Fixing Participation Failure in “Approval vs B2R”

[cfrank]

DISCLAIMER: This is conjectural and needs adjustments, but at least participation compliance can be improved. I have had LLMs run thousands of examples of this iteration, and have found no examples of participation failure. Still, it needs to be audited closely.

To understand this method, you should also know about Bottom Two Runoff (B2R), which you can learn about here for example:

https://www.votingtheory.org/forum/topic/564/bottom-n-and-bottom-2-runoffs-are-equivalent

The method is a development of this concept:

https://www.votingtheory.org/forum/topic/563/direct-independent-condorcet-validation/20

Consider the following method, which I’m just calling Approval vs. B2R, similar to Approval-seeded Llull as per @Jack-Waugh ---it's a bit technical, but there's a reason for that, so bear with me:

DEFINITION

(1) Voters submit rank ballots with approval cutoffs.
(2) Candidates are sorted by approval rates, with rank-based head-to-head breaking ties if possible. If approval ties cannot be broken by head-to-heads (when they are also tied), introduce authoritative participant(s) to break the ties.
(3) According to this sorting order, run B2R and identify the B2R winner, with head-to-head ties broken by the sorting order.
(4) Compute the maximum approval rating. If the B2R winner attains this rate, and NO OTHER (non Condorcet loser?) candidates do, then elect the B2R winner (who is also the approval winner).
(5) Otherwise, among top-approved candidates who are different from the B2R winner, select the one with the tightest margin against the B2R winner (possibly who is also not the Condorcet loser?), with ties determined by the sorting. They will be the B2R winner's adversary.
(6) Run a secondary, independent head-to-head election between the B2R winner and their adversary, with the following caveats:

--> Voters are not tied down in any way to their original preference between the B2R winner and the adversary, and can freely vote for either in the independent head-to-head. Also, voters who did not participate in the first round are fully allowed to participate in the final round. By default, voters' original ballots will be used to determine the preference, but voters may opt in to swap their rating either 0 or 1 times, whichever amount is necessary to indicate an advantage that they wish to disclose.
--> However, based on these swaps, we can count the net number of swaps that are advantageous to the adversary over the B2R winner compared with the original ballots. If this number is positive, the election proceeds as you would expect, with ties broken by the sort order. However, if the number is not positive, if the original head-to-head was in favor of the B2R survivor, and if a material difference would be incurred, then the adversary will be conferred an automatic +1 head-to-head advantage, and will also automatically win ties.

I conjecture that these caveats about the runoff—including setting aside the B2R winner, increasing the Approval contender’s advantage by +1, and giving them ties—together restore the participation criterion under fully sincere ballots, at the expense of the full Condorcet criterion. However, when the margins are not close, the Condorcet criterion is still satisfied if ballots are sincere.

Here’s what else is fascinating about this: we can directly control the tradeoff between participation and Condorcet—in particular, when applicable, we can choose to increase the B2R adversary's margin over the B2R winner by +1 with probability P. Then the method satisfies participation under sincere ballots with probability at least P, and simultaneously satisfies the Condorcet criterion under sincere ballots with probability at least 1-P.

IN SUMMARY:

This method I believe is participation compliant, which it is supposed to be by intentional design. This still needs to be proved. But as a consequence of this intention, it was also designed to fail Condorcet compliant in a controlled way. As per my comprehension, it will fail Condorcet compliance under these exact conditions:

(1) The Condorcet winner C exists (and will therefore be the B2R survivor);
(2) The approval winner Y is different from C;
(3) The head-to-head rank-based margin of C over Y is either 0 or +1; and
(4) The +1 boost to Y is applied, with ties going to Y.

More generally, it will fail the Smith criterion under these exact conditions:

(1) The approval winner Y is not in the Smith set;
(2) The head-to-head rank-based margin of the B2R survivor over Y is either 0 or +1; and
(3) The +1 boost to Y is applied, with ties going to Y.

In all other cases, it satisfies the Smith criterion. Thus in a sense, this method tries to be as close to Smith compliant as possible while enforcing participation compliance as non-negotiable. It unconditionally satisfies a weakened version of the Condorcet criterion: If the Condorcet winner exists and its weakest margin of victory is at least 2, then the Condorcet winner is elected. It also unconditionally satisfies a more significantly weakened version of the Smith criterion: If every member of the Smith set has a weakest margin of victory against non-Smith set members of at least 2, then the election winner will belong to the Smith set.

Not that it's necessary to frame the properties in terms of Smith or Condorcet criterion-sounding conditions, but possibly those weak conditions can be strengthened. It is what it is.

Future work will be to prove participation and to refine the mechanisms for guaranteeing participation.