FBPPAR

The following system is called FBPPAR voting, for "Favorite-Betrayal-Proof Prefer Accept Reject". It is a version of PAR voting, with an extra "compromise" option in order to pass the favorite betrayal criterion (FBC).

1. Voters can Prefer, Accept, or Reject each candidate. Default is "Accept"; except that for voters who do not explicitly reject any candidates, default is "Reject". Voters can also mark a global option that says: "I believe that voters like me should be the first to compromise."
2. Candidates with a majority of Reject, or with under 25% Prefer, are eliminated, unless that would eliminate all candidates. If a candidate would have been eliminatable considering all the "prefer" votes they got on "compromise" ballots as "rejects", then they are considered "eager to compromise"
3. The winner is the non-eliminated candidate with the highest score. Voters give 1 point to each candidate whom they prefer; and, if all the candidates they gave points to are "eager to compromise", they also give 1 point to each candidate whom they accept.

This is largely a theoretical proposal. In real-world elections, the "compromise" option would probably never be useful.

For instance, consider the voting scenarios which meet the following restrictions:

1. Each candidate either comes from one of no more than 3 "ideological categories", or is "nonviable".
2. No "nonviable" candidate is preferred by more than 25%.
3. Each voter rejects at least one of the 3 "ideological categories" (that is, rejects all candidates in that category).

If the above restrictions hold, then the "compromise" option would never be strategically favored, and so simple PAR voting would meet FBC. It is arguably likely that real-world voting scenarios will meet the above restrictions, except for a negligible fraction of "ideologically atypical" voters. Thus, as a real-world proposal, PAR voting's greater simplicity makes it better than FBPPAR.