On Single-peaked Domains and Min-max Rules

Achuthankutty, Gopakumar and Roy, Souvik (2017): On Single-peaked Domains and Min-max Rules.

Abstract

We consider social choice problems where different agents can have different sets of admissible single-peaked preferences. We show every unanimous and strategy-proof social choice function on such domains satisfies Pareto property and tops-onlyness. Further, we characterize all domains on which (i) every unanimous and strategy-proof social choice function is a min-max rule, and (ii) every min-max rule is strategy-proof. As an application of our result, we obtain a characterization of the unanimous and strategy-proof social choice functions on maximal single-peaked domains (Moulin (1980), Weymark (2011)), minimally rich single-peaked domains (Peters et al. (2014)), maximal regular single-crossing domains (Saporiti (2009), Saporiti (2014)), and distance based single-peaked domains.

• Item Type: MPRA Paper
• Language: English
• Keywords: Strategy-proofness, single-peaked preferences, min-max rules, min-max domains, top-connectedness, Pareto property, tops-onlyness.
• Subjects: D - Microeconomics > D7 - Analysis of Collective Decision-Making > D71 - Social Choice ; Clubs ; Committees ; Associations; D - Microeconomics > D8 - Information, Knowledge, and Uncertainty > D82 - Asymmetric and Private Information ; Mechanism Design

References

ARRIBILLAGA, R. P. AND J.MASSÓ (2016): “Comparing generalized median voter schemes according to their manipulability,” Theoretical Economics, 11, 547–586.

ARROW, K.J. (1969): “Tullock and an Existence Theorem,” Public Choice, 6, 105–111.

ASWAL,N., S.CHATTERJI , AND A.SEN (2003): “Dictatorial domains,” Economic Theory, 22, 45–62.

BARBERÀ, S., F.GUL, AND E.STACCHETTI (1993): “Generalized Median Voter Schemes and Committees,” Journal of Economic Theory, 61, 262 – 289.

BARBERÀ, S. AND M.O.JACKSON (2004): “Choosing How to Choose: Self-Stable Majority Rules and Constitutions,” The Quarterly Journal of Economics, 119, 1011–1048.

BARBERÀ, S., J.MASSÒ, AND A.NEME (1999): “Maximal domains of preferences preserving strategy-proofness for generalized median voter schemes,” Social Choice and Welfare, 16, 321–336.

BARBERÀ, S. AND B.PELEG (1990): “Strategy-proof voting schemes with continuous preferences,” Social Choice and Welfare, 7, 31–38.

BLACK, D. (1948): “On the Rationale of Group Decision-making,” Journal of Political Economy, 56, 23–34.

CHATTERJI , S. AND A. SEN (2011): “Tops-only domains,” Economic Theory, 46, 255–282.

CHATTERJI, S., A. SEN , AND H. ZENG (2014): “Random dictatorship domains,” Games and Economic Behavior, 86, 212 – 236.

CHATTERJI, S. AND H.ZENG (2015): “On Random Social Choice Functions with the Tops-only Property,” Working paper.

CHING, S. (1997): “Strategy-proofness and “median voters”,” International Journal of Game Theory,26, 473–490.

DEMANGE, G. (1982): “Single-peaked orders on a tree,” Mathematical Social Sciences, 3, 389 – 396.

——— (1994): “Intermediate preferences and stable coalition structures,” Journal of Mathematical Economics, 23, 45 – 58.

ELKIND,E., P.FALISZEWSKI, AND P. SKOWRON (2014): “A Characterization of the Single-Peaked Single-Crossing Domain.” in AAAI, vol. 14, 654–660.

EPPLE, D. AND G.J.PLATT (1998): “Equilibrium and Local Redistribution in an Urban Economy when Households Differ in both Preferences and Incomes,” Journal of Urban Economics, 43, 23–51.

EPPLE, D., R.ROMANO, AND H. SIEG (2006): “Admission, Tuition, and Financial Aid Policies in the Market for Higher Education,” Econometrica, 74, 885–928.

EPPLE, D. AND T. ROMER (1991): “Mobility and Redistribution,” Journal of Political Economy, 99, 828–858.

EPPLE, D., T.ROMER , AND H. SIEG (2001): “Interjurisdictional Sorting and Majority Rule: An Empirical Analysis,” Econometrica, 69, 1437–1465.

GIBBARD, A. (1973): “Manipulation of Voting Schemes: A General Result,” Econometrica, 41, 587–601.

HAMADA, K. (1973): “A simple majority rule on the distribution of income,” Journal of Economic Theory, 6, 243–264.

HETTICH, W. (1979): “A Theory of Partial Tax Reform,” The Canadian Journal of Economics / Revue canadienne d’Economique, 12, 692–712.

HOTELLING, H. (1929): “Stability in Competition,” The Economic Journal, 41–57.

KUNG, F.-C. (2006): “An Algorithm for Stable and Equitable Coalition Structures with Public Goods,” Journal of Public Economic Theory, 8, 345–355.

MELTZER, A.H. AND S.F.RICHARD (1981): “A Rational Theory of the Size of Government,” Journal of Political Economy, 89, 914–927.

MOULIN, H. (1980): “On strategy-proofness and single peakedness,” Public Choice, 35, 437–455.

NEHRING, K. AND C.PUPPE (2007a): “The structure of strategy-proof social choice Part I: General characterization and possibility results on median spaces,” Journal of Economic Theory, 135, 269 – 305.

——— (2007b): “Efficient and strategy-proof voting rules: A characterization,” Games and Economic Behavior, 59, 132 – 153.

PETERS, H., S. ROY , A. SEN , AND T. STORCKEN (2014): “Probabilistic strategy-proof rules over single-peaked domains,” Journal of Mathematical Economics, 52, 123 – 127.

PUPPE , C. (2016): “The single-peaked domain revisited: A simple global characterization,” Working Paper Series in Economics, Karlsruher Institut für Technologie (KIT), 97.

RABINOWITZ , G. (1978): “On the Nature of Political Issues: Insights from a Spatial Analysis,” American Journal of Political Science, 22, 793–817.

RABINOWITZ, G. AND S. E. MACDONALD (1989): “A Directional Theory of Issue Voting,” American Political Science Review, 83, 93–121.

RABINOWITZ, G., J.W. PROTHRO, AND W. JACOBY (1982): “Salience as a Factor in the Impact of Issues on Candidate Evaluation,” The Journal of Politics, 44, 41–63.

ROBERTS , K. W. (1977): “Voting over income tax schedules,” Journal of Public Economics, 8, 329–340.

ROMER, T. AND H. ROSENTHAL (1979): “Bureaucrats Versus Voters: On the Political Economy of Resource Allocation by Direct Democracy,” The Quarterly Journal of Economics, 93, 563–587.

ROY, S. AND T. STORCKEN (2016): “Unanimity, Pareto optimality and strategy-proofness on connected domains,” Working Paper.

SAPORITI, A. (2009): “Strategy-proofness and single-crossing,” Theoretical Economics, 4, 127–163.

——— (2014): “Securely implementable social choice rules with partially honest agents,” Journal of Economic Theory, 154, 216 – 228.

SATTERTHWAITE , M.A. (1975): “Strategy-proofness and Arrow’s conditions: Existence and correspondence theorems for voting procedures and social welfare functions,” Journal of Economic Theory, 10, 187 – 217.

SCHUMMER, J. AND R.V. VOHRA (2002): “Strategy-proof Location on a Network,” Journal of Economic Theory, 104, 405 – 428.