Repeated balloting is when a voting method is used multiple times with the same voters. It can be done until a certain stopping condition is reached i.e. a majority are in agreement, and other constraints can be added, such as eliminating some candidates in between ballotings or weakening the stopping condition over time.
One variation of repeated balloting is to require a winner to win two consecutive rounds, rather than only one. An anecdote that explains the reasoning for this is as follows: the cardinals who vote for the Pope once used Approval voting with a 2/3rds majority threshold and repeated balloting. In one election, a cardinal managed to convince several other cardinals to give his friend, a candidate, some "pity approvals" so that he wouldn't feel bad at losing overwhelmingly. That friend ended up nearly winning because so many cardinals were about to approve him.
- "RangeVoting.org - Catholic Popes elected via Approval Voting - election stories". www.rangevoting.org. Retrieved 2020-05-14.
One strategem which almost worked was the following. Cardinal B. Cueva and his assistant went around privately meeting a large number of cardinals asking them to vote for Cueva just so the poor fellow could know he had at least one friend, or as a "personal message of honor and respect." Thirty-two "kind polite" pseudo-votes were thus solicited, enough to assure his election, but the effort was discovered just before the ballot. It was laughed off as a joke when it didn't quite work.
- FairVote.org. "Voting method debates go way back: Electing the Pope". FairVote. Retrieved 2020-05-14.
The Church eventually abolished the approval voting element of the process in after the 1621 conclave. One reason apparently was some gaming of the system. In 1559, for example, a cardinal almost won when an ally confidentially met one-on-one with many participants and asked them to cast a "token" approval vote for his friend to avoid him being shut out -- instead the cardinal won votes on 17 of 32 ballots, close to what it took to win.