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Acyclic Domains of Linear Orders: A Survey

In: The Mathematics of Preference, Choice and Order

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  • Bernard Monjardet

    (CES, Université Paris I Panthéon Sorbonne)

Abstract

A = {1,2…i, j,k…n} is a finite set of n elements that I will generally call alternatives (but which could also be called issues, decisions, outcomes, candidates, objects, etc.). The elements of A will be also denoted by letters like x,y, z etc. A subset of cardinality p of A will be called a p-set. A 2 (respectively, A 3) denotes the set of all ordered pairs (x,y) (respectively, ordered triples (x,y, z) written for convenience as xyz) of A. When the elements of A are denoted by the n first integers, P 2(n) denotes the set of the n(n- 1)/2 ordered pairs (i

Suggested Citation

  • Bernard Monjardet, 2009. "Acyclic Domains of Linear Orders: A Survey," Studies in Choice and Welfare, in: Steven J. Brams & William V. Gehrlein & Fred S. Roberts (ed.), The Mathematics of Preference, Choice and Order, pages 139-160, Springer.
    • Handle: RePEc:spr:stcchp:978-3-540-79128-7_8
    DOI: 10.1007/978-3-540-79128-7_8
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    References listed on IDEAS

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    1. B. Monjardet, 1978. "An Axiomatic Theory of Tournament Aggregation," Mathematics of Operations Research, INFORMS, vol. 3(4), pages 334-351, November.
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    Cited by:

    1. Chatterji, Shurojit & Zeng, Huaxia, 2018. "On random social choice functions with the tops-only property," Games and Economic Behavior, Elsevier, vol. 109(C), pages 413-435.
    2. Olivier Hudry & Bernard Monjardet, 2010. "Consensus theories: An oriented survey," Documents de travail du Centre d'Economie de la Sorbonne 10057, Université Panthéon-Sorbonne (Paris 1), Centre d'Economie de la Sorbonne.
    3. Liu, Peng, 2020. "Random assignments on sequentially dichotomous domains," Games and Economic Behavior, Elsevier, vol. 121(C), pages 565-584.
    4. Bredereck, Robert & Chen, Jiehua & Woeginger, Gerhard J., 2016. "Are there any nicely structured preference profiles nearby?," Mathematical Social Sciences, Elsevier, vol. 79(C), pages 61-73.
    5. Chatterji, Shurojit & Roy, Souvik & Sadhukhan, Soumyarup & Sen, Arunava & Zeng, Huaxia, 2022. "Probabilistic fixed ballot rules and hybrid domains," Journal of Mathematical Economics, Elsevier, vol. 100(C).
    6. Liu, Peng & Zeng, Huaxia, 2019. "Random assignments on preference domains with a tier structure," Journal of Mathematical Economics, Elsevier, vol. 84(C), pages 176-194.
    7. Bernard Monjardet, 2008. ""Mathématique Sociale" and Mathematics. A case study: Condorcet's effect and medians," Post-Print halshs-00309825, HAL.
    8. Roy, Souvik & Sadhukhan, Soumyarup, 2021. "A unified characterization of the randomized strategy-proof rules," Journal of Economic Theory, Elsevier, vol. 197(C).
    9. Puppe, Clemens, 2018. "The single-peaked domain revisited: A simple global characterization," Journal of Economic Theory, Elsevier, vol. 176(C), pages 55-80.
    10. Chatterji, Shurojit & Zeng, Huaxia, 2023. "A taxonomy of non-dictatorial unidimensional domains," Games and Economic Behavior, Elsevier, vol. 137(C), pages 228-269.
    11. Clemens Puppe & Arkadii Slinko, 2019. "Condorcet domains, median graphs and the single-crossing property," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 67(1), pages 285-318, February.
    12. Li, Guanhao & Puppe, Clemens & Slinko, Arkadii, 2021. "Towards a classification of maximal peak-pit Condorcet domains," Mathematical Social Sciences, Elsevier, vol. 113(C), pages 191-202.
    13. Alexander Karpov & Arkadii Slinko, 2023. "Constructing large peak-pit Condorcet domains," Theory and Decision, Springer, vol. 94(1), pages 97-120, January.
    14. Gilbert Laffond & Jean Lainé, 2014. "Triple-consistent social choice and the majority rule," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 22(2), pages 784-799, July.
    15. Shurojit Chatterji & Huaxia Zeng, 2022. "A Taxonomy of Non-dictatorial Unidimensional Domains," Papers 2201.00496, arXiv.org, revised Oct 2022.
    16. Alexander Karpov, 2019. "On the Number of Group-Separable Preference Profiles," Group Decision and Negotiation, Springer, vol. 28(3), pages 501-517, June.
    17. Slinko, Arkadii, 2019. "Condorcet domains satisfying Arrow’s single-peakedness," Journal of Mathematical Economics, Elsevier, vol. 84(C), pages 166-175.
    18. Bernard Monjardet, 2006. "Condorcet domains and distributive lattices," Cahiers de la Maison des Sciences Economiques b06072, Université Panthéon-Sorbonne (Paris 1).
    19. Li, Guanhao & Puppe, Clemens & Slinko, Arkadii, 2020. "Towards a classification of maximal peak-pit Condorcet domains," Working Paper Series in Economics 144, Karlsruhe Institute of Technology (KIT), Department of Economics and Management.
    20. Chatterji, Shurojit & Zeng, Huaxia, 2019. "Random mechanism design on multidimensional domains," Journal of Economic Theory, Elsevier, vol. 182(C), pages 25-105.
    21. Li, Guanhao, 2023. "A classification of peak-pit maximal Condorcet domains," Mathematical Social Sciences, Elsevier, vol. 125(C), pages 42-57.
    22. Ping Zhan, 2019. "A simple construction of complete single-peaked domains by recursive tiling," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 90(3), pages 477-488, December.
    23. Saari, Donald G., 2014. "Unifying voting theory from Nakamura’s to Greenberg’s theorems," Mathematical Social Sciences, Elsevier, vol. 69(C), pages 1-11.
    24. Shurojit Chatterji & Souvik Roy & Soumyarup Sadhukhan & Arunava Sen & Huaxia Zeng, 2021. "Probabilistic Fixed Ballot Rules and Hybrid Domains," Papers 2105.10677, arXiv.org, revised Jan 2022.

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    More about this item

    Keywords

    Distributive Lattice; Linear Order; Social Choice; Coxeter Group; Maximal Chain;
    All these keywords.

    JEL classification:

    • D71 - Microeconomics - - Analysis of Collective Decision-Making - - - Social Choice; Clubs; Committees; Associations

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