This continuation of the Properties article-series resumes topic #4 (methods and their properties and criterion-compliance). There have been some answers, at the Electorama election-methods mailing-list, to my question about mitigating properties of traditional unimproved Condorcet. Those answers involved a few criteria that I hadn’t included in the set by which I compare methods but I will add them now.
So I’ll start this article by comparing the methods covered so far including Approval, Score, and traditional unimproved Condorcet by those recently cited criteria. Then I’ll resume the intended procedure, by evaluating Improved Condorcet by all of the properties and criteria that I’m using, including the recently cited ones. First, I’ll define these recently cited criteria:
Suppose the ballot consists of two vertical strips of paper. The strip on the left lists in a vertical column, equally-spaced, the designations of the rank positions or rating positions. The strip adjacent to it on the right lists, in a vertical column, equally-spaced, the candidates’ names, with each name opposite the name of a rank position or rating position. Pick up the right-hand strip of paper, turn it upside-down, and replace it where it initially was. Now the candidate names (though upside-down and more difficult to read) are next to different rank-position or rating-position names, the ballot has been reversed.
Some TUC methods comply with Clone Independence. What benefit does that bring?
Some TUC methods meet Clone-Independence. I’m told that, under the particular special conditions stipulated in the criterion’s premise, compliance with Clone-Independence avoids some strategy problem. But in general, the TUC methods have big strategy problems nevertheless:
For an election whose candidate-set includes three candidates, Favorite, Compromise, and Worst, say that the method is Beatpath, the most popular TUC method and Beatpath’s definition is long so I won’t include it here. It can be found at Electowiki. Beatpath is also known as the Schulze method, Schwartz Sequential Dropping (SSD),Cloneproof Schwartz Sequential Dropping (CSSD), Beatpath Winner, Path Voting, and Path Winner.
Suppose that you, and a few others, rank Compromise alone in first place, with Favorite below Compromise. In the count, Compromise wins, as the only unbeaten candidate. But then, suppose you and a few others move Favorite up to first place, with Compromise. Now maybe, with fewer people ranking Compromise over Favorite, Favorite beats Compromise. Say Worst beats Favorite and Compromise beats Worst. Now there’s no unbeaten candidate. Instead there’s a cycle (Favorite>Compromise>Worst>Favorite). Maybe, by Beatpath’s count rule, Worst is the winner in that cycle. Because you decided to not rank Compromise over Favorite, you let Worst win instead of Compromise.
But if you rank Compromise over Favorite, you could be causing Favorite to be beaten, where Favorite could otherwise win as the unbeaten candidate. Additionally, you’re increasing Favorite’s defeat by Compromise, which could additionally contribute to Favorite’s defeat, by Beatpath’s count rule.
So which should you do? No one knows. Even in a u/a election (defined in previous articles), Beatpath’s strategy is unknown. You’ll just have to guess what to do. Maybe try to top-rank only the acceptable candidates who seem especially likely to win, and rank below top the acceptable candidates (possibly including your favorite) who are more likely to just spoil the win of another top-ranked candidate.
So, even if Beatpath’s clone-independence avoids some strategy problem, in some particular special situation, it certainly doesn’t prevent Beatpath (and other TUC methods) from being a strategic mess, without known strategy, even in a u/a election, and with favorite-burial need. In those regards, Beatpath and the other TUC methods compare very unfavorably with Approval, Score, and the Improved Condorcet methods that will be defined in this article.
And that’s just at top-end. TUC methods have a bottom-end strategy-need, even in the simple u/a situation: Optimal u/a strategy for TUC calls for trying to rank the unacceptable methods in reverse order of win-ability.
In contrast, as I said before, the optimal u/a strategy of Approval is just to approve all of the acceptables and none of the unacceptables. That’s the simplest and easiest u/a strategy. The optimal u/a strategy of Score is really the same: Top rate the acceptables and bottom-rate the unacceptables.
Compliance with Reversal Symmetry, and its meaning
Approval and Score comply with Reversal Symmetry. The Improved Condorcet methods that will be defined in this article don’t comply with Reversal-Symmetry.
What does Reversal-Symmetry failure mean?
According to the person who cited that failure at the election-methods mailing list, when a method fails Reversal-Symmetry, that method is saying that the best candidate is also the worst candidate. That depends on the assumption that, if the not reversed ballots choose the best candidate, then the reversed ballots should choose the worst candidate.
That last assumption is unsupported and unwarranted. For one thing, it depends on an assumption that, when reversed, a ballot is saying the exact opposite of what it said before reversal. Is that true?
Suppose that you and I are candidates in a rank-balloting election. One ballot ranks me in first place and you in second-to-last place. Another ballot ranks me last, and ranks you in 2nd place. Are those two ballots saying exactly opposite things?
No, because there’s something special and unique about favorite-ness. For one thing, favorite-ness has always been a standard in its own right. It means something if a candidate is the favorite of more people than is any other candidate. The Plurality voting system has given that standard a bad name but ICT (defined below) doesn’t share any of Plurality’s problems.
For another thing, a top-count—a count of how many people rank a candidate in first place—is a measure of faction-size, and that faction-size measure is ICT’s way of automatically avoiding the chicken dilemma. (I’ve defined the chicken dilemma before, but it will be clear when, below, I define the Chicken Dilemma Criterion.
So, first place rank position has special meaning and significance, beyond mere order or distance above another candidate. That’s why reversing a ballot doesn’t necessarily make it say the exact opposite of what it said before reversal—depending on the voting system and what it looks at. While I’m defining more criteria, I should define “0-info Sincerity”:
Approval passes 0-info Sincerity. Score doesn’t pass 0-info Sincerity. Neither does the TUC methods, nor the Improved Condorcet methods defined below. I mention 0-info Sincerity because I’ve heard it proposed. It isn’t one of the criteria by which I criticize TUC, though TUC fails it.
Now, definitions of some Improved Condorcet methods:
Resuming section #4 (methods and their properties and criterion-compliances):
I neglected to include Later-No-Help in some of the method evaluations up to now, and so I’ll repeat its definition and mention which methods meet it:
Now, definitions of a few more criteria:
But rank methods can automatically avoid that nuisance. ICT and Symmetrical ICT automatically avoid it. Because the chicken dilemma is easily automatically avoided by rank method, there is no excuse for a rank method to have the chicken dilemma.
That chicken dilemma nuisance is the nearest thing to a problem that Approval and Score have. Therefore, any method that doesn’t automatically get rid of that nuisance doesn’t significantly improve on Approval and Score. For that reason, no method more elaborate than Approval and Score should be considered for any purpose unless it automatically gets rid of the chicken dilemma. In other words, this is unless it passes CD. TUC fails CD. ICT and Symmetrical ICT pass CD.
ZLNHe could be called a weakening of LNHe. But calling it “weak LNHe” would be misleading, because it is only very slightly weaker than LNHe. It’s easier to refer to a 0-info election than to try to name different kinds of voting-predictive information and stipulate them to be absent. But the information that actually must be absent in order for complying methods to meet that criterion’s requirement is information that is usually or always at least mostly absent even in non-0-info elections.
Therefore, ZLNHe is nearly the same thing as LNHe, and the word “weakening” hardly even applies. I suggest that, with a voting system complying with ZLNHe or Strong ZLNHe (defined below), there’s no need to vote for unacceptable candidates (just as can be said for methods complying with the slightly stronger LNHe). That’s a valuable bottom-end strategy simplification.
Someone could argue that a compliance with Strong 0-info probabilistic Later-No-Help could, and should more properly, be called a failure of a 0-info probabilistic Later-No-Harm. Could? Sure. More properly? No.
Things are different when we’re talking about probabilities in a 0-info election. When, in that 0-info election, the probability of electing an unacceptable isn’t reduced by ranking unacceptables—no improvement is gained by ranking unacceptables—then obviously there is no loss if there’s a cost that prevents us from ranking unacceptables. We didn’t want to anyway.
I name Strong ZLNHe in terms of LNHe because the relevant thing about it is the absence of need to rank unacceptable candidates. Strong ZLNHe simply achieves what ZLNHe achieves, but more so. If there were a little not-so-reliable information about the relative Win-abilities of unacceptables X and Y, then there could begin to be some incentive to rank one over the other. Compliance with Strong ZLNHe instead of just ZLNHe would more strongly outweigh that incentive to rank unacceptables—this could delay its becoming important, as there begins to be a little not-very-reliable win-ability information.
So, instead of a failure of a 0-info probabilistic Later-No-Harm, a compliance with Strong ZLNHe is more relevantly regarded as a compliance with a stronger and more reassuring 0-info probabilistic Later-No-Help.
Resuming the evaluation of ICT and Symmetrical: A Commentary
LNHe is relevant to bottom-end strategy. For example, many rank methods that fail LNHe have bottom-end strategy that calls for ranking unacceptable candidates in reverse order of win-ability. A method that meets LNHe doesn’t have such a strategy-need. LNHe-complying methods don’t need bottom-end strategy.
As I’ve said, I claim that meeting ZLNHe and/or Strong ZLNHe is, for practical purposes, just as good as meeting LNHe. There is no need for bottom-end strategy in a u/a election. It isn’t necessary to rank any unacceptable candidates in a u/a election. That means that Symmetrical ICT’s u/a strategy is as simple as that of Approval and Score.
Resuming the evaluation of ICT and Symmetrical ICT (SICT) by properties and criteria:
I re-emphasize here that TUC’s top-end strategy is a mess, in comparison to ICT and SICT. In fact, TUC’s top-end strategy isn’t even known.
That’s why even a good Condorcet method isn’t suitable for official public elections. For official public elections, I recommend only Approval and Score. I recommend ICT and (especially) SICT for informational polling, to inform and guide voting in an upcoming official public Plurality election. That’s valuable because it could help to bring immediate improvement long before we get a better voting system enacted.