Guarantees for the Success Frequency of an Algorithm for Finding Dodgson-Election Winners

In the year 1876 the mathematician Charles Dodgson, who wrote fiction under the now more famous name of Lewis Carroll, devised a beautiful voting system that has long fascinated political scientists. However, determining the winner of a Dodgson election is known to be complete for the p 2 level of the polynomial hierarchy. This implies that unless P = NP no polynomial-time solution to this problem exists, and unless the polynomial hierarchy collapses to NP the problem is not even in NP. Nonetheless, we prove that when the number of voters is much greater than the number of candidates— although the number of voters may still be polynomial in the number of candidates—a simple greedy algorithm very frequently finds the Dodgson winners in such a way that it “knows” that it has found them, and furthermore the algorithm never incorrectly declares a nonwinner to be a winner

Improving Dodgson scoring techniques

The Dodgson score problem is part of the Dodgson election scheme invented by Charles Dodgson and presented in his manuscript. One of the system\u27s strengths (and motivations for its study) is that it satisfies the Condorcet criterion (which states that any candidate who defeats all other candidates in pairwise elections will be declared the winner). It is unfortunate, though, that in a given election no Condorcet winner may exist. Dodgson\u27s election system chooses the winner closest to being the Condorcet winner in the sense that it requires the shortest sequence of edits (swapping of adjacent candidates in the voters\u27 preference rankings) to the votes in order to make it one. The length of this sequence is known as the Dodgson score. The problem of finding the Dodgson score of a candidate is computationally intractable. Thus an approximation is necessary. This paper puts forth MCDodgsonScore, a polynomialtime computable (ln(m) + 1)-approximation of that problem. This approximation is optimal, meaning that an approximation with an asymptotically tighter error bound does not exist. MCDodgsonScore builds on a technique introduced by Homan and Hemaspaandra in 2006. A nice feature of MCDodgsonScore is that, when treated as its own voting rule, it will also satisfy the Condorcet criterion