Condorcet with Borda Runoff
This is a minor attempt to modify Condorcet methods in a simple way to become more responsive to broader consensus and supermajority power. It’s sort of like the reverse of STAR and may already be a system that I don’t know the name of. In my opinion, the majority criterion is not necessarily a good thing in itself, since it enables tyrannical majorities to force highly divisive candidates to win elections, which is why I’ve been trying pretty actively to find some way to escape it.
For the moment I will assume that a Condorcet winner exists in every relevant case, and otherwise defer the replacement to another system.
First, find the Condorcet winner, which will be called the “primary” Condorcet winner. Next, find the “secondary” Condorcet winner, which is the Condorcet winner from the same ballots where the primary Condorcet winner is removed everywhere.
Define the Borda difference from B to A on a ballot as the signed difference in their ranks. For example, the Borda difference from B to A on the ballot A>B>C>D is +1, and on C>B>D>A is -2.
If A and B are the primary and secondary Condorcet winners, respectively, then we tally all of the Borda differences from B to A. If the difference is positive (or above some threshold), then A wins, and if it is negative or zero (or not above the threshold), then B wins.
For example, consider the following election:
In this case, A is a highly divisive majoritarian candidate and is the primary Condorcet winner. B is easily seen to be the secondary Condorcet winner. The net Borda difference from B to A is
Therefore B would be chosen as the winner in this case.
Some notes about this method:
It certainly does not satisfy the Condorcet criterion, nor does it satisfy the majority criterion. These are both necessarily sacrificed in an attempt to prevent highly divisive candidates from winning the election. It does reduce to majority rule in the case of two candidates, and it does satisfy the Condorcet loser criterion, as well as monotonicity and is clearly polynomial time. It can also be modified to use some other metric in the runoff based on the ballot-wise Borda differences.
Continuing with the above example, suppose that the divisive majority attempts to bury B, which is the top competitor to A.
This will change the ballots to something like
And if the described mechanism is used in this case, we will find instead that C is elected. So burial has backfired if B is "honestly" preferred over C by the divisive majority, and they would have been better off indicating their honest preference and electing B.
And again, suppose that the divisive majority decides to bury the top two competitors to A, namely B and C, below D, keeping the order of honest preference between them. We will find
In this case, the secondary Condorcet winner is D, and the mechanism will in fact elect D, again a worse outcome for the tactical voters.
Finally, suppose that they swap the order of honest preference and vote as
Still this elects D.
As a general description, this method will elect the Condorcet winner unless they are too divisive, in which case it will elect the secondary Condorcet winner, which will necessarily be less divisive. I believe that choosing the runoff to be between the primary and secondary Condorcet winners should maintain much of the stability of Condorcet methods, while the Borda runoff punishes burial and simultaneously addresses highly divisive candidates.
These are sacrificed in an attempt to prevent highly divisive candidates from winning the election.
I can certainly appreciate this goal, as anti-divisiveness is my primary reason for caring about any of this.
My concern that any step away from Condorcet tends to reduce its stability (I believe an election that elects the Condorcet winner tends to be a Nash equilibrium when everyone votes sincerely). Regardless, this seems similar to my suggestion of trying to measure as directly as possible how divisive a candidate is (larger standard deviation, for instance), and down-weighting them one way or another.
@rob thanks. And apologies for being snippy before. You and @Andy-Dienes pretty much convinced me that score voting isn’t going to work.
@cfrank All good and likewise.
@rob you are probably right to be careful about stability, I’m trying to stay very near to Condorcet. I think this system may still be vulnerable to burial but my guess is that it’s still highly resistant to tactics in general. I could be wrong, I’m going to code it up and see what happens.
Is that because it is too hard to figure out the better voting decision procedure?
I don't understand your question.
It is a Nash equilibrium when everyone votes sincerely (*), because it is specifically designed to be.
If you are wondering why I think that is a good thing, ask.... although I think it should be fairly obvious to most people, and I think I just explained it in another thread.
* not 100% perfect, but close enough
@rob Well, let's suppose everyone knows a more effective strategy for serving that person's values. Then voting "sincerely" is not a Nash equilibrium, because once everyone does it and the outcome is observed, some voters will regret that they gave up power, so next time around, they will change to insincerity.
@jack-waugh Then it isn't a Nash equilibrium. A Nash equilibrium is when no one can get better results by changing their behavior, without anyone else changing their behavior. (*) The scenario you described is the opposite of that.
/* note that "better results" is rather tricky with voting, since no individual voter will tend to change the outcome. So there are reasonable ways of defining "better results," that is a discussion of its own.
@jack-waugh This is one of the strongest selling points of Approval actually. Given complete information, the election of a Condorcet winner (when one exists) is a Nash equilibrium in Approval voting.
@andy-dienes said in Condorcet with Borda Difference Runoff:
Given complete information,
That's the tricky part, though, isn't it?
So I want to deny that sincere voting can be a Nash equilibrium.
You are mixing issues. Gibbard is one thing, that applies to discrete candidates, and is a tiny corner case of imperfection, rather than a thing that generalizes to all cases.
This is why I go back to voting for a number and choosing a median, because you can speak of all this stuff about Nash equilibria and strategic voting and such without being thrown off by Gibbard or Arrow or what have you.
If you wish to deny that for any given election method, sincere voting and strategic voting are not 100.00000% identical, fine. For temperature voting as I described, I will argue, no, they actually are identical, you gain exactly zero value by voting with anything other than your preferred temperature. (*) It is indeed a case of "everyone voting sincerely is a Nash equilibrium."
For good Condorcet methods, this is so close to being true that I think it is good enough, but if you want to spin on Gibbard, knock yourself out. I don't think it is a productive use of time, though. It is black and white thinking, basically like "even the safest airplanes crash some percentage of the time, therefore I might as well fly in this thing"
For Score, Approval, and choose-one, it is obviously not close to true that sincere voting is a Nash equilibrium.
/* please don't try to find contrived counterexamples such as people who like both 68 and 72 better than 70.
For temperature voting as I described, I will argue, no, they actually are identical, you gain exactly zero value by voting with anything other than your preferred temperature.
"Temperature" voting is usually referred to in academic literature as the "single-peaked preference model" and in fact median is strategyproof! (as is approval, and Condorcet methods)***
***pretty sure this is true at least. been a while since I read the relevant papers.
You are mixing issues. Gibbard is one thing, that applies to discrete candidates,
How are human candidates for political office not discrete?
and is a tiny corner case of imperfection, rather than a thing that generalizes to all cases.
What does it take to bring about the tiny corner case?
@andy-dienes said in Condorcet with Borda Difference Runoff:
"single-peaked preference model" and in fact median is strategyproof!
Awesome. I've never seen any discussion of it elsewhere, it's good to know that others have used it as an example.
@andy-dienes right, I'm familiar with that one, I took it a bit further by having, rather that discrete choices along a spectrum, an infinite number of potential choices along the same spectrum. Works well for temperature, club dues, etc, and the extreme simplicity of the voting and tabulating serves the example well.
@rob Here is a further modification, which you may dislike perhaps, but I think it produces a candidate that is very broadly consensual. One can continue defining tertiary and quaternary and Nth-order Condorcet winners, etc. Rather than having a Borda difference runoff or something of that nature, one can try to find the first N such that the Nth order Condorcet winner positionally dominates the (N+1)st order Condorcet winner, and by that criterion try to elect the Nth order Condorcet winner.
For example, if the ballots are
Then the primary, secondary, tertiary and quaternary Condorcet winners are A, C, B and D in that order. B is the lowest-order Condorcet winner who positionally dominates the (N+1)st Condorcet winner, and is in some ways a potentially better candidate than C. If you look at the ballots, B is in the top two positions for 78% of the electorate (although the remaining 22% scored them minimally). That's just a different method but I think it could be interesting to look into.
This method conforms to the results of the first example given as well, since in that case B is the lowest-order Condorcet winner who positionally dominates their successor. One issue is that it's possible that none of the Nth-order Condorcet winners positionally dominate their successor.