Condorcet with Borda Runoff

rob

@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

@rob said in Condorcet with Borda Difference Runoff:

Then it isn't a Nash equilibrium.

Exactly. So I want to deny that sincere voting can be a Nash equilibrium. There exists a better way to vote, per Gibbard. All the voters have to do is find it.

@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.

rob

@andy-dienes said in Condorcet with Borda Difference Runoff:

Given complete information,

That's the tricky part, though, isn't it?

rob

@jack-waugh said in Condorcet with Borda Difference Runoff:

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.

@rob said in Condorcet with Borda Difference Runoff:

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)***

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***pretty sure this is true at least. been a while since I read the relevant papers.

Jack Waugh

@rob said in Condorcet with Borda Difference Runoff:

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?

rob

@jack-waugh said in Condorcet with Borda Difference Runoff:

How are human candidates for political office not discrete?

They are discrete, I was contrasting them with the opposite extreme, which is where they lie along a continuum.

rob

@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.

@rob yep, in particular this is the Median Voter Theorem

rob

@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.

cfrank

@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

C>B>D>A [38%]
A>B>C>D [30%]
A>C>D>B [22%]
B>D>A>C [10%]

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.

@cfrank This is quite similar to the construction here, where generalized Condorcet winners are defined by positional dominance over progressively larger subsets of candidates. I think you would enjoy that paper.

cfrank

@andy-dienes I think you may have sent this or a similar paper before, and I did enjoy it and would definitely like to read about this concept again, I think something like it is a good idea.

Casimir

This idea is quite similar to MARS. And in the examples you list, MARS would yield the same results each time (assuming a range 0-3). The main difference is that it seems you imply strict rankings. This creates a teaming effect. The A voters fail to win strategically, because of the number of candidates supported by the other group, not because of a resistance to polarization inherent in the method. When you think about it, it's quite odd if a majority has no way to vote that ensures them winning (majority criterion).

cfrank

@casimir I don't think avoiding the majority criterion is odd if it means that the slim majority is required in some way to make concessions to the minority and thereby somehow build a broad supermajority consensus. Unfortunately you are correct that this method is vulnerable to strategic nomination and trying to address that while preserving "minority consent" gets complicated. Sometimes I think that single-winner systems are not ideal at all, and it would just be better to have proportional representation and some kind of sortition system.

@cfrank said in Condorcet with Borda Runoff:

I don't think avoiding the majority criterion is odd if it means that the slim majority is required in some way to make concessions to the minority and thereby somehow build a broad supermajority consensus.

Let's say the electorate is split into polarized groups of 52% and 48%, each of which is thoroughly unwilling to compromise with the other. Do you agree that the larger group should get to choose the winner? Supermajorities just aren't possible all the time, and it still doesn't make sense to try to force the issue in a single-winner method.

it would just be better to have proportional representation and some kind of sortition system.

yes yes yes. this is precisely what I mean by suggesting the philosophy guide the design at a higher level than actual algorithm mechanics.

cfrank

@andy-dienes oh absolutely, when there is no hope of compromise the majority should win out, there isn't a better option. In my opinion that's the worst-case scenario. Ideally there would be some mechanism of finding some level of supermajority consensus if one exists. I think the path of least resistance for the majority to take control should be for them to make reasonable concessions to the minority whenever possible. They could in principle take control by force, but hopefully they won't be motivated enough to do that.

I was trying to think of finding the primary Condorcet winner A, and then ignoring the ballots that rank A as highly as appears and finding the "minority" Condorcet winner B, then doing some kind of comparison of the rankings between the voters who marked A as highly as appears and the voters who marked B as highly as appears. But that's probably getting too complicated.

@cfrank said in Condorcet with Borda Runoff:

I was trying to think of finding the primary Condorcet winner A, and then ignoring the ballots that rank A first and finding the "minority" Condorcet winner B, then doing some kind of comparison of the rankings between the voters who marked A first and the voters who marked B first. But that's getting too complicated.

I mean, this is basically what IRV does though lol (and as it happens, IRV has one of the strongest 'veto powers' aka minority protection of any voting method)

cfrank

@andy-dienes that is sort of what IRV does, but I think it's a bit different from what I'm considering since IRV doesn't operate on Condorcet winners but rather on elimination by anti-pluralities. For example, if the "majority" and "minority" Condorcet winners were identical, the election could end.