Llull Voting

Llull Voting is a single winner voting system that attempts to find a compromise between approval voting and the Condorcet criterion. There are two different versions. It is named in honor of Ramon Llull, a discoverer of the Borda Count and Condorcet Method.

Version 1: Llull-Smith Voting
Assume that there are m number of voters and n number of alternatives. Each voter submits a total preference order containing all n alternatives. Note that voters are allowed to express indifference between alternatives in their total preference order. If there exists an alternative that is approved by a majority of all m voters, then the alternative with the greatest number of approvals wins the election. However, if no alternative is approved of by a majority of voters, then the member of the Smith set with the greatest number of approvals is elected the winning alternative.

Version 2: Llull-Schwartz Voting
Assume that there are m number of voters and n number of alternatives. Each voter submits a total preference order containing all n alternatives. Note that voters are allowed to express indifference between alternatives in their total preference order. If there exists an alternative that is approved by a majority of all m voters, then the alternative with the greatest number of approvals wins the election. However, if no alternative is approved of by a majority of voters, then the member of the Schwartz set with the greatest number of approvals is elected the winning alternative.

A Note Regarding Both Versions
If voters are allowed to express indifference, then when voter preferences are aggregated into paired comparison competitions, the expression of indifference between two alternatives by a voter will be understood as an abstention in voting. Thus, the winner of the paired comparison competition will simply be the alternative that is preferred by more voters than the other. For example, suppose there are six voters voting on alternatives x and y. If three are indifferent, two prefer x over y, and one prefers y over x, then x beats y in terms of social preference. However, if the six voters had voted differently, and two were indifferent and two preferred x over y, and two preferred y over x, then there is a tie between x and y in terms of social preference.