1932

Abstract

Kenneth J. Arrow was one of the most important intellectuals of the twentieth century, and his “impossibility theorem” is arguably the starting point of modern, axiomatic social choice theory. In this review, we begin with a brief discussion of Arrow's theorem and subsequent work that extended the result. We then discuss its implications for voting and constitutional systems, including a number of seminal results—both positive and negative—that characterize what such systems can accomplish and why. We then depart from this narrow interpretation of the result to consider more varied institutional design questions such as apportionment and geographical districting. Following this, we address the theorem's implications for measurement of concepts of fundamental interest to political science such as justice and inequality. Finally, we address current work applying social choice concepts and the axiomatic method to data analysis more generally.

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2019-05-11
2024-03-28
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Literature Cited

  1. Abdulkadiroğlu A, Angrist JD, Narita Y, Pathak PA 2017. Research design meets market design: using centralized assignment for impact evaluation. Econometrica 85:1373–432
    [Google Scholar]
  2. Abdulkadiroğlu A, Sönmez T 2003. School choice: a mechanism design approach. Am. Econ. Rev. 93:729–47
    [Google Scholar]
  3. Ackerman M, Ben-David S 2009. Measures of clustering quality: a working set of axioms for clustering. Advances in Neural Information Processing Systems12128 Red Hook, NY: Curran Assoc.
  4. Altman A, Tennenholtz M 2005. Ranking systems: the PageRank axioms. Proceedings of the 6th ACM Conference on Electronic Commerce18 New York: ACM
  5. Arrow KJ 1951. Social Choice and Individual Values New York: Wiley
  6. Arrow KJ 1963. Social Choice and Individual Values New York: Wiley. 2nd ed.
  7. Atkinson AB 1970. On the measurement of inequality. J. Econ. Theory 2:244–63
    [Google Scholar]
  8. Austen-Smith D, Banks JS 1999. Positive Political Theory I: Collective Preference Ann Arbor: Univ. Mich. Press
  9. Austen-Smith D, Banks JS 2004. Positive Political Theory II: Strategy and Structure Ann Arbor: Univ. Mich. Press
  10. Balinski ML, Young HP 2001. Fair Representation: Meeting the Ideal of One Man, One Vote Brookings Inst. Press. 2nd ed.
  11. Ballester MA, Haeringer G 2011. A characterization of the single-peaked domain. Soc. Choice Welfare 36:305–22
    [Google Scholar]
  12. Bandyopadhyay S, Murty MN, Narayanam R 2017. A generic axiomatic characterization of centrality measures in social network. arXiv:1703.07580
  13. Barberà S, Beviá C 2002. Self-selection consistent functions. J. Econ. Theory 105:263–77
    [Google Scholar]
  14. Barberà S, Jackson MO 2004. Choosing how to choose: self-stable majority rules and constitutions. Q. J. Econ. 119:1011–48
    [Google Scholar]
  15. Bervoets S, Merlin V 2012. Gerrymander-proof representative democracies. Int. J. Game Theory 41:473–88
    [Google Scholar]
  16. Bhattacharya M 2018. Constitutionally consistent voting rules over single-peaked domains Typescript, GREQAM, Aix-Marseille School Econ., Aix-Marseille Univ., Marseille, France
  17. Black D 1948. On the rationale of group decision-making. J. Political Econ. 56:23–34
    [Google Scholar]
  18. Blackorby C, Bossert W 2008. Interpersonal comparisons of well-being. The Oxford Handbook of Political Economy BR Weingast, DA Wittman40824 Oxford, UK: Oxford Univ. Press
  19. Blackorby C, Bossert W, Donaldson D 2002. Utilitarianism and the theory of justice. Handbook of Social Choice and Welfare 1 KJ Arrow, AK Sen, K Suzumura54396 New York: Elsevier
  20. Blin JM, Satterthwaite M 1976. Strategy-proofness and single-peakedness. Public Choice 26:51–58
    [Google Scholar]
  21. Boldi P, Luongo A, Vigna S 2017. Rank monotonicity in centrality measures. Network Sci. 5:529–50
    [Google Scholar]
  22. Boldi P, Vigna S 2014. Axioms for centrality. Internet Math. 10:222–62
    [Google Scholar]
  23. Borgatti SP, Everett MG 2006. A graph-theoretic perspective on centrality. Soc. Netw. 28:466–84
    [Google Scholar]
  24. Brams SJ, Fishburn PC 1978. Approval voting. Am. Political Sci. Rev. 72:831–47
    [Google Scholar]
  25. Brams SJ, Fishburn PC 1984. Some logical defects of the single transferable vote. Choosing an Electoral System: Issues and Alternatives A Lijphart, B Grofman New York: Praeger
  26. Brams SJ, Fishburn PC 2002. Voting procedures. Handbook of Social Choice and Welfare KJ Arrow, AK Sen, K Suzumura 1173236 Amsterdam: Elsevier
  27. Brown DJ 1975. Aggregation of preferences. Q. J. Econ. 89:456–69
    [Google Scholar]
  28. Butts CT 2000. An axiomatic approach to network complexity. J. Math. Sociol. 24:273–301
    [Google Scholar]
  29. Casella A 2005. Storable votes. Games Econ. Behav. 51:391–419
    [Google Scholar]
  30. Chambers C 2008. Consistent representative democracy. Games Econ. Behav. 62:348–63
    [Google Scholar]
  31. Chambers CP 2009. An axiomatic theory of political representation. J. Econ. Theory 1:375–89
    [Google Scholar]
  32. Cohen S, Zohar A 2015. An axiomatic approach to link prediction. Proceedings of the Twenty-Ninth AAAI Conference on Artificial Intelligence5864 Palo Alto: AAAI Press
  33. Dalton H 1920. The measurement of the inequality of incomes. Econ. J. 30:348–61
    [Google Scholar]
  34. Dasgupta P, Hammond P, Maskin E 1979. The implementation of social choice rules: some general results on incentive compatibility. Rev. Econ. Stud. 46:185–216
    [Google Scholar]
  35. Dasgupta P, Maskin E 2008. On the robustness of majority rule. J. Eur. Econ. Assoc. 6:949–73
    [Google Scholar]
  36. d'Aspremont C 1985. Axioms for social welfare orderings. Social Goals and Social Organization: Essays in Memory of Elisha Pazner L Hurwicz, D Schmeidler, H Sonnenschein1975 Cambridge, UK: Cambridge Univ. Press
  37. d'Aspremont C, Gevers L 1977. Equity and the informational basis of collective choice. Rev. Econ. Stud. 44:199–209
    [Google Scholar]
  38. d'Aspremont C, Gevers L 2002. Social welfare functionals and interpersonal comparability. Handbook of Social Choice and Welfare 1 KJ Arrow, AK Sen, K Suzumura459541 New York: Elsevier
  39. Doron G, Kronick R 1977. Single transferrable vote: an example of a perverse social choice function. Am. J. Political Sci.21:30311
  40. Duggan J 2013. Uncovered sets. Soc. Choice Welfare 41:489–535
    [Google Scholar]
  41. Duggan J 2017. May's theorem in one dimension. J. Theor. Politics 291321
  42. Duggan J, Schwartz T 2000. Strategic manipulability without resoluteness or shared beliefs: Gibbard-Satterthwaite generalized. Soc. Choice Welfare 17:85–93
    [Google Scholar]
  43. Eliaz K 2004. Social aggregators. Soc. Choice Welfare 22:317–30
    [Google Scholar]
  44. Emerson P 2013. The original Borda count and partial voting. Soc. Choice Welfare 40:353–58
    [Google Scholar]
  45. Ferejohn JA, Grether DM 1974. On a class of rational social decision procedures. J. Econ. Theory 8:471–82
    [Google Scholar]
  46. Fey M 2008. Choosing from a large tournament. Soc. Choice Welfare 31:301–9
    [Google Scholar]
  47. Fishburn PC 1978. Symmetric and consistent aggregation with dichotomous voting. Aggregation and Revelation of Preferences20118 Amsterdam: North Holland
  48. Foster JE 1985. Inequality measurement. Fair Allocation 33 Proceedings of Symposia in Applied Mathematics HP Young3168 Providence, RI: Am. Math. Soc.
  49. Gaertner W 2001. Domain Conditions in Social Choice Theory Cambridge, UK: Cambridge Univ. Press
  50. Gailmard S, Patty JW, Penn EM 2008. Arrow's theorem on single-peaked domains. The Political Economy of Democracy E Aragonés, C Beviá, H Llavador, N Schofield33542 Bilbao, Spain: Fundación BBVA
  51. Gale D, Shapley LS 1962. College admissions and the stability of marriage. Am. Math. Monthly 69:9–15
    [Google Scholar]
  52. Gärdenfors P 1973. Positionalist voting functions. Theory Decis. 4:1–24
    [Google Scholar]
  53. Gibbard A 1973. Manipulation of voting schemes: a general result. Econometrica 414587601
  54. Gibbard A 1974. A Pareto-consistent libertarian claim. J. Econ. Theory 7:388–410
    [Google Scholar]
  55. Gibbard AF 2014. Intransitive social indifference and the Arrow dilemma. Rev. Econ. Des. 18:3–10
    [Google Scholar]
  56. Gini C 1912. Variabilità e mutabilità Contributo allo Studio delle Distribuzioni e delle Relazioni Statistiche Bologna, Italy: C. Cuppini
  57. Gini C 1921. Measurement of inequality of incomes. Econ. J. 31:124–26
    [Google Scholar]
  58. Greenberg J 1979. Consistent majority rules over compact sets of alternatives. Econometrica 47:627–36
    [Google Scholar]
  59. Grossi D, Pigozzi G 2014. Judgment aggregation: a primer. Synth. Lect. Artif. Intel. Mach. Learn. 8:1–151
    [Google Scholar]
  60. Hammond PJ 1979. Equity in two person situations: some consequences. Econometrica112735
  61. Holzman R 1990. An axiomatic approach to location on networks. Math. Oper. Res. 15:553–63
    [Google Scholar]
  62. Hortala-Vallve R 2012. Qualitative voting. J. Theor. Politics 24:526–54
    [Google Scholar]
  63. Houy N 2004. A note on the impossibility of a set of constitutions stable at different levels Work. Pap. v04039, Maison Sci. Econ., Univ. Panthéon-Sorbonne, Paris
  64. Jin R, Lee VE, Hong H 2011. Axiomatic ranking of network role similarity. Proceedings of the 17th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining92230 New York: ACM
  65. Kleinberg JM 2003. An impossibility theorem for clustering. NIPS'02: Proceedings of the 15th International Conference on Neural Information Processing Systems46370 Cambridge, MA: MIT Press
  66. Koray S 2000. Self-selective social choice functions verify Arrow and Gibbard-Satterthwaite theorems. Econometrica 68:981–96
    [Google Scholar]
  67. Koray S, Slinko A 2008. Self-selective social choice functions. Soc. Choice Welfare 31:129–49
    [Google Scholar]
  68. Kornhauser LA 1992. Modeling collegial courts. II. Legal doctrine. J. Law Econ. Organ. 8:441–70
    [Google Scholar]
  69. Kornhauser LA, Sager LG 1986. Unpacking the court. Yale Law J. 96:82–117
    [Google Scholar]
  70. Lagerspetz E 1996. Paradoxes and representation. Electoral Stud. 15:83–92
    [Google Scholar]
  71. Laslier JF 1997. Tournament Solutions and Majority Voting New York: Springer-Verlag
  72. Lax JR 2011. The new judicial politics of legal doctrine. Annu. Rev. Political Sci. 14:131–57
    [Google Scholar]
  73. List C 2013. Social choice theory. The Stanford Encyclopedia of Philosophy EN Zalta Metaphys. Res. Lab., Stanford Univ., Winter 2013 ed. https://plato.stanford.edu/archives/win2013/entries/social-choice/
  74. List C, Puppe C 2009. Judgment aggregation: a survey. Handbook of Rational and Social Choice C List, C Puppe45782 New York: Oxford Univ. Press
  75. Lorenz MO 1905. Methods of measuring the concentration of wealth. Publ. Am. Stat. Assoc. 9:209–19
    [Google Scholar]
  76. Ma J 1994. Strategy-proofness and the strict core in a market with indivisibilities. Int. J. Game Theory 23:75–83
    [Google Scholar]
  77. Maskin E 1999. Nash equilibrium and welfare optimality. Rev. Econ. Stud. 66:23–38
    [Google Scholar]
  78. May KO 1952. A set of independent necessary and sufficient conditions for simple majority decision. Econometrica 20:680–84
    [Google Scholar]
  79. Meila M 2005. Comparing clusterings: an axiomatic view. Proceedings of the 22nd International Conference on Machine Learning57784 New York: ACM
  80. Moser S, Patty JW, Penn EM 2009. The structure of heresthetical power. J. Theor. Politics 21:139–59
    [Google Scholar]
  81. Moulin H 1980. On strategy-proofness and single peakedness. Public Choice 35:437–55
    [Google Scholar]
  82. Muller E, Satterthwaite MA 1977. The equivalence of strong positive association and strategy-proofness. J. Econ. Theory 14:412–18
    [Google Scholar]
  83. Myerson RB 1977. Graphs and cooperation in games. Math. Oper. Res. 2:225–29
    [Google Scholar]
  84. Myerson RB 1979. Incentive compatibility and the bargaining problem. Econometrica 47:61–73
    [Google Scholar]
  85. Nakamura K 1979. The vetoers in a simple game with ordinal preferences. Int. J. Game Theory 815561
  86. Nash JF Jr 1950. Equilibrium points in n-person games. PNAS 36:48–49
    [Google Scholar]
  87. Newman M 2010. Networks: An Introduction New York: Oxford Univ. Press
  88. Nurmi H 1997. Compound majority paradoxes and proportional representation. Eur. J. Political Econ. 13:443–54
    [Google Scholar]
  89. Nurmi H 1999. Voting Paradoxes and How to Deal with Them New York: Springer-Verlag
  90. Nurmi H, Meskanen T 2000. Voting paradoxes and MCDM. Group Decis. Negotiation 9:297–313
    [Google Scholar]
  91. Palfrey TR, Srivastava S 1991. Nash implementation using undominated strategies. Econometrica 592479501
  92. Patty JW, Penn EM 2014. Social Choice and Legitimacy: The Possibilities of Impossibility New York: Cambridge Univ. Press
  93. Patty JW, Penn EM 2015a. Aggregation, evaluation, and social choice theory. Good Soc. 24:4972
    [Google Scholar]
  94. Patty JW, Penn EM 2015b. Analyzing big data: social choice and measurement. PS: Political Sci. Politics 48:95–101
    [Google Scholar]
  95. Patty JW, Penn EM 2019. A defense of Arrow's independence of irrelevant alternatives. Public Choice 179:145–64
  96. Penn EM 2006a. Alternate definitions of the uncovered set, and their implications. Soc. Choice Welfare 27:83–87
    [Google Scholar]
  97. Penn EM 2006b. The Banks set in infinite spaces. Soc. Choice Welfare 27:531–43
    [Google Scholar]
  98. Penn EM 2011. Impossibility theorems and voting paradoxes in collective choice theory. Wiley Encyclopedia of Operations Research and Management Science JJ Cochran New York: Wiley https://onlinelibrary.wiley.com/doi/10.1002/9780470400531.eorms0398
  99. Penn EM, Patty JW, Gailmard S 2011. Manipulation and single-peakedness: a general result. Am. J. Political Sci. 55:436–49
    [Google Scholar]
  100. Pettit P 2001. Deliberative democracy and the discursive dilemma. Philos. Issues 11:268–99
    [Google Scholar]
  101. Pigou AC 1912. Wealth and Welfare London: Macmillan
  102. Plott CR 1976. Axiomatic social choice theory: an overview and interpretation. Am. J. Political Sci.
  103. Puppe C, Tasnádi A 2015. Axiomatic districting. Soc. Choice Welfare 44:31–50
    [Google Scholar]
  104. Quinn KM 2012. The academic study of decision making on multimember courts. Calif. Law Rev. 100:1493–501
    [Google Scholar]
  105. Reny PJ 2001. Arrow's theorem and the Gibbard-Satterthwaite theorem: a unified approach. Econ. Lett. 70:99–105
    [Google Scholar]
  106. Roberts F 1984. Measurement Theory: With Applications to Decisionmaking, Utility, and the Social Sciences Cambridge, UK: Cambridge Univ. Press
  107. Saari DG 1989. A dictionary for voting paradoxes. J. Econ. Theory 48:443–75
    [Google Scholar]
  108. Saari DG 1990. Susceptibility to manipulation. Public Choice 64:21–41
    [Google Scholar]
  109. Saari DG 2000. Mathematical structure of voting paradoxes. Econ. Theory 15:1–53
    [Google Scholar]
  110. Saari DG 2010. Systematic analysis of multiple voting rules. Soc. Choice Welfare 34:217–47
    [Google Scholar]
  111. Saari DG 2011. The geometry of voting. Handbook of Social Choice and Welfare KJ Arrow, A Sen, K Suzumura 2897946 Oxford, UK: Elsevier
  112. Saari DG, Barney S 2003. Consequences of reversing preferences. Math. Intel. 25:17–31
    [Google Scholar]
  113. Saari DG, Van Newenhizen J 1988. Is approval voting an ‘unmitigated evil’? A response to Brams, Fishburn, and Merrill. Public Choice 59:133–47
    [Google Scholar]
  114. Saporiti A 2009. Strategy-proofness and single-crossing. Theor. Econ. 4:127–63
    [Google Scholar]
  115. Satterthwaite MA 1975. Strategy-proofness and Arrow's conditions: existence and correspondence theorems for voting procedures and social welfare functions. J. Econ. Theory 10:187–217
    [Google Scholar]
  116. Schofield N 1984. Social equilibrium and cycles on compact sets. J. Econ. Theory 33:59–71
    [Google Scholar]
  117. Schwartz T 1986. The Logic of Collective Choice New York: Columbia Univ. Press
  118. Sen AK 1970a. Collective Choice and Social Welfare San Francisco: Holden-Day
  119. Sen AK 1970b. The impossibility of a Paretian liberal. J. Political Econ. 78:152–57
    [Google Scholar]
  120. Sen AK 1973. On Economic Inequality Oxford, UK: Oxford Univ. Press
  121. Sen AK 1986. Social choice theory. Handbook of Mathematical Economics 31073181 Oxford, UK: Elsevier
  122. Sen AK, Pattanaik PK 1969. Necessary and sufficient conditions for rational choice under majority decision. J. Econ. Theory 1:178–202
    [Google Scholar]
  123. Shapley L, Scarf H 1974. On cores and indivisibility. J. Math. Econ. 1:23–37
    [Google Scholar]
  124. Shepsle KA 1975. Congressional committee assignments: an optimization model with institutional constraints. Public Choice 22:55–78
    [Google Scholar]
  125. Simon HA 1953. Notes on the observation and measurement of political power. J. Politics 15:500–16
    [Google Scholar]
  126. Suzumura K 2011. Welfarism, individual rights, and procedural fairness. Handbook of Social Choice and Welfare KJ Arrow, AK Sen, K Suzumura 260585 Oxford, UK: Elsevier
  127. Tasnádi A 2011. The political districting problem: a survey. Soc. Econ. 33:543–54
    [Google Scholar]
  128. Thomson W 2011. Fair allocation rules. Handbook of Social Choice and Welfare 2393506 Oxford, UK: Elsevier
  129. van den Brink R, Gilles RP 2003. Ranking by outdegree for directed graphs. Discrete Math. 271:261–70
    [Google Scholar]
  130. Vohra R 1996. An axiomatic characterization of some locations in trees. Eur. J. Oper. Res. 90:78–84
    [Google Scholar]
  131. Vorsatz M 2007. Approval voting on dichotomous preferences. Soc. Choice Welfare 28:127–41
    [Google Scholar]
  132. Ward MD, Stovel K, Sacks A 2011. Network analysis and political science. Annu. Rev. Political Sci. 14:245–64
    [Google Scholar]
  133. Wilson R 1972. Social choice theory without the Pareto principle. J. Econ. Theory 5:478–86
    [Google Scholar]
  134. Young HP 1974. An axiomatization of Borda's rule. J. Econ. Theory 9:43–52
    [Google Scholar]
  135. Zadeh RB, Ben-David S 2009. A uniqueness theorem for clustering. Proceedings of the Twenty-fifth Conference on Uncertainty in Artificial Intelligence63946 Arlington, VA: AUAI Press
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