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Arrow’s decisive coalitions

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  • Wesley H. Holliday

    (University of California)

  • Eric Pacuit

    (University of Maryland)

Abstract

In his classic monograph, Social Choice and Individual Values, Arrow introduced the notion of a decisive coalition of voters as part of his mathematical framework for social choice theory. The subsequent literature on Arrow’s Impossibility Theorem has shown the importance for social choice theory of reasoning about coalitions of voters with different grades of decisiveness. The goal of this paper is a fine-grained analysis of reasoning about decisive coalitions, formalizing how the concept of a decisive coalition gives rise to a social choice theoretic language and logic all of its own. We show that given Arrow’s axioms of the Independence of Irrelevant Alternatives and Universal Domain, rationality postulates for social preference correspond to strong axioms about decisive coalitions. We demonstrate this correspondence with results of a kind familiar in economics—representation theorems—as well as results of a kind coming from mathematical logic—completeness theorems. We present a complete logic for reasoning about decisive coalitions, along with formal proofs of Arrow’s and Wilson’s theorems. In addition, we prove the correctness of an algorithm for calculating, given any social rationality postulate of a certain form in the language of binary preference, the corresponding axiom in the language of decisive coalitions. These results suggest for social choice theory new perspectives and tools from logic.

Suggested Citation

  • Wesley H. Holliday & Eric Pacuit, 2020. "Arrow’s decisive coalitions," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 54(2), pages 463-505, March.
    • Handle: RePEc:spr:sochwe:v:54:y:2020:i:2:d:10.1007_s00355-018-1163-z
    DOI: 10.1007/s00355-018-1163-z
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    as
    1. Blau, Julian H., 1979. "Semiorders and collective choice," Journal of Economic Theory, Elsevier, vol. 21(1), pages 195-206, August.
    2. John Weymark, 1984. "Arrow's theorem with social quasi-orderings," Public Choice, Springer, vol. 42(3), pages 235-246, January.
    3. Marc Fleurbaey & Philippe Mongin, 2005. "The news of the death of welfare economics is greatly exaggerated," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 25(2), pages 381-418, December.
    4. Blair, Douglas H. & Bordes, Georges & Kelly, Jerry S. & Suzumura, Kotaro, 1976. "Impossibility theorems without collective rationality," Journal of Economic Theory, Elsevier, vol. 13(3), pages 361-379, December.
    5. Maurice Salles, 2017. "On Quine on Arrow," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 48(4), pages 877-886, April.
    6. Saari,Donald G., 2008. "Disposing Dictators, Demystifying Voting Paradoxes," Cambridge Books, Cambridge University Press, number 9780521731607, October.
    7. James M. Buchanan, 1954. "Social Choice, Democracy, and Free Markets," Journal of Political Economy, University of Chicago Press, vol. 62(2), pages 114-114.
    8. Sen, Amartya K, 1979. "Personal Utilities and Public Judgements: Or What's Wrong with Welfare Economics?," Economic Journal, Royal Economic Society, vol. 89(355), pages 537-558, September.
    9. Kirman, Alan P. & Sondermann, Dieter, 1972. "Arrow's theorem, many agents, and invisible dictators," Journal of Economic Theory, Elsevier, vol. 5(2), pages 267-277, October.
    10. List, Christian & Polak, Ben, 2010. "Introduction to judgment aggregation," Journal of Economic Theory, Elsevier, vol. 145(2), pages 441-466, March.
    11. Patty,John W. & Penn,Elizabeth Maggie, 2014. "Social Choice and Legitimacy," Cambridge Books, Cambridge University Press, number 9780521138338, October.
    12. Rubinstein, Ariel & Fishburn, Peter C., 1986. "Algebraic aggregation theory," Journal of Economic Theory, Elsevier, vol. 38(1), pages 63-77, February.
    13. Amartya Sen, 1969. "Quasi-Transitivity, Rational Choice and Collective Decisions," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 36(3), pages 381-393.
    14. Herzberg, Frederik & Eckert, Daniel, 2012. "The model-theoretic approach to aggregation: Impossibility results for finite and infinite electorates," Mathematical Social Sciences, Elsevier, vol. 64(1), pages 41-47.
    15. Reny, Philip J., 2001. "Arrow's theorem and the Gibbard-Satterthwaite theorem: a unified approach," Economics Letters, Elsevier, vol. 70(1), pages 99-105, January.
    16. Rubinstein, Ariel, 1984. "The Single Profile Analogues to Multi Profile Theorems: Mathematical Logic's Approach," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 25(3), pages 719-730, October.
    17. Ehud Kalai & Eitan Muller & Mark Satterthwaite, 1979. "Social welfare functions when preferences are convex, strictly monotonic, and continuous," Public Choice, Springer, vol. 34(1), pages 87-97, March.
    18. Wilson, Robert, 1972. "Social choice theory without the Pareto Principle," Journal of Economic Theory, Elsevier, vol. 5(3), pages 478-486, December.
    19. Campbell, Donald E. & Kelly, Jerry S., 2002. "Impossibility theorems in the arrovian framework," Handbook of Social Choice and Welfare, in: K. J. Arrow & A. K. Sen & K. Suzumura (ed.), Handbook of Social Choice and Welfare, edition 1, volume 1, chapter 1, pages 35-94, Elsevier.
    20. Guha, Ashok, 1972. "Neutrality, Monotonicity, and the Right of Veto," Econometrica, Econometric Society, vol. 40(5), pages 821-826, September.
    21. Dryzek, John S. & List, Christian, 2003. "Social Choice Theory and Deliberative Democracy: A Reconciliation," British Journal of Political Science, Cambridge University Press, vol. 33(1), pages 1-28, January.
    22. Andreu Mas-Colell & Hugo Sonnenschein, 1972. "General Possibility Theorems for Group Decisions," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 39(2), pages 185-192.
    23. Allan Gibbard, 2014. "Intransitive social indifference and the Arrow dilemma," Review of Economic Design, Springer;Society for Economic Design, vol. 18(1), pages 3-10, March.
    24. Hammond, Peter J, 1976. "Equity, Arrow's Conditions, and Rawls' Difference Principle," Econometrica, Econometric Society, vol. 44(4), pages 793-804, July.
    25. Blair, Douglas H. & Pollak, Robert A., 1979. "Collective rationality and dictatorship: The scope of the arrow theorem," Journal of Economic Theory, Elsevier, vol. 21(1), pages 186-194, August.
    26. Sen, Amartya Kumar, 1970. "The Impossibility of a Paretian Liberal," Scholarly Articles 3612779, Harvard University Department of Economics.
    27. Sen, Amartya, 1970. "The Impossibility of a Paretian Liberal," Journal of Political Economy, University of Chicago Press, vol. 78(1), pages 152-157, Jan.-Feb..
    28. Patty,John W. & Penn,Elizabeth Maggie, 2014. "Social Choice and Legitimacy," Cambridge Books, Cambridge University Press, number 9780521191012, October.
    29. Ning Yu, 2012. "A one-shot proof of Arrow’s impossibility theorem," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 50(2), pages 523-525, June.
    30. Sen, Amartya, 1993. "Internal Consistency of Choice," Econometrica, Econometric Society, vol. 61(3), pages 495-521, May.
    31. Lauwers, Luc & Van Liedekerke, Luc, 1995. "Ultraproducts and aggregation," Journal of Mathematical Economics, Elsevier, vol. 24(3), pages 217-237.
    32. Saari,Donald G., 2008. "Disposing Dictators, Demystifying Voting Paradoxes," Cambridge Books, Cambridge University Press, number 9780521516051, October.
    33. Bengt Hansson, 1976. "The existence of group preference functions," Public Choice, Springer, vol. 28(1), pages 89-98, December.
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