>B>A>D
5D>C>A>B
1D>A>C>B

whereupon A wins using every one of these Condorcet methods: Tideman ranked pairs, Basic Condorcet, Simpson-Kramer min-max, and Schulze beatpaths. (Success!) The new defeats matrix is

 Option A B C D A 0 14 10 13 B 7 0 11 13 C 11 10 0 9 D 8 8 12 0

## How common is this?

Some people think maybe it is very rare for Condorcet methods to suffer from this problem. But there are theorems to the contrary. Here we shall work out a crude estimate for the rarity.

We shall look at the (admittedly simplified versus reality, but at least it is clearly defined and not biased) random elections model. In this model, each voter casts a random vote chosen uniformly independently from among all possible votes. (E.g, with 4 candidates, there are 4!=24 possible votes. Each voter chooses one by flipping a dodecahedral die and a coin.)

Equivalent interpretation: "All V-voter elections are considered equally likely" where an "election" means a table saying, for each voter, how they voted. (That is what I mean by "unbiased." Clearly I cannot be inserting any evil personal biases favoring one particular kind of elections, because all elections are equally likely.) We assume a large number V→∞ of voters.

In this model, it is known that a "Condorcet winner" candidate C exists (in 4-candidate elections) with probability 82.452%.

ESTIMATE: With the "Tideman ranked pairs" Condorcet method, with probability≈45% (or more) in the V→∞ limit, some voter-type whose vote was "X>C>the others" (for some X, where C is the honest-voting Condorcet winner), can force victory for X (assuming all the other voter-types stay with their honest votes) by strategically changing their vote to "bury" C, like this "X>others>C."

DERIVATION: Restricted to the three non-C candidates, there is somebody who would win (since ties have probability→0). Call that somebody X. This strategic move by the X>C>others voters will, with probability→100% (which follows from standard Gaussian statistics), cause C to become pairwise-defeated by every candidate except X, and always by the 2 largest pairwise margins.

Thanks to this strategic move, we now have a situation in which X would win if C were omitted from the race, and in which C loses to every rival but X by the hugest two margins. With Tideman's Condorcet method, C's defeats are "locked in" first. Now X will win if either of the one or two X>somebody defeats is large enough to get locked in before the C>X defeat. That happens with probability≈2/3 because there are 6 possible orderings of these 3 defeats by severity, and only 2 of those orderings stop X from winning – or in the case where X only defeats one of his non-C rivals we get probability≈1/2. So on average say 58%. (This is really cheating because the orderings are not equally likely because X and C are not random candidates, etc. So this argument is only approximate, which is why we call it an "estimate" not a "theorem." One could do a computer Monte-Carlo measurement to overcome that objection and thus get the exact probability to within 0.01%. One can also see this for genuine theorems.) Finally, 0.8245×0.58≈45%. Q.E.D.

Conclusion: This whole "burial strategy works" phenomenon seems very common. In the random elections model using Tideman ranked pairs, it happens at least about 45% of the time. With many other Condorcet methods and other probabilistic models, your mileage may differ, but it ought still to be quite common.

If you want a precise conjecture: I suspect every commonly-accepted Condorcet method will exhibit "burial strategy works" over 30% of the time in random elections with 4 candidates and #voters→∞; and with some number more than 4 candidates, I believe burial will work even more often. In the Dirichlet model, I suspect again we will have >30% occurrence rates provided the number of candidates is right.

Also, if we were to add in the chances that some voter-class could cause the election result to impove in their view (despite not necessarily electing their favorite) by burying the winner, then the estimate "45%" would rise, and if there were the right number of candidates, it'd rise by a lot.