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.

However, Symmetrical ICT doesn't actually pass the favorite betrayal criterion (as shown below).

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 Condorcet compliance and a low rate of FBC failure into absolute FBC compliance and a low rate of Condorcet criterion failures (along with absolute Majority Condorcet compliance).

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 Condorcet, 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 chicken dilemma criterion. It fails the Condorcet criterion.

It was intended to pass the favorite betrayal criterion, but doesn't succeed in doing so due to the "(X>Y) > (Y>X)" term in the definition. It is possible that a voter can lower their favorite from the top and thereby make their compromise the only candidate who isn't "beaten."

Favorite betrayal example

0.389: B>A=C
0.290: A>C>B
0.241: C>B>A
0.079: A=C>B ⇒ C>A=B

In this case, the voters on the bottom row can elect a better candidate, C, by reversing their preference for A.

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. Mike considered current United States voters to have near-dichotomous preferences: that each voter has a much wider gap between acceptable candidates and unacceptable ones than between candidates of the same category.