Sample Complexity for Winner Prediction in Elections

Predicting the winner of an election is a favorite problem both for news media pundits and computational social choice theorists. Since it is often infeasible to elicit the preferences of all the voters in a typical prediction scenario, a common algorithm used for winner prediction is to run the election on a small sample of randomly chosen votes and output the winner as the prediction. We analyze the performance of this algorithm for many common voting rules. More formally, we introduce the (e, δ)-winner determination problem, where given an election on n voters and m candidates in which the margin of victory is at least en votes, the goal is to determine the winner with probability at least 1-δ. The margin of victory of an election is the smallest number of votes that need to be modified in order to change the election winner. We show interesting lower and upper bounds on the number of samples needed to solve the (e, δ)-winner determination problem for many common voting rules, including scoring rules, approval, maximin, Copeland, Bucklin, plurality with runoff, and single transferable vote. Moreover, the lower and upper bounds match for many common voting rules in a wide range of practically appealing scenarios.

[1]  Piotr Faliszewski,et al.  Probabilistic Possible Winner Determination , 2010, AAAI.

[2]  Craig Boutilier,et al.  Robust Approximation and Incremental Elicitation in Voting Protocols , 2011, IJCAI.

[3]  Moshe Tennenholtz,et al.  On Coalitions and Stable Winners in Plurality , 2012, WINE.

[4]  Noga Alon,et al.  A combinatorial characterization of the testable graph properties: it's all about regularity , 2006, STOC '06.

[5]  Jianhua Lin,et al.  Divergence measures based on the Shannon entropy , 1991, IEEE Trans. Inf. Theory.

[6]  Craig Boutilier,et al.  Multi-Winner Social Choice with Incomplete Preferences , 2013, IJCAI.

[7]  Ran Canetti,et al.  Lower Bounds for Sampling Algorithms for Estimating the Average , 1995, Inf. Process. Lett..

[8]  M. Trick,et al.  Voting schemes for which it can be difficult to tell who won the election , 1989 .

[9]  Lirong Xia,et al.  Computing the margin of victory for various voting rules , 2012, EC '12.

[10]  Dana Ron,et al.  Property Testing , 2000 .

[11]  Francesco Bonchi,et al.  Voting in social networks , 2009, CIKM.

[12]  Y. Narahari,et al.  Scalable Preference Aggregation in Social Networks , 2013, HCOMP.

[13]  Craig Boutilier,et al.  Vote Elicitation with Probabilistic Preference Models: Empirical Estimation and Cost Tradeoffs , 2011, ADT.

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

[15]  Craig Boutilier,et al.  Efficient Vote Elicitation under Candidate Uncertainty , 2013, IJCAI.

[16]  Vincent Conitzer,et al.  Vote elicitation: complexity and strategy-proofness , 2002, AAAI/IAAI.

[17]  Sarit Kraus,et al.  Evaluation of election outcomes under uncertainty , 2008, AAMAS.

[18]  László Lovász,et al.  Graph limits and parameter testing , 2006, STOC '06.

[19]  Ziv Bar-Yossef,et al.  Sampling lower bounds via information theory , 2003, STOC '03.

[20]  Arnab Bhattacharyya.  A pr 2 01 6 Sample Complexity for Winner Prediction in Elections , 2022 .

[21]  Vincent Conitzer.  Eliciting single-peaked preferences using comparison queries , 2007, AAMAS '07.

[22]  Edith Hemaspaandra,et al.  The complexity of Kemeny elections , 2005, Theor. Comput. Sci..

[23]  Edith Elkind,et al.  On elections with robust winners , 2013, AAMAS.

[24]  Eric Horvitz,et al.  Social Choice Theory and Recommender Systems: Analysis of the Axiomatic Foundations of Collaborative Filtering , 2000, AAAI/IAAI.

[25]  Huaiyu Zhu.  On Information and Sufficiency , 1997 .

[26]  Jörg Rothe,et al.  Exact analysis of Dodgson elections: Lewis Carroll's 1876 voting system is complete for parallel access to NP , 1997, JACM.

[27]  Johan Bollen,et al.  Smartocracy: Social Networks for Collective Decision Making , 2007, 2007 40th Annual Hawaii International Conference on System Sciences (HICSS'07).

[28]  Ravi Kumar,et al.  Sampling algorithms: lower bounds and applications , 2001, STOC '01.

[29]  Bo Waggoner,et al.  Lp Testing and Learning of Discrete Distributions , 2014, ITCS.

[30]  Sarit Kraus,et al.  On the evaluation of election outcomes under uncertainty , 2008, Artif. Intell..

[31]  Craig Boutilier,et al.  Robust Winners and Winner Determination Policies under Candidate Uncertainty , 2014, AAAI.

[32]  J. Kiefer,et al.  Asymptotic Minimax Character of the Sample Distribution Function and of the Classical Multinomial Estimator , 1956 .