On the approximability of Dodgson and Young elections

The voting rules proposed by Dodgson and Young are both designed to find the alternative closest to being a Condorcet winner, according to two different notions of proximity; the score of a given alternative is known to be hard to compute under either rule. In this paper, we put forward two algorithms for approximating the Dodgson score: an LP-based randomized rounding algorithm and a deterministic greedy algorithm, both of which yield an O(log m) approximation ratio, where m is the number of alternatives; we observe that this result is asymptotically optimal, and further prove that our greedy algorithm is optimal up to a factor of 2, unless problems in NP have quasi-polynomial time algorithms. Although the greedy algorithm is computationally superior, we argue that the randomized rounding algorithm has an advantage from a social choice point of view. Further, we demonstrate that computing any reasonable approximation of the ranking produced by Dodgson’s rule is NP-hard. This result provides a complexity-theoretic explanation of sharp discrepancies that have been observed in the Social Choice Theory literature when comparing Dodgson elections with simpler voting rules. Finally, we show that the problem of calculating the Young score is NP-hard to approximate by any factor. This leads to an inapproximability result for the Young ranking. ∗Research Academic Computer Technology Institute and Department of Computer Engineering and Informatics, University of Patras, 26500 Rio, Greece, email: [email protected] †Department of Computer Science, Rochester Institute of Technology, 102 Lomb Memorial Drive, Rochester, NY 146235603, email: [email protected] ‡School of Business Administration and Center for the Study of Rationality, The Hebrew University of Jerusalem, Jerusalem 91904, Israel, email: [email protected] §Department of Computer Science, Rochester Institute of Technology, 102 Lomb Memorial Drive Rochester, NY 146235603, email: [email protected] ¶Research Academic Computer Technology Institute and Department of Computer Engineering and Informatics, University of Patras, 26500 Rio, Greece, email: [email protected] ‖Research Academic Computer Technology Institute and Department of Computer Engineering and Informatics University of Patras, 26500 Rio, Greece, email: [email protected] ∗∗Microsoft Israel R&D Center, Herzeliya 46725, Israel, email: [email protected] ††School of Computer Science and Engineering, The Hebrew University of Jerusalem, Jerusalem 91904, Israel, email: [email protected]