Michael Ossipoff’s Election Methods Properties Series
Main Categories of Properties and Criteria.
2. List of Pass/Fail Criteria That Will Be Used (Without Definitions)
3. Cast of Characters
List of the Methods Being Compared (Without Definitions).
The Section on Each Method Will Conclude With a Compact List of the Names of its Compliances and Desirable Properties.
In topic 4, properties, including criterion-compliances, will be discussed in order of importance starting with the criteria most relevant to strategy. Beginning with a general description of the method’s strategy, I will then discuss other important properties, such as count-fraud vulnerability and enactability, ending with some of the less important criteria.
5. (As Cross-Referencing) Properties and Criteria, With Complying and Failing Methods
Definitions of Any Discussed Properties Not Previously Defined.
6. Conclusions
Where We Are in Voting System Reform. The End of the Proposal-Selection Period is Overdue. Time for the Outreach and Enactment Period.
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:
Clone-Independence:
A candidate-set, S, is a clone-set if, for every candidate, x, not in S, if a voter votes a member of S over x, then that voter votes all of S over x; and if a voter votes x over a member of S, then that voter votes x over all of S. Removing, from the election and the ballots, one or more (but not all) candidates from S shouldn’t change the matter of whether the winner comes from S.
Reversal-Symmetry:
Reversing all the ballots should change the winner.
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”:
The 0-info Sincerity Criterion:
A 0-info election is an election about which nothing is known other than the candidates and the voting system. Nothing is known about how the voters are likely to vote.
A voter votes sincerely if s/he doesn’t falsify a preference (vote an unfelt preference) or fail to vote a felt preference that the balloting system in use would have allowed him/her to vote in addition to the preferences that s/he actually does vote.
The optimal strategy in a 0-info election should be sincere.
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:
Improved-Condorcet-Top (ICT):
Ballots consist of rankings of the candidates. At any rank position, any number of candidates can be ranked. Voters may use as many rank positions as desired.
(X>Y) means the number of ballots ranking X over Y.
(Y>X) means the number of ballots ranking Y over X
(X=Y)T means the number of ballots ranking both X and Y at top. To rank a candidate at top means to not rank anyone over him/her.
“iff’ means “if and only if”.
“>” means “is greater than”.
X beats Y iff (X>Y) > (Y>X) + (X=Y)T
1. If there is only one unbeaten candidate, then s/he wins.
2. If everyone or no one is unbeaten, then the winner is the candidate top-ranked on the most ballots.
3. If some, but not all, candidates are unbeaten, then the winner is the unbeaten candidate who is top-ranked on the most ballots.
Symmetrical ICT (sometimes abbreviated SICT):
Same as ICT except:
(X=Y)B means the number of ballots ranking both X and Y at bottom.
A ballot ranks a candidate at bottom if it doesn’t rank him/her over anyone.
X beats Y iff (X>Y) + (X=Y)B > (Y>X) + (X=Y)T.
The exception is that two candidates can’t beat each other. If, by the above beat-condition, two candidates beat each other, then only one of them beats the other. The one that beats the other is the one that is ranked over the other by more voters than vice-versa.
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:
Later-No-Help (LNHe):
While you’re making out your ballot, when you’ve voted for some candidates, you shouldn’t need to vote for additional candidates in order to fully help the candidates for whom you’ve already voted. To vote for a candidate is to vote him/her over at least one other candidate. To fully help a candidate is to vote in a way that does as much as possible toward electing that candidate.
Approval and Score meet LNHe.
TUC fails LNHe.
ICT and Symmetrical ICT fail LNHe.
But, though Symmetrical ICT doesn’t strictly pass LNHe, it does pass some criteria that will be defined below, which is effectively just as good as passing LNHe.
Now, definitions of a few more criteria:
Chicken Dilemma Criterion (CD):
Supporting definitions:
- The A voters are the voters who prefer candidate A to everyone else. The B voters are the voters who prefer candidate B to everyone else.
- The other candidates are the candidates other than A and B.
- A particular voter votes sincerely if s/he doesn’t falsify a preference, or fail to vote a felt preference that the balloting system in use would have allowed him/her to vote in addition to the preferences that s/he actually votes.
Premise:
- The A voters and the B voters, combined, add up to more than half of the voters in the election.
- The A voters and the B voters all prefer both A and B to the other candidates.
- The A voters are more numerous than are the B voters.
- Voting is sincere, except that the B voters refuse to vote A over anyone.
- Candidate A would be the unique winner under sincere voting (in other words, if the B voters voted sincerely, as do all the other voters).
Requirement: B doesn’t win.
In the chicken dilemma scenario described in the premise of the Chicken Dilemma Criterion (CD) defined above, if B won, then the B voters would have successfully taken advantage of the A voters’ cooperativeness. The A voters wanted to vote both A and B over the candidates disliked by both the A voters and B voters, to ensure that the winner comes from {A,B}. Thereby they helped {A,B} against worse candidates. But because they helped, B, voters, by not helping, could make B win instead of A. With methods that fail CD, the message is “You help, you lose.”
ICT and Symmetrical ICT meet CD.
Approval and Score don’t meet CD.
TUC doesn’t meet CD.
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.
Definition of Zero-info LNHe (ZLNHe):
Supporting definitions:
- A zero-information (0-info) election is an election about which all that is known is the candidates and the voting system. There’s no information about the voters, their preferences, or how they’re likely to vote.
- To vote a candidate at bottom is to not vote that candidate over anyone. To vote a candidate above bottom is to vote that candidate over someone.
Zero-Info LNHe (ZLNHe):
In a 0-info election, voting above bottom one or more of some certain set of candidates shouldn’t decrease the probability that the winner will come from that set, as compared to voting them all at bottom.
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.
Definition of Strong ZLNHe:
Same as ZLNHe, except that voting one or more members of that set over bottom should increase the probability that the winner will come from that set (instead of just not decreasing that probability).
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.
Compliance with ZLNHe and Strong ZLNHe:
Of course all methods that meet LNHe also meet ZLNHe. Methods that comply with LNHe include Approval voting, Score Voting (also called Range voting), and IRV. Symmetrical ICT meets Strong ZLNHe, though it doesn’t strictly meet LNHe. Ordinary ICT, and traditional unimproved Condorcet methods don’t comply with LNHe, ZLNHe, or Strong ZLNHe.
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:
Important Strategy Criteria:
ICT and SICT meet FBC and CD.
SICT additionally meets ZLNHe and Strong ZLNHe.
General Strategy Description of ICT and SICT:
In a u/a election, top-rank all of the acceptable candidates.
In ICT (as in TUC) rank the unacceptables in reverse order of win-ability.
In SICT, simply don’t rank any unacceptable candidates.
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.
Social Optimizations:
I don’t regard any Condorcet methods, including TUC, ICT and SICT, as having social optimizations. I’ve described the important social optimizations of Approval and Score.
Ease of Implementation, and Count-Fraud-Vulnerability:
ICT and SICT share the computation-intensive count, and therefore the great count-fraud vulnerability, of TUC.
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.
Enact-ability:
ICT and SICT share the enact-ability disadvantage of TUC. There are infinitely many ways to count ranked ballots, and there’s no agreement about which to propose. The rank methods don’t share Approval’s advantages of naturalness, obviousness, minimal change, and simplicity. Therefore any rank method would be much more difficult to enact than Approval or Score.
Embarrassment Criteria:
Like TUC, ICT and SICT fail Participation, Consistency, and IIAC.
ICT and SICT pass Monotonicity and Plurality.
They don’t pass Reversal-Symmetry, but I’ve stated above why a method shouldn’t be expected to pass Reversal-Symmetry. They don’t pass Clone-Independence, but I’ve told why that’s an insignificant strategy disadvantage with respect to Beatpath, in comparison to Beatpath’s (and TUC’s in general) egregious and pitiable strategy mess.
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