Achieving fully proportional representation: Approximability results

We study the complexity of (approximate) winner determination under the Monroe and Chamberlin-Courant multiwinner voting rules, which determine the set of representatives by optimizing the total satisfaction or dissatisfaction of the voters with their representatives. The total (dis)satisfaction is calculated either as the sum of individual (dis)satisfactions in the utilitarian case or as the (dis)satisfaction of the worst off voter in the egalitarian case. We provide good approximation algorithms for the satisfaction-based utilitarian versions of the Monroe and Chamberlin-Courant rules, and inapproximability results for the dissatisfaction-based utilitarian versions of these rules and also for all egalitarian cases. Our algorithms are applicable and particularly appealing when voters submit truncated ballots. We provide experimental evaluation of the algorithms both on real-life preference-aggregation data and on synthetic preference data. These experiments show that our simple and fast algorithms can, in many cases, find near-perfect solutions.

[1]  A. Slinko,et al.  Exploratory Analysis of Similarities Between Social Choice Rules , 2006 .

[2]  Toshihiro Kamishima,et al.  Nantonac collaborative filtering: recommendation based on order responses , 2003, KDD '03.

[3]  Nadja Betzler,et al.  On the Computation of Fully Proportional Representation , 2011, J. Artif. Intell. Res..

[4]  Craig Boutilier,et al.  Social Choice : From Consensus to Personalized Decision Making , 2011 .

[5]  Ariel D. Procaccia,et al.  On the approximability of Dodgson and Young elections , 2009, Artif. Intell..

[6]  Lane A. Hemaspaandra,et al.  Guarantees for the success frequency of an algorithm for finding Dodgson-election winners , 2005, J. Heuristics.

[7]  H. Simon,et al.  Rational choice and the structure of the environment. , 1956, Psychological review.

[8]  S. Berg.  Paradox of voting under an urn model: The effect of homogeneity , 1985 .

[9]  Toby Walsh,et al.  Where are the hard manipulation problems? , 2010, J. Artif. Intell. Res..

[10]  Edith Elkind,et al.  Multiwinner Elections Under Preferences That Are Single-Peaked on a Tree , 2013, IJCAI.

[11]  Craig Boutilier,et al.  Learning Mallows Models with Pairwise Preferences , 2011, ICML.

[12]  Nir Ailon,et al.  Aggregating inconsistent information: Ranking and clustering , 2008 .

[13]  Rolf Niedermeier,et al.  Parameterized computational complexity of Dodgson and Young elections , 2008, Inf. Comput..

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

[15]  Piotr Faliszewski,et al.  Multimode Control Attacks on Elections , 2009, IJCAI.

[16]  Piotr Faliszewski,et al.  Campaigns for lazy voters: truncated ballots , 2012, AAMAS.

[17]  A. Slinko,et al.  EXPLORATORY ANALYSIS OF SIMILARITIES BETWEEN COMMON SOCIAL CHOICE RULES , 2022 .

[18]  Steven J. Brams,et al.  Proportional Representation , 1998 .

[19]  Atri Rudra,et al.  Ordering by weighted number of wins gives a good ranking for weighted tournaments , 2006, SODA '06.

[20]  Burt L. Monroe,et al.  Fully Proportional Representation , 1995, American Political Science Review.

[21]  Henning Fernau,et al.  Facility location problems: A parameterized view , 2011, Discret. Appl. Math..

[22]  A. A. J. Marley,et al.  Behavioral Social Choice - Probabilistic Models, Statistical Inference, and Applications , 2006 .

[23]  Jörg Flum,et al.  Parameterized Complexity Theory , 2006, Texts in Theoretical Computer Science. An EATCS Series.

[24]  Jörg Rothe,et al.  Exact Complexity of the Winner Problem for Young Elections , 2001, Theory of Computing Systems.

[25]  Michael R. Fellows,et al.  Parameterized Complexity , 1998 .

[26]  Evangelos Markakis,et al.  Approximation Algorithms and Mechanism Design for Minimax Approval Voting , 2010, AAAI.

[27]  Geoffrey Pritchard,et al.  Approximability of Dodgson’s rule , 2008, Soc. Choice Welf..

[28]  Piotr Faliszewski,et al.  Properties of multiwinner voting rules , 2014, Social Choice and Welfare.

[29]  John R. Chamberlin,et al.  Representative Deliberations and Representative Decisions: Proportional Representation and the Borda Rule , 1983, American Political Science Review.

[30]  Nicholas Mattei,et al.  An Empirical Study of Voting Rules and Manipulation with Large Datasets , 2012 .

[31]  Piotr Faliszewski,et al.  The complexity of fully proportional representation for single-crossing electorates , 2013, Theor. Comput. Sci..

[32]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[33]  Jörg Flum,et al.  Parameterized Complexity Theory (Texts in Theoretical Computer Science. An EATCS Series) , 2006 .

[34]  Martin Lackner.  Incomplete Preferences in Single-Peaked Electorates , 2014, AAAI.

[35]  Ariel D. Procaccia,et al.  On the complexity of achieving proportional representation , 2008, Soc. Choice Welf..

[36]  C. L. Mallows.  NON-NULL RANKING MODELS. I , 1957 .

[37]  Yann Chevaleyre,et al.  Issues in Multiagent Resource Allocation , 2006, Informatica.

[38]  D. Marc Kilgour,et al.  Approval Balloting for Multi-winner Elections , 2010 .

[39]  Piotr Faliszewski,et al.  Fully Proportional Representation with Approval Ballots: Approximating the MaxCover Problem with Bounded Frequencies in FPT Time , 2015, AAAI.

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

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

[42]  Nicholas Mattei,et al.  A behavioral perspective on social choice , 2012, Annals of Mathematics and Artificial Intelligence.

[43]  Toby Walsh,et al.  The Computational Impact of Partial Votes on Strategic Voting , 2014, ECAI.

[44]  Joachim Gudmundsson,et al.  Computational Aspects of Multi-Winner Approval Voting , 2014, MPREF@AAAI.

[45]  Gerhard J. Woeginger,et al.  Group Activity Selection Problem , 2012, WINE.

[46]  Rolf Niedermeier,et al.  Fixed-parameter algorithms for Kemeny rankings , 2009, Theor. Comput. Sci..

[47]  Ulrich Endriss,et al.  Multiagent resource allocation with sharable items: simple protocols and Nash equilibria , 2010, AAMAS.

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

[49]  Dimitrios M. Thilikos,et al.  Invitation to fixed-parameter algorithms , 2007, Comput. Sci. Rev..

[50]  Nicholas Mattei,et al.  Empirical Evaluation of Voting Rules with Strictly Ordered Preference Data , 2011, ADT.

[51]  Yoav Shoham,et al.  Multiagent Systems - Algorithmic, Game-Theoretic, and Logical Foundations , 2009 .

[52]  Meir Kalech,et al.  Preference elicitation for narrowing the recommended list for groups , 2014, RecSys '14.

[53]  Aranyak Mehta,et al.  Some results on approximating the minimax solution in approval voting , 2007, AAMAS '07.

[54]  Friedrich Pukelsheim,et al.  Proportional Representation: Apportionment Methods and Their Applications , 2013 .

[55]  N. Tideman.  The Single Transferable Vote , 1995 .

[56]  M. L. Fisher,et al.  An analysis of approximations for maximizing submodular set functions—I , 1978, Math. Program..

[57]  Claire Mathieu,et al.  Electronic Colloquium on Computational Complexity, Report No. 144 (2006) How to rank with few errors A PTAS for Weighted Feedback Arc Set on Tournaments , 2006 .

[58]  Rolf Niedermeier,et al.  Fixed-Parameter Algorithms for Kemeny Scores , 2008, AAIM.

[59]  Adrian Vetta,et al.  The Demand Matching Problem , 2002, IPCO.

[60]  Ariel D. Procaccia,et al.  Socially desirable approximations for Dodgson's voting rule , 2010, EC '10.