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Dodgson's method is anelectoral system based on a proposal by mathematician Charles Dodgson, better known asLewis Carroll. The method searches for amajority-preferred winner; if no such winner is found, the method proceeds by finding the candidate who could be transformed into a Condorcet winner with the smallest number ofballot edits possible, where a ballot edit switches two neighboring candidates on a voter's ballot.[1]
This classic Condorcet-consistent system, though computationally complex, was defined in Dodgson'sA Method of Taking Votes on More an Two Issues pamphlet. It appeared in March 1876, printed by the Clarendon Press, Oxford and headed “not yet published”.[2][3]
In Dodgson's method, each voter submits an ordered list of all candidates according to their own preference (from best to worst). The winner is defined to be the candidate for whom we need to perform the minimum number of pairwise swaps in each ballot (added over all candidates) before they become aCondorcet winner.
In short, we must find the voting profile with minimumKendall tau distance from the input, such that it has a Condorcet winner; then, the Condorcet winner is declared the victor. Computing the winner or even the Dodgson score of a candidate (the number of swaps needed to make that candidate a winner) is anNP-hard problem[4] by reduction fromExact Cover by 3-Sets (X3C).[5]
Given an integerk and an election, it isNP-complete to determine whether a candidate can become a Condorcet winner with fewer thank swaps.
