Smith//Minimax

The Smith//Minimax method is a version of Minimax where the winner of the election is limited to the Smith set.

To determine the winner of an election, use the following algorithm:

• Determine the smallest set of candidates in a particular election who, when paired off in pairwise elections, can beat all other candidates outside the set. This is known as the Smith set This set might consist of only one candidate, the Condorcet winner. However, when the electorate is conflicted (as in Condorcet's paradox), the set usually has at least one cycle of candidates for whom A beats B, B beats C, and C beats A.
• If there is more than one candidate remaining, determine the winner using one of the two following methods (see the defeat strength article):
  • "winning votes": elects the candidate whose greatest pairwise loss to another candidate is the least, when the strength of a pairwise loss is measured as the number of voters who voted for the winning side.
  • "margins": elects the candidate whose greatest pairwise loss is measured as the number of votes for the winning side minus the number of votes for the losing side.

Example using winning votes:

19: C>A>B>D
17: D>C>A>B
17: B>C>A>D
16: D>B>C>A
16: A>B>C>D
15: D>A>B>C

A beats B (67 to 33) beats C (64 to 36) beats A (69 to 31), but all three beat D (all 52 to 48), so the three are in the Smith set. Smith//Minimax first eliminates D, and then C wins for having the fewest winning votes against them in any of their defeats (C was beaten only by someone with 64 winning votes, whereas the others are beaten by 67 and 69 wv respectively). Regular Minimax would elect D for having the weakest defeat here (only 52 wv against them), resulting in a failure of the mutual majority criterion.

Notes

Regular Minimax is one of the most well-known Condorcet methods that fails the mutual majority criterion. Smith//Minimax, being Smith-efficient, passes.

This page is a stub - please add to it.