Symmetrical ICT

Symmetrical ICT, short for Symmetrical Improved Condorcet, Top is a voting method designed by Michael Ossipoff. It is based on Kevin Venzke's concept of "Improved Condorcet", which is a modification of pairwise comparison logic that enables methods to pass the favorite betrayal criterion at the cost of sometimes failing the Condorcet criterion.

Definition

(Note: This is not actually a Condorcet method. It is a Condorcet method only when using a modified definition of what a Condorcet method is.)

(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 X and Y in 1st place.

(X=Y)B means the number of ballots ranking X and Y at bottom, i.e. not ranking either X or Y above anyone else.

Let the partial beat relation b(X, Y) be true if (X>Y) + (X=Y)B > (Y>X) + (X=Y)T. Then X beats Y if:

• p(X,Y) and not p(Y, X), or
• p(X,Y) and p(Y, X) and (X>Y) > (Y>X).

The winner is chosen as follows:

1. If only one candidate is unbeaten, then s/he wins.
2. If everyone or no one is unbeaten, then the winner is the candidate ranked in first place on the most ballots.
3. If some, but not all, candidates are unbeaten, then the winner is the unbeaten candidate ranked in first place on the most ballots.

Improved Condorcet

Condorcet methods usually have a low but nonzero rate of favorite betrayal failures. Improved Condorcet is a modification of pairwise comparisons in an otherwise Condorcet-compliant method to turn absolute Conrocet compliance and a low rate of FBC failure into absolute FBC compliance and a low rate of Condorcet criterion failures.

Mike Ossipoff argued that improved Condorcet allows a voter who wants one of X and Y to win, and who ranks X first, to change a ranking of X>Y into X=Y without undue risk that this will change the winner from Y to someone lower ranked by that voter; and thus that it's better to satisfy the IC version of Condorcet than the actual Condorcet criterion.

History

The tied-at-the-top rule and Improved Condorcet ideas were devised by Kevin Venzke in an effort to create a Minmax variant that passes the FBC. Then, later, Chris Benham proposed completion by top-count, to avoid the chicken dilemma and thus achieve defection-resistance. Mike Ossipoff shortened the name of this method to "Improved Condordet, Top".

Mike later proposed that the ICT tied-at-the-top rule also be applied to the bottom end, to almost achieve later-no-help compliance, which then led to Symmetrical ICT.

Criterion compliances

Symmetrical ICT passes the favorite betrayal criterion and the chicken dilemma criterion. It fails the Condorcet criterion.

Notes

Omitting the (X=Y)B term would turn this method into ordinary ICT. In Mike's opinion, ordinary ICT has the most important properties of Symmetrical ICT, but Symmetrical ICT adds a somewhat less important improvement, consisting of simpler bottom-end strategy.

In an election with dichotomous preferences, the best ICT strategy is Approval strategy: equal-rank all approved candidates first and all unapproved candidates last.

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A few improved properties of ICT and Symmetrical ICT:

I already mentioned that ICT and Symmetrical ICT meet FBC. That's the main, most important, difference between Symmetrical ICT and traditional, unimproved Condorcet.

But, additionally, ICT and Symmetrical ICT, automatically avoid the chicken dilemma. They meet CD, the Chicken Dilemma Criterion.

Comparison of strategy in a u/a election:

When there are unacceptable candidates who could win, that can greatly simplify voting strategy. I call such a situation a u/a election (standing for unacceptable/acceptable). It's a situaiton in which all that matters is that the winner be an acceptable rather than an unacceptable. ...and that's incomparably more imporant than the matter of which acceptable or which unacceptable wins. You could say that the candidates can be divided into two sets, such that the merit within the sets is negligible compared to the merit difference between the sets.

That's a u/a election. Voting strategy can be much simpler in a u/a election. For instance, in Approval, just approve all of the acceptables, and none of the unacceptables.

Top-end u/a strategy for ICT, Symmetrical ICT, and traditional unimproved Condorcet:

Traditional unimproved Condorcet? It isn't merely more complicated than Approval. It's unknown.

ICT and Symmetrical ICT:

Top-rank all of the acceptables and none of the acceptables.

In traditional unimproved Condorcet, you'd still have that same need to top-rank all of the unacceptbles, but there's a risk and penalty for doing so: Anyone you top-rank could pairbeat another top-ranked candidate and thereby give the win to your last choice (ad described above). Of course you'd have to try to guess which acceptable(s) has the best chance to win, and which would sufficiently unlikely to win that the main effect of top-ranking them would be to risk spoiling another top-ranked candidate's win. You'd be guessing. You wouldn't know what to do. Not even in a u/a election.

Bottom-End strategy for ICT, Symmetrical ICT and traditional unimproved Condorcet:

ICT and traditional unimproved Condorcet:

Rank the unacceptable candidates in reverse order of winnability.

Of course you don't have to, but it's optimal. Because it's only bottom-end strategy, it isn't an important problem. Merely a nuisance.

But it's avoided by Symmetrical ICT:

Bottom-Ennd strategy for Symmetrical ICT:

Don't rank any unacceptables.

In other words, u/a strategy in Symmetrical ICT is as simple as that of Approval.

Let me add here that I suggest that all of our official public elections are u/a. So what does it tell us, when the best that a rank method can do is no different from Approval? It suggests that there's no need or reason to bother with rank methods in official public elections. Well, there is one way in which ICT and Symmetrical ICT improve on Approval, even in a u/a election. They automatically avoid the chicken dilemma, as said above. The chicken dilemma is easily dealt with in Approval and Score, and, for a number of other reasons too, isn't really a problem with Approval and Score. Only a nuisance. But it's nice that ICT and Symmetrical ICT automatically get rid of that nuisance. You don't improve on Approval without doing that. Don't even consider a rank-method that doesn't automatically avoid the chicken dilemma.

The count-computation-intensiveness of rank methods, and the consequent count-fraud vulnerabiity, make rank methods unsuitable for official public elections. But I propose ICT and Symmetrical ICT for informational polling, to inform and guide strategy in an upcoming official public election by Plurality--until we can replace Plurality with Approval or Score (Range).

I don't claim that Symmetrical ICT actually strictly meets Later-No-Help (LNHe). Approval strictly meets LNHe.

LNHe says that, when you've voted for some candidates on your ballot (you vote for a candidate if you vote him/her over someone), then you don't need to vote for additional candidates on that ballot in order to help as much as you can the candidates for whom you've already voted.

In Symmetrical ICT, you knew that the unacceptable X was going to be beaten by someone other than unacceptable Y, but that Y could be inbeaten and win, then you'd have reason to rank X, to help beat Y. But there's no such information available. One thing that Symmetrical ICT guarantees is that, by leaving X and Y unranked, you're doing everything that you can do to make one beat the other. Not ranking any unacceptable is good u/a strategy in Symmetrical ICT.

Michael Ossipoff