The Smoothed Possibility of Social Choice

We develop a framework to leverage the elegant "worst average-case" idea in smoothed complexity analysis to social choice, motivated by modern applications of social choice powered by AI and ML. Using our framework, we characterize the smoothed likelihood of some fundamental paradoxes and impossibility theorems as the number of agents increases. For Condrocet's paradox, we prove that the smoothed likelihood of the paradox either vanishes at an exponential rate, or does not vanish at all. For the folklore impossibility on the non-existence of voting rules that satisfy anonymity and neutrality, we characterize the rate for the impossibility to vanish, to be either polynomially fast or exponentially fast. We also propose a novel easy-to-compute tie-breaking mechanism that optimally preserves anonymity and neutrality for even number of alternatives in natural settings. Our results illustrate the smoothed possibility of social choice---even though the paradox and the impossibility theorem hold in the worst case, they may not be a big concern in practice in certain natural settings.

[1]  Carl D. Meyer,et al.  Matrix Analysis and Applied Linear Algebra , 2000 .

[2]  R. Durrett.  Probability: Theory and Examples , 1993 .

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

[4]  Nicolas de Condorcet.  Essai Sur L'Application de L'Analyse a la Probabilite Des Decisions Rendues a la Pluralite Des Voix , 2009 .

[5]  David Kempe,et al.  Communication, Distortion, and Randomness in Metric Voting , 2019, AAAI.

[6]  Shang-Hua Teng,et al.  Smoothed analysis: an attempt to explain the behavior of algorithms in practice , 2009, CACM.

[7]  Joseph A. Gallian,et al.  Contemporary Abstract Algebra , 2021 .

[8]  David P. Woodruff,et al.  Efficient and Thrifty Voting by Any Means Necessary , 2019, NeurIPS.

[9]  Ilan Nehama,et al.  Approximately classic judgement aggregation , 2010, Annals of Mathematics and Artificial Intelligence.

[10]  A. Sen,et al.  The Possibility of Social Choice , 1999 .

[11]  Luca Trevisan,et al.  Average-Case Complexity , 2006, Found. Trends Theor. Comput. Sci..

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

[13]  W. Hoeffding.  Probability Inequalities for sums of Bounded Random Variables , 1963 .

[14]  Kirk Pruhs,et al.  The Price of Stochastic Anarchy , 2008, SAGT.

[15]  I. Gilboa,et al.  Maxmin Expected Utility with Non-Unique Prior , 1989 .

[16]  Ashish Khetan,et al.  Data-driven Rank Breaking for Efficient Rank Aggregation , 2016, J. Mach. Learn. Res..

[17]  Yuval Filmus,et al.  AND testing and robust judgement aggregation , 2019, STOC.

[18]  Lirong Xia,et al.  Minimizing Time-to-Rank: A Learning and Recommendation Approach , 2019, IJCAI.

[19]  Lirong Xia.  Improving Group Decision-Making by Artificial Intelligence , 2017, IJCAI.

[20]  H. Robbins.  A Remark on Stirling’s Formula , 1955 .

[21]  W. Gehrlein.  Condorcet's paradox and the likelihood of its occurrence: different perspectives on balanced preferences* , 2002 .

[22]  M. Satterthwaite,et al.  Strategy-proofness and single-peakedness , 1976 .

[23]  Edith Hemaspaandra,et al.  Bypassing Combinatorial Protections: Polynomial-Time Algorithms for Single-Peaked Electorates , 2010, AAAI.

[24]  J. Berger.  Statistical Decision Theory and Bayesian Analysis , 1988 .

[25]  A. V. Deemen,et al.  On the empirical relevance of Condorcet’s paradox , 2014 .

[26]  Lawrence G. Sager.  Handbook of Computational Social Choice , 2015 .

[27]  Onur Doğan,et al.  Anonymous and Neutral Social Choice:Existence Results on Resoluteness , 2015 .

[28]  H. Moulin,et al.  Axioms of Cooperative Decision Making. , 1990 .

[29]  Elchanan Mossel,et al.  A quantitative Gibbard-Satterthwaite theorem without neutrality , 2011, STOC '12.

[30]  Martin J. Wainwright,et al.  Estimation from Pairwise Comparisons: Sharp Minimax Bounds with Topology Dependence , 2015, J. Mach. Learn. Res..

[31]  Shang-Hua Teng,et al.  Smoothed analysis of algorithms: why the simplex algorithm usually takes polynomial time , 2001, STOC '01.

[32]  Dorothea Baumeister,et al.  Towards Reality: Smoothed Analysis in Computational Social Choice , 2020, AAMAS.

[33]  H. Moulin.  The strategy of social choice , 1983 .

[34]  Lirong Xia,et al.  Learning and Decision-Making from Rank Data , 2019, Synthesis Lectures on Artificial Intelligence and Machine Learning.

[35]  Peter C. Fishburn,et al.  The probability of the paradox of voting: A computable solution , 1976 .

[36]  Elchanan Mossel,et al.  A quantitative Arrow theorem , 2009, 0903.2574.

[37]  Benjamin Radcliff,et al.  Condorcet Winners and the Paradox of Voting: Probability Calculations for Weak Preference Orders , 1995, American Political Science Review.

[38]  Noam Nisan,et al.  A Quantitative Version of the Gibbard-Satterthwaite Theorem for Three Alternatives , 2011, SIAM J. Comput..

[39]  H. Moulin.  Axioms of Cooperative Decision Making , 1988 .

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

[41]  C. Plott,et al.  The Probability of a Cyclical Majority , 1970 .

[42]  Ariel D. Procaccia,et al.  The Distortion of Cardinal Preferences in Voting , 2006, CIA.

[43]  Felix Brandt,et al.  Analyzing the Practical Relevance of Voting Paradoxes via Ehrhart Theory, Computer Simulations, and Empirical Data , 2016, AAMAS.

[44]  Iyad Rahwan,et al.  A Voting-Based System for Ethical Decision Making , 2017, AAAI.

[45]  David P. Woodruff,et al.  Optimal Communication-Distortion Tradeoff in Voting , 2020, EC.

[46]  Felix Brandt,et al.  Exploring the No-Show Paradox for Condorcet Extensions Using Ehrhart Theory and Computer Simulations , 2019, AAMAS.

[47]  Florian Heiss,et al.  Discrete Choice Methods with Simulation , 2016 .

[48]  Edith Elkind,et al.  Preference Restrictions in Computational Social Choice: Recent Progress , 2016, IJCAI.

[49]  Anindya De,et al.  A size-free CLT for poisson multinomials and its applications , 2015, STOC.

[50]  G. Thompson,et al.  The Theory of Committees and Elections. , 1959 .

[51]  Robert M. May,et al.  Some mathematical remarks on the paradox of voting , 1971 .

[52]  L. A. Goodman,et al.  Social Choice and Individual Values , 1951 .

[53]  Elliot Anshelevich,et al.  Approximating Optimal Social Choice under Metric Preferences , 2015, AAAI.

[54]  Ariel D. Procaccia,et al.  Optimal Aggregation of Uncertain Preferences , 2016, AAAI.

[55]  William V. Gehrlein,et al.  Elections, Voting Rules and Paradoxical Outcomes , 2017 .

[56]  Nathan Keller,et al.  On the probability of a rational outcome for generalized social welfare functions on three alternatives , 2009, J. Comb. Theory, Ser. A.

[57]  Gregory Valiant,et al.  Estimating the unseen: an n/log(n)-sample estimator for entropy and support size, shown optimal via new CLTs , 2011, STOC '11.

[58]  Gil Kalai,et al.  A Fourier-theoretic perspective on the Condorcet paradox and Arrow's theorem , 2002, Adv. Appl. Math..

[59]  Michel Regenwetter,et al.  The impartial culture maximizes the probability of majority cycles , 2003, Soc. Choice Welf..

[60]  William J. Cook,et al.  Sensitivity theorems in integer linear programming , 1986, Math. Program..

[61]  Donald E. Campbell,et al.  The finer structure of resolute, neutral, and anonymous social choice correspondences , 2015 .

[62]  Daniel M. Kane,et al.  The fourier transform of poisson multinomial distributions and its algorithmic applications , 2015, STOC.

[63]  T. Nicolaus Tideman,et al.  Statistical evaluation of voting rules , 2016, Soc. Choice Welf..

[64]  Ariel D. Procaccia,et al.  When do noisy votes reveal the truth? , 2013, EC '13.

[65]  Lirong Xia.  Bayesian Estimators As Voting Rules , 2016, UAI.

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

[67]  Amartya Sen,et al.  A Possibility Theorem on Majority Decisions , 1966 .

[68]  A. Ozkes,et al.  Anonymous, Neutral, and Resolute Social Choice Revisited , 2020, SSRN Electronic Journal.

[69]  Nicholas Mattei,et al.  A Cost-Effective Framework for Preference Elicitation and Aggregation , 2018, UAI.