Least-bad Single-winner Ranking Method?
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The question came up in the context of a new method proposed in the present forum.
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@Jack-Waugh I would say either Minimax Condorcet or Borda
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Objectively, the answer is Game Theory Voting. However, it is basically impossible to explain to, well, anyone.
The idea is that, given any two ranked-ballot voting methods, we can compare how many voters prefer the winner of one over the winner of the other, and vice versa. Essentially, treating the ballots as giving head-to-head votes between the candidates who would win under each method.
On average, in the long run, game theory voting will produce winners who are preferred by as many or more voters than the winner produced by any other method. In other words, there is no ranked-ballot voting system whose winners are preferred by more voters than the game-theory winner, over the long run.
It is a Condorcet method, and indeed a Smith method, however the actual tiebreaking procedure is a linear optimization problem, which can be solved using, eg. the simplex method, to find each candidate’s optimal probability of winning. The winner is then selected with those probabilities.
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@NevinBR said in Least-bad Single-winner Ranking Method?:
On average, in the long run, game theory voting will produce winners who are preferred by as many or more voters than the winner produced by any other method. In other words, there is no ranked-ballot voting system whose winners are preferred by more voters than the game-theory winner, over the long run.
So "optimal" means that the expected value of the margin between the winner selected by GTV and by some other system will always be non-negative?
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@NevinBR Thank you for the Game Theory Voting link. I see that there is an assumption that when a Condorcet selection exists that it's the best choice. I think there are situations where that is NOT a correct assumption. I tried to modify the scenario that the authors, Rivest and Shen, used, but it wasn't working out very well, so I simplified one of the scenarios that I had previously come across, which was from using randomly generated numbers.
Scenario: There are 4 candidates and 13, 138 voters. With FPTP, candidate B gets 3,900 votes, C 3,547, D 3,456 and A 2,325. B wins Condorcet buy 2 votes, 6570/6568, over each of the other candidates. At first blush, B seems to be clearly the best choice.
I exaggerated the randomly generated numbers into a scenario where ALL of the voters that picked B as their first choice picked D as their second choice. So, in a head to head with B and D, B got 2 more votes than D, but D was the second choice of all 3,900 supporters of B. Conversely, only 639 of the 3,456 D supporters chose B as their second choice.
It seems to me that candidate D is preferred by the most people to the greatest degree.
I didn't try to understand all of the Game Theory details, but it seems to me that there is a way to modify it so that it considers this in the weighting and sets the probabilities accordingly.
I'm wondering what people think of this. (Both of the assertion of D being the best choice, and the potential of Game Theory to account for this.)
Here are the full numbers for the scenario:
0:Option A
819:Option A>Option B>Option C>Option D
52:Option A>Option B>Option D>Option C
141:Option A>Option C>Option B>Option D
898:Option A>Option C>Option D>Option B
116:Option A>Option D>Option B>Option C
299:Option A>Option D>Option C>Option B
0:Option B
0:Option B>Option A>Option C>Option D
0:Option B>Option A>Option D>Option C
0:Option B>Option C>Option A>Option D
0:Option B>Option C>Option D>Option A
1800:Option B>Option D>Option A>Option C
2100:Option B>Option D>Option C>Option A
0:Option C
988:Option C>Option A>Option B>Option D
658:Option C>Option A>Option D>Option B
3:Option C>Option B>Option A>Option D
667:Option C>Option B>Option D>Option A
473:Option C>Option D>Option A>Option B
668:Option C>Option D>Option B>Option A
0:Option D
1044:Option D>Option A>Option B>Option C
445:Option D>Option A>Option C>Option B
382:Option D>Option B>Option A>Option C
257:Option D>Option B>Option C>Option A
635:Option D>Option C>Option A>Option B
693:Option D>Option C>Option B>Option A -
I've just been looking through old threads and found this one.
@nevinbr said in Least-bad Single-winner Ranking Method?:
Objectively, the answer is Game Theory Voting. However, it is basically impossible to explain to, well, anyone.
This method was also discussed at greater length on the old CES Google Group here. But the point is that while it is optimal in the sense that you might want to optimise this one particular thing, you might not want to optimise that one particular thing, so it would be wrong to say that it's objectively the best method.
In fact, when you look at what it is trying to achieve, it becomes quite clear that it's really just a bit of game theory fun rather than a method seriously trying to optimise something that a human would want to optimise. And you can see that by an example that you gave in the CES thread.
To expand on that last point, consider the following election with 3 candidates and 100 voters:
49 ABC
48 CAB
3 BCA
This has a Condorcet cycle:
A beats B by 94
B beats C by 4
C beats A by 2
At first glance, we notice that A has the largest victory and the smallest defeat, as well as the highest Borda total. However, the GT method elects A just 4% of the time.
Can that really be optimal?
Let’s think about it.
Okay, B gets crushed by A and barely squeaks by C. We don’t want to get crushed because that is bad for our long-term average, so B probably should not win. And if B loses (or didn’t run at all) then C defeats A.
It seems that A’s victory over B is mostly ephemeral. As much as we would like to score that +94 to improve our long-term average, the only way it happens is if we pick A and the system we’re up against picks B. But since B shouldn’t win, we expect the other system won’t pick B either.
In particular, if we usually pick A and the other system usually picks C, then we are going to lose frequently. We would rather be that other system and pick C most often, which is exactly what GT does. The optimal distribution is:
A wins 4%
B wins 2%
C wins 94%
Moreover, in any 3-way Condorcet cycle, the probability of each candidate winning is always proportional to the margin of victory between the other two candidates. And this is provably optimal.
In the example ballots, it seems clear that A is the best winner. However, under this method, because B definitely isn't a good winner, other methods won't select B so the A>B pairwise win might as well be ignored. Better concentrate on A v C instead and since C wins that, C is the overall best pick.
So what this method does is not pick the candidate that is somehow judged to be best for society, but the candidate that has the best average margin of victory against a candidate picked by another method playing this same game. And this method would never lose on average against another method (though obviously might tie - e.g. against itself). I don't see how this optimality relates to real life at all and why it would be good for us to adopt it.
To make this clear, the ranked pairs method is a Condorcet method that a lot of people like. But I could devise a method that elects a candidate that pairwise beats the ranked pairs winner whenever one exists. Otherwise elect the ranked pairs winner. According the metric used here, this method is better than ranked pairs - when these two methods are viewed as the only choices at least. But is this method better tban ranked pairs by any reasonable measure? Of course not.
An interesting academic exercise, but nothing more. Certainly not objectively the best single-winner voting method.
