Dodgson's Rule Approximations and Absurdity

TL;DR

This paper introduces new polynomial-time computable rules related to Dodgson's voting rule, addressing its absurdities and showing they approximate the original rule closely under various voter behavior assumptions.

Contribution

The paper proposes the DC, DR, and D& rules as better realizations and approximations of Dodgson's rule, with polynomial-time computability and convergence properties.

Findings

DC, DR, and D& scores differ from Dodgson scores by a fixed amount

These rules converge to Dodgson's rule under various voter behavior assumptions

D& is especially effective as an approximation

Abstract

With the Dodgson rule, cloning the electorate can change the winner, which Young (1977) considers an "absurdity". Removing this absurdity results in a new rule (Fishburn, 1977) for which we can compute the winner in polynomial time (Rothe et al., 2003), unlike the traditional Dodgson rule. We call this rule DC and introduce two new related rules (DR and D&). Dodgson did not explicitly propose the "Dodgson rule" (Tideman, 1987); we argue that DC and DR are better realizations of the principle behind the Dodgson rule than the traditional Dodgson rule. These rules, especially D&, are also effective approximations to the traditional Dodgson's rule. We show that, unlike the rules we have considered previously, the DC, DR and D& scores differ from the Dodgson score by no more than a fixed amount given a fixed number of alternatives, and thus these new rules converge to Dodgson under any reasonable assumption on voter behaviour, including the Impartial Anonymous Culture assumption.