New characterizations of strategy-proofness under single-peakedness

We provide novel representations of strategy-proof voting rules applicable when voters have uni-dimensional single-peaked preferences. In particular, we introduce a ‘grading curve’ representation which is particularly useful when introducing variable electorates. Our analysis recovers, links and uni...

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Published in	Mathematical programming Vol. 203; no. 1-2; pp. 207 - 238
Main Authors	Jennings, Andrew B., Laraki, Rida, Puppe, Clemens, Varloot, Estelle M.
Format	Journal Article
Language	English
Published	Berlin/Heidelberg Springer Berlin Heidelberg 01.01.2024 Springer Springer Nature B.V Springer Verlag
Subjects	Analysis Calculus of Variations and Optimal Control; Optimization Combinatorics Computer Science Full Length Paper Mathematical and Computational Physics Mathematical Methods in Physics Mathematics Mathematics and Statistics Mathematics of Computing Numerical Analysis Participation Quantitative Finance Representations Theoretical Voting Participation Proportionality 91B12 Voting Theory Single-peaked preferences Voting Linear/uniform median Consistency 91B14 Social Choice Strategy-proofness Mathematics Subject Classification 91B12 Voting Theory • 91B14 Social Choice
Online Access	Get full text
ISSN	0025-5610 1436-4646
DOI	10.1007/s10107-023-02010-x

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Abstract	We provide novel representations of strategy-proof voting rules applicable when voters have uni-dimensional single-peaked preferences. In particular, we introduce a ‘grading curve’ representation which is particularly useful when introducing variable electorates. Our analysis recovers, links and unifies existing results in the literature, and provides new characterizations when strategy-proofness is combined with other desirable properties such as ordinality, participation, consistency, and proportionality. Finally, the new representations are used to compute the strategy-proof methods that maximize the ex-ante social welfare for the L 2 -norm and a uniform prior. The resulting strategy-proof welfare maximizer is the linear median (or ‘uniform median’), that we also characterize as the unique proportional strategy-proof voting rule.
AbstractList	We provide novel representations of strategy-proof voting rules applicable when voters have uni-dimensional single-peaked preferences. In particular, we introduce a 'grading curve' representation which is particularly useful when introducing variable electorates. Our analysis recovers, links and unifies existing results in the literature, and provides new characterizations when strategy-proofness is combined with other desirable properties such as ordinality, participation, consistency, and proportionality. Finally, the new representations are used to compute the strategy-proof methods that maximize the ex-ante social welfare for the L 2-norm and a uniform prior. The resulting strategy-proof welfare maximizer is the linear median (or 'uniform median'), that we also characterize as the unique proportional strategy-proof voting rule. We provide novel representations of strategy-proof voting rules applicable when voters have uni-dimensional single-peaked preferences. In particular, we introduce a ‘grading curve’ representation which is particularly useful when introducing variable electorates. Our analysis recovers, links and unifies existing results in the literature, and provides new characterizations when strategy-proofness is combined with other desirable properties such as ordinality, participation, consistency, and proportionality. Finally, the new representations are used to compute the strategy-proof methods that maximize the ex-ante social welfare for the $$L_2$$ L 2 -norm and a uniform prior. The resulting strategy-proof welfare maximizer is the linear median (or ‘uniform median’), that we also characterize as the unique proportional strategy-proof voting rule. We provide novel representations of strategy-proof voting rules applicable when voters have uni-dimensional single-peaked preferences. In particular, we introduce a ‘grading curve’ representation which is particularly useful when introducing variable electorates. Our analysis recovers, links and unifies existing results in the literature, and provides new characterizations when strategy-proofness is combined with other desirable properties such as ordinality, participation, consistency, and proportionality. Finally, the new representations are used to compute the strategy-proof methods that maximize the ex-ante social welfare for the L 2 -norm and a uniform prior. The resulting strategy-proof welfare maximizer is the linear median (or ‘uniform median’), that we also characterize as the unique proportional strategy-proof voting rule. We provide novel representations of strategy-proof voting rules applicable when voters have uni-dimensional single-peaked preferences. In particular, we introduce a ‘grading curve’ representation which is particularly useful when introducing variable electorates. Our analysis recovers, links and unifies existing results in the literature, and provides new characterizations when strategy-proofness is combined with other desirable properties such as ordinality, participation, consistency, and proportionality. Finally, the new representations are used to compute the strategy-proof methods that maximize the ex-ante social welfare for the L2-norm and a uniform prior. The resulting strategy-proof welfare maximizer is the linear median (or ‘uniform median’), that we also characterize as the unique proportional strategy-proof voting rule. We provide novel representations of strategy-proof voting rules applicable when voters have uni-dimensional single-peaked preferences. In particular, we introduce a 'grading curve' representation which is particularly useful when introducing variable electorates. Our analysis recovers, links and unifies existing results in the literature, and provides new characterizations when strategy-proofness is combined with other desirable properties such as ordinality, participation, consistency, and proportionality. Finally, the new representations are used to compute the strategy-proof methods that maximize the ex-ante social welfare for the [Formula omitted]-norm and a uniform prior. The resulting strategy-proof welfare maximizer is the linear median (or 'uniform median'), that we also characterize as the unique proportional strategy-proof voting rule.
Audience	Academic
Author	Laraki, Rida Varloot, Estelle M. Jennings, Andrew B. Puppe, Clemens
Author_xml	– sequence: 1 givenname: Andrew B. surname: Jennings fullname: Jennings, Andrew B. organization: Public Integrity Foundation – sequence: 2 givenname: Rida surname: Laraki fullname: Laraki, Rida organization: Department of Computer Science, University of Liverpool, CNRS, LAMSADE, Université Paris Dauphine-PSL, UM6P – sequence: 3 givenname: Clemens orcidid: 0000-0002-2110-1017 surname: Puppe fullname: Puppe, Clemens organization: Karlsruhe Institute of Technology, Higher School of Economics – sequence: 4 givenname: Estelle M. surname: Varloot fullname: Varloot, Estelle M. email: estelle.varloot@liverpool.ac.uk organization: Department of Computer Science, University of Liverpool
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Keywords	Participation Proportionality 91B12 Voting Theory Single-peaked preferences Voting Linear/uniform median Consistency 91B14 Social Choice Strategy-proofness Mathematics Subject Classification 91B12 Voting Theory • 91B14 Social Choice
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References	BorderKCJordanJSStraightforward elections, unanimity and phantom votersRev. Econ. Stud.198350115317069598810.2307/2296962 NehringKPuppeCThe structure of strategy-proof social choice - part I: general characterization and possibility results on median spacesJ. Econ. Theory20071351269305288838010.1016/j.jet.2006.04.008 Caragiannis, I., Procaccia, A.D., Shah, N.: Truthful univariate estimators. In: Balcan, M., Weinberger, K.Q. (eds.) Proceedings of the 33nd International Conference on Machine Learning, ICML 2016, New York City, NY, USA, June 19-24, 2016. JMLR Workshop and Conference Proceedings, vol. 48, pp. 127–135. JMLR.org, (2016). http://proceedings.mlr.press/v48/caragiannis16.html NehringKPuppeCEfficient and strategy-proof voting rules: a characterizationGames Econom. Behav.2007591132153231766010.1016/j.geb.2006.05.002 Weymark, J.: A unified approach to strategy-proofness for single-peaked preferences. SERIEs: J. Spanish Econ. Assoc. (2), 529–550 (2011) EhlersLPetersHStorckenTStrategy-proof probabilistic decision schemes for one-dimensional single-peaked preferencesJ. Econ. Theory20021052408434192186710.1006/jeth.2001.2829 BalinskiMLarakiRMajority Judgment Measuring, Ranking, and Electing2010CambridgeMIT Press MoulinHOn strategy-proofness and single peakednessPublic Choice198035443745510.1007/BF00128122 MoulinHCondorcet’s principle implies the no show paradoxJ. Econ. Theory1988451536494906810.1016/0022-0531(88)90253-0 BarberàSGulFStacchettiEGeneralized median voter schemes and committeesJ. Econ. Theory1993612262289125233310.1006/jeth.1993.1069 MüllerMPuppeCStrategy-proofness implies minimal participation under single-peakednessEcon. Theory Bull.2023111131151458507110.1007/s40505-023-00247-5 Balinski, M., Laraki, R.: Election by majority judgement: Experimental evidence. In Situ and Laboratory Experiments on Electoral Law Reform: French Presidential Elections. Springer (2010) SmithJHAggregation of preferences with variable electorateEconometrica19734161027104144122710.2307/1914033 Young, H.P.: Social choice scoring functions. In: SIAM Journal on Applied Mathematics Vol. 28 vol. 4, pp. 824–838. Society for Industrial and Applied Mathematics (1975) Satterthwaite, M.A.: Strategy-proofness and arrow’s conditions: existence and correspondence theorems for voting procedures and social welfare functions. J. Econ. Theory, 187–217 (1975) Jennings, A.B.: Monotonicity and manipulability of ordinal and cardinal social choice functions. In: PhD of A. Jennings (2010) Freeman, R., Pennock, D.M., Peters, D., Vaughan, J.W.: Truthful aggregation of budget proposals. In: Proceedings of the 2019 ACM Conference on Economics and Computation, EC 2019, Phoenix, AZ, USA, June 24–28, 2019, pp. 751–752 (2019). https://doi.org/10.1145/3328526.3329557 GibbardAManipulation of voting schemes: a general resultEconometrica197341458760144140710.2307/1914083 A Gibbard (2010_CR3) 1973; 41 JH Smith (2010_CR10) 1973; 41 S Barberà (2010_CR6) 1993; 61 M Müller (2010_CR18) 2023; 11 L Ehlers (2010_CR13) 2002; 105 KC Border (2010_CR5) 1983; 50 2010_CR2 2010_CR1 2010_CR8 2010_CR12 K Nehring (2010_CR15) 2007; 59 H Moulin (2010_CR17) 1988; 45 2010_CR9 2010_CR11 H Moulin (2010_CR4) 1980; 35 2010_CR14 M Balinski (2010_CR16) 2010 K Nehring (2010_CR7) 2007; 135
References_xml	– reference: SmithJHAggregation of preferences with variable electorateEconometrica19734161027104144122710.2307/1914033 – reference: BorderKCJordanJSStraightforward elections, unanimity and phantom votersRev. Econ. Stud.198350115317069598810.2307/2296962 – reference: BarberàSGulFStacchettiEGeneralized median voter schemes and committeesJ. Econ. Theory1993612262289125233310.1006/jeth.1993.1069 – reference: NehringKPuppeCThe structure of strategy-proof social choice - part I: general characterization and possibility results on median spacesJ. Econ. Theory20071351269305288838010.1016/j.jet.2006.04.008 – reference: Freeman, R., Pennock, D.M., Peters, D., Vaughan, J.W.: Truthful aggregation of budget proposals. In: Proceedings of the 2019 ACM Conference on Economics and Computation, EC 2019, Phoenix, AZ, USA, June 24–28, 2019, pp. 751–752 (2019). https://doi.org/10.1145/3328526.3329557 – reference: Weymark, J.: A unified approach to strategy-proofness for single-peaked preferences. SERIEs: J. Spanish Econ. Assoc. (2), 529–550 (2011) – reference: GibbardAManipulation of voting schemes: a general resultEconometrica197341458760144140710.2307/1914083 – reference: Young, H.P.: Social choice scoring functions. In: SIAM Journal on Applied Mathematics Vol. 28 vol. 4, pp. 824–838. Society for Industrial and Applied Mathematics (1975) – reference: MoulinHOn strategy-proofness and single peakednessPublic Choice198035443745510.1007/BF00128122 – reference: Satterthwaite, M.A.: Strategy-proofness and arrow’s conditions: existence and correspondence theorems for voting procedures and social welfare functions. J. Econ. Theory, 187–217 (1975) – reference: MoulinHCondorcet’s principle implies the no show paradoxJ. Econ. Theory1988451536494906810.1016/0022-0531(88)90253-0 – reference: EhlersLPetersHStorckenTStrategy-proof probabilistic decision schemes for one-dimensional single-peaked preferencesJ. Econ. Theory20021052408434192186710.1006/jeth.2001.2829 – reference: NehringKPuppeCEfficient and strategy-proof voting rules: a characterizationGames Econom. Behav.2007591132153231766010.1016/j.geb.2006.05.002 – reference: BalinskiMLarakiRMajority Judgment Measuring, Ranking, and Electing2010CambridgeMIT Press – reference: Caragiannis, I., Procaccia, A.D., Shah, N.: Truthful univariate estimators. In: Balcan, M., Weinberger, K.Q. (eds.) Proceedings of the 33nd International Conference on Machine Learning, ICML 2016, New York City, NY, USA, June 19-24, 2016. JMLR Workshop and Conference Proceedings, vol. 48, pp. 127–135. JMLR.org, (2016). http://proceedings.mlr.press/v48/caragiannis16.html – reference: MüllerMPuppeCStrategy-proofness implies minimal participation under single-peakednessEcon. Theory Bull.2023111131151458507110.1007/s40505-023-00247-5 – reference: Jennings, A.B.: Monotonicity and manipulability of ordinal and cardinal social choice functions. In: PhD of A. Jennings (2010) – reference: Balinski, M., Laraki, R.: Election by majority judgement: Experimental evidence. In Situ and Laboratory Experiments on Electoral Law Reform: French Presidential Elections. Springer (2010) – ident: 2010_CR11 doi: 10.1137/0128067 – ident: 2010_CR14 – volume-title: Majority Judgment Measuring, Ranking, and Electing year: 2010 ident: 2010_CR16 – volume: 59 start-page: 132 issue: 1 year: 2007 ident: 2010_CR15 publication-title: Games Econom. Behav. doi: 10.1016/j.geb.2006.05.002 – volume: 50 start-page: 153 issue: 1 year: 1983 ident: 2010_CR5 publication-title: Rev. Econ. Stud. doi: 10.2307/2296962 – volume: 41 start-page: 1027 issue: 6 year: 1973 ident: 2010_CR10 publication-title: Econometrica doi: 10.2307/1914033 – volume: 41 start-page: 587 issue: 4 year: 1973 ident: 2010_CR3 publication-title: Econometrica doi: 10.2307/1914083 – ident: 2010_CR9 – volume: 135 start-page: 269 issue: 1 year: 2007 ident: 2010_CR7 publication-title: J. Econ. Theory doi: 10.1016/j.jet.2006.04.008 – ident: 2010_CR12 doi: 10.1007/s13209-011-0064-5 – ident: 2010_CR2 doi: 10.1016/0022-0531(75)90050-2 – volume: 61 start-page: 262 issue: 2 year: 1993 ident: 2010_CR6 publication-title: J. Econ. Theory doi: 10.1006/jeth.1993.1069 – volume: 11 start-page: 131 issue: 1 year: 2023 ident: 2010_CR18 publication-title: Econ. Theory Bull. doi: 10.1007/s40505-023-00247-5 – volume: 45 start-page: 53 issue: 1 year: 1988 ident: 2010_CR17 publication-title: J. Econ. Theory doi: 10.1016/0022-0531(88)90253-0 – ident: 2010_CR1 doi: 10.1007/978-1-4419-7539-3_2 – volume: 105 start-page: 408 issue: 2 year: 2002 ident: 2010_CR13 publication-title: J. Econ. Theory doi: 10.1006/jeth.2001.2829 – volume: 35 start-page: 437 issue: 4 year: 1980 ident: 2010_CR4 publication-title: Public Choice doi: 10.1007/BF00128122 – ident: 2010_CR8 doi: 10.1145/3328526.3329557
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SubjectTerms	Analysis Calculus of Variations and Optimal Control; Optimization Combinatorics Computer Science Full Length Paper Mathematical and Computational Physics Mathematical Methods in Physics Mathematics Mathematics and Statistics Mathematics of Computing Numerical Analysis Participation Quantitative Finance Representations Theoretical Voting
Title	New characterizations of strategy-proofness under single-peakedness
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