The following system is called FBPPAR voting, for "Favorite-Betrayal-Proof Prefer Accept Reject". It is a version of PAR voting, with an extra "stand aside" option in order to pass the favorite betrayal criterion (FBC).
Here's the procedure. Note that the two steps with extra indentation (1.1 and 3.1) only rarely matter, so it's best to understand the system without them first.
- Voters can Prefer, Accept, or Reject each candidate. Default is "Reject" for voters who do not explicitly reject any candidates, and "Accept" otherwise.
- For any candidate they prefer, voters may also check "stand aside". (This is rarely useful; it is only worthwhile if they think that the candidate might become the leader in step 3 and stand in the way of a stronger compromise leader.) "Stand aside" has no effect if it is checked along with any option other than "prefer", or with no option.
- Candidates get 1 point for every ballot that prefers them.
- Candidates with over 25% Prefer, and less than 50% Reject, are called viable. The viable candidate (if any) with the most non-stand-aside preferences is given the label of leader.
- If the leader would not have been viable counting all "prefer/stand aside" votes as "reject", then the label of leader switches to the next lower viable candidate (if any) in non-stand-aside preferences. Repeat this step as many times as necessary.
- Viable candidates get 1 point for every ballot that accepts them and does not prefer the leader.
- Winner is the highest score.
This is largely a theoretical proposal. In real-world elections, the "stand aside" option would probably almost never be useful; certainly not enough to justify the extra complexity.
For instance, consider the voting scenarios which meet the following restrictions:
- Each candidate either comes from one of no more than 3 "ideological categories", or is "nonviable".
- No "nonviable" candidate is preferred by more than 25%.
- Each voter rejects at least one of the 3 "ideological categories" (that is, rejects all candidates in that category), and accepts or prefers all candidates in some other category.
- No honest Condorcet cycles.
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.