[EM] Someone thinks that Approval should meet the Mutual Majority Criterion

[Abd ul-Rahman Lomax]

Another issue that was left a bit hanging in discussions on the CES list:

Does top-two Approval fail the Favorite Betrayal Criterion? There are 
really two forms of top-two Approval to be considered, plus a third detail.

1. Top two approval where two candidates advance to the general election.
2. Top two approval where a candidate with a majority can win, 
otherwise two candidates advance.
3. If write-in votes are allowed in the runoff, the primary is 
actually a nomination device, not the actual election. The actual 
election being Approval, the combination must satisfy FBC if Approval 
does, and it does.

(If write-in votes are allowed, in this concept, the runoff must also 
be Approval.)

Arizona had a method up for legislative passage that would have 
allowed municipalities to use a two-stage voting system with an 
Approval primary, top-two advancing to the general election with 
ballot placement, and, apparently, write-ins allowed in the general 
election (as well as in the primary). The primary has no majority 
test, it is top-two plurality, but voters may vote for as many 
candidates as they choose. The runoff is standard vote-for-one.

So, first of all, does this method fail FBC? If so, is the scenario 
plausible for real voters? These are nonpartisan elections.