Best-is-worst paradox

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In social choice theory, the best-is-worst paradox occurs when a voting rule declares the same candidate to be both the best and worst possible winner. The worst candidate can be identified by reversing each voter's ballot (to rank candidates from worst-to-best), then applying the voting rule to the reversed ballots find a new "anti-winner".^[1][2]

Rules that never exhibit a best-is-worst paradox are said to satisfy the reversal criterion, which states that if every voter's opinions on each candidate are perfectly reversed (i.e. they rank candidates from worst to best), the outcome of the election should be reversed as well, meaning the first- and last- place finishers should switch places.^[2] In other words, the results of the election should not depend arbitrarily on whether voters rank candidates from best to worst (and then select the best candidate), or whether we ask them to rank candidates from worst to best (and then select the least-bad candidate).

Methods that satisfy reversal symmetry include the Borda count, ranked pairs, Kemeny–Young, and Schulze. Most rated voting systems, including approval and score voting, satisfy the criterion as well. Best-is-worst paradoxes can occur in ranked-choice runoff voting (RCV) and minimax. A well-known example is the 2022 Alaska special election, where candidate Mary Peltola was both the winner and anti-winner.