Tied at the top

From electowiki

The tied at the top rule for pairwise count methods says that when two candidates X and Y are ranked equal at the top of a ballot, both candidates are counted as receiving a vote for them against the other. Also, when pairwise wins (and not just "votes for") are determined, if adding the number of votes tying X and Y "tied at the top" to one side or the other can determine which candidate wins pairwise against the other, then this contest is interpreted to be a pairwise tie.

In some cases, use of the tied at the top rule can allow a method to satisfy the Favorite Betrayal Criterion (FBC). Kevin Venzke devised the rule to be used with Condorcet//Approval (yielding Improved Condorcet Approval) and Minmax (winning votes).


One possible issue with the tied at the top rule in its formulation mentioned above is that it forces voters who don't want to create pairwise ties between the candidates they equally rank 1st to do so. For example, a Democrat voter wishing to indicate no preference between two Democrat candidates while helping both of them pairwise beat a Republican candidate may simply wish to allow other voters to choose which of the Democrats should win, rather than preventing one from pairwise beating the other. Thus, one possible improvement to the rule is to allow voters to explicitly indicate whether or not they want the candidates they rank 1st to be in a pairwise tie or not. One way to do this is to, in addition to allowing voters to rank candidates 1st, 2nd, 3rd, etc, allow voters to indicate an "above-1st"/"0th" rank which is counted as superior to all ranks, including the 1st rank. Another way would be to allow voters to check a box indicating they want all top-ranked candidates to pairwise tie if possible.

The tied at the top rule could potentially be used for all ranks, though this should probably be explicitly indicated by the voter.

Example for Smith//Approval:

25 A>B| >C

40 B>C| >A

35 C>A| >B

There is an A>B>C>A cycle, with the approvals being A 60, B 65, and C 75. C is elected for having the most approvals in the Smith set. The easiest way for A-top voters to get someone they prefer to C under regular Smith//Approval is for at least 11 of them to swap A and B, making B a Condorcet winner. With a modification of the tied at the top rule, they could instead vote A=B, and this would yield a matchup of 49 votes for A, 40 for B, and 11 tied votes. Since the 11 tied votes is greater than the margin of 9 votes, the matchup could simply be dropped, meaning that B becomes a CW because B pairwise beats C. N

Note that C-top voters can vote C=A to make A have no pairwise defeats toothat , so A and B would both be in the Smith set, wistill th B winning with 65 approvals to A's 60.