The John and Jane dialogue must have given you the wrong impression that my other discussions of the possibilities were limited to the three candidate case. Please read them again with the idea in mind that they apply to any finite number of candidates.
It was understood, although I admit my proof, that a Compromise>Favorite>Worst>Compromise cycle cannot produce favorite betrayal incentive in winning votes, only applies when the cycle contains three candidates.
I won't quibble about whether an exhaustive consideration of cases is more thorough than a course of simulations. They are both valuable and complement each other, because each can give insights to improve the next version of the other.
I would agree with that. But looking at cases and subcases and trying to divvy up the possibilities this way can lead to strange conclusions. Imagine we come up with two cases that point to voting method A being better than B, and a third case that shows the opposite. The conclusion you may draw (especially if you like Copeland ;D ) is that method A is better. But what really matters is how LIKELY these situations are. If the third case is ten times more likely, then B seems like the better method (at least from the standard of favorite betrayal).
Before the dialogue, I considered two cases: (1) Compromise is sure to beat Favorite. (2) Compromise is not sure to beat Favorite.
So far these two cases exhaust the possibilities, but that won't stop us from considering some subcases later.
In case (1) DMC gives no incentive at all for Favorite Betrayal, because approval of Compromise already reinforces the Compromise>Favorite defeat to the max.
However, in the same case under Shulze, Ranked Pairs, or River (whether margins or wv) it sometimes helps to betray Favorite by ranking Compromise strictly ahead of Favorite. I'll give an example below, for those that have never seen this before.
All such examples will require a cycle which contains more than three candidates. I find such many-candidate cycle examples uncompelling, because they seem unlikely. I suppose if there was already a sincere cycle in the preferences, then it wouldn't be unreasonable to imagine strategic voting could generate a larger cycle. But it does not concern me NEARLY as much as the example I gave, in which a strategic vote by one faction in a linear political spectrum with three major candidates produced favorite betrayal incentive in DMC.
Near the end of my previous message (in a part not quoted by Adam) I showed that (in the Bubble Sorted Approval formulaton of DMC) only in (what I called) case 2d would Favorite Betrayal payoff,
It's difficult to translate DMC to the cases Demorep lists there, since the way DMC is evaluated is not strictly the same as what he mentions there. Does Demorep's bubble sort go top-down or bottom-up?
In terms of the cycles and the approval ranks, my example seems most like 2c, only it is, in fact, a problem for DMC.
and that case 2d is not only unlikely,
Unlikely, absolutely. More likely than the example you included in your message? Yes -- just as clearly in my opinion. Of course it is only my opinion. You could perhaps convince me that it is a larger threat by showing a set of possible votes that would lead to that cycle, and arguing that that set of votes are plausible, or could be reached from a plausible set of votes through a strategic vote by one of the factions.
but also very difficult to trust the polls on.
I could construct the example so that the strategy is correct assuming the polls for pairwise margins and approval counts are accurate to 5%; probably more, if I pushed it.
(I would appreciate a similarly thorough analysis of cases (1) and (2) from the wv folks.)
As hard as it is to translate those cases into DMC, it seems outright impossible to do so for a method that doesn't consider approval counts at all. It seems like all we can really consider are permutations of pairwise defeats. If you had something else in mind, get me started and I'll try to follow through.
-Adam
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