Computing the margin of victory for various voting rules

The margin of victory of an election, defined as the smallest number k such that k voters can change the winner by voting differently, is an important measurement for robustness of the election outcome. It also plays an important role in implementing efficient post-election audits, which has been widely used in the United States to detect errors or fraud caused by malfunctions of electronic voting machines. In this paper, we investigate the computational complexity and (in)approximability of computing the margin of victory for various voting rules, including approval voting, all positional scoring rules (which include Borda, plurality, and veto), plurality with runoff, Bucklin, Copeland, maximin, STV, and ranked pairs. We also prove a dichotomy theorem, which states that for all continuous generalized scoring rules, including all voting rules studied in this paper, either with high probability the margin of victory is Θ(√n), or with high probability the margin of victory is Θ(n), where n is the number of voters. Most of our results are quite positive, suggesting that the margin of victory can be efficiently computed. This sheds some light on designing efficient post-election audits for voting rules beyond the plurality rule.

[1]  Ariel D. Procaccia,et al.  Average-case tractability of manipulation in voting via the fraction of manipulators , 2007, AAMAS '07.

[2]  Toby Walsh,et al.  How Hard Is It to Control an Election by Breaking Ties? , 2013, ECAI.

[3]  Ariel D. Procaccia,et al.  Algorithms for the coalitional manipulation problem , 2008, SODA '08.

[4]  Vincent Conitzer,et al.  Finite Local Consistency Characterizes Generalized Scoring Rules , 2009, IJCAI.

[5]  Michael A. Trick,et al.  How hard is it to control an election? Math , 1992 .

[6]  Moni Naor,et al.  Rank aggregation methods for the Web , 2001, WWW '01.

[7]  P. Stark.  A Sharper discrepancy measure for post-election audits , 2008, 0811.1697.

[8]  John R. Chamberlin.  An investigation into the relative manipulability of four voting systems , 1985 .

[9]  Felix A. Fischer,et al.  The Price of Neutrality for the Ranked Pairs Method , 2012, AAAI.

[10]  Philip B. Stark,et al.  Risk-Limiting Postelection Audits: Conservative $P$-Values From Common Probability Inequalities , 2009, IEEE Transactions on Information Forensics and Security.

[11]  Piotr Faliszewski,et al.  How Hard Is Bribery in Elections? , 2006, J. Artif. Intell. Res..

[12]  Bezalel Peleg.  A note on manipulability of large voting schemes , 1979 .

[13]  Eric Wustrow,et al.  Security analysis of India's electronic voting machines , 2010, CCS '10.

[14]  Ronald L. Rivest,et al.  Computing the Margin of Victory in IRV Elections , 2011, EVT/WOTE.

[15]  Anand D. Sarwate,et al.  Risk-limiting Audits for Nonplurality Elections , 2011 .

[16]  Piotr Faliszewski,et al.  Using complexity to protect elections , 2010, Commun. ACM.

[17]  John J. Bartholdi,et al.  Single transferable vote resists strategic voting , 2015 .

[18]  Shmuel Nitzan,et al.  The vulnerability of point-voting schemes to preference variation and strategic manipulation , 1985 .

[19]  Piotr Faliszewski,et al.  AI's War on Manipulation: Are We Winning? , 2010, AI Mag..

[20]  Sandip Sen,et al.  Voting for movies: the anatomy of a recommender system , 1999, AGENTS '99.

[21]  David C. Mcgarvey.  A THEOREMI ON THE CONSTRUCTION OF VOTING PARADOXES , 1953 .

[22]  Vincent Conitzer,et al.  When are elections with few candidates hard to manipulate? , 2007, J. ACM.

[23]  Arkadii M. Slinko,et al.  How large should a coalition be to manipulate an election? , 2004, Math. Soc. Sci..

[24]  David Cary,et al.  Estimating the Margin of Victory for Instant-Runoff Voting , 2011, EVT/WOTE.

[25]  P. Stark.  Conservative statistical post-election audits , 2008, 0807.4005.

[26]  Ariel D. Procaccia,et al.  Complexity of unweighted coalitional manipulation under some common voting rules , 2009, IJCAI 2009.

[27]  Jörg Rothe,et al.  Challenges to complexity shields that are supposed to protect elections against manipulation and control: a survey , 2013, Annals of Mathematics and Artificial Intelligence.

[28]  Arkadii Slinko,et al.  On Asymptotic Strategy-Proofness of Classical Social Choice Rules , 2002 .

[29]  Geoffrey Pritchard,et al.  Asymptotics of the minimum manipulating coalition size for positional voting rules under impartial culture behaviour , 2009, Math. Soc. Sci..

[30]  Z. Neeman,et al.  The asymptotic strategyproofness of scoring and Condorcet consistent rules , 2002 .

[31]  Geoffrey Pritchard,et al.  On the Average Minimum Size of a Manipulating Coalition , 2006, Soc. Choice Welf..

[32]  Vincent Conitzer,et al.  Generalized scoring rules and the frequency of coalitional manipulability , 2008, EC '08.

[33]  Eithan Ephrati,et al.  The Clarke Tax as a Consensus Mechanism Among Automated Agents , 1991, AAAI.

[34]  Philip B. Stark.  Efficient Post-Election Audits of Multiple Contests: 2009 California Tests , 2009 .

[35]  Geoffrey Pritchard,et al.  Exact results on manipulability of positional voting rules , 2007, Soc. Choice Welf..

[36]  Philip B. Stark,et al.  Super-Simple Simultaneous Single-Ballot Risk-Limiting Audits , 2010, EVT/WOTE.