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Arrow on domain conditions: a fruitful road to travel

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Abstract

We stress the importance that Arrow attached to studying the role of domain conditions in determining the validity of his impossibility theorem, a subject to which he devoted two chapters of Social Choice and Individual Values. Then we partially survey recent results about the role of domain conditions on the possibility to design satisfactory aggregation rules and social choice functions, as a proof of the continued vitality of this subject, that he pioneered, as he did with so many others.

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Notes

  1. Notice that many other variants of collective rules could be adopted when defining how individual preferences are combined. Domains could be enlarged by not requiring individual preferences to be transitive, and codomains could allow for different expressions of aggregation, including sets of alternatives, choice functions, lotteries, or different sorts of binary relations. We restrict attention to the three basic forms defined above, which are enough to make our points.

  2. We refer the reader to Le Breton and Weymark (2010) for definitions and results related to this section.

  3. Single crossing is equivalent to order restriction, a condition also used in different contexts. See Gans and Smart (1996).

  4. Here again this loose statement should be qualified depending on whether the number of alternatives is odd or even. The precise statement of Moulin (1980) will clarify any possible ambiguity. Similar remarks regarding strategy proofness of the median rule under single crossing preferences are contained in Saporiti and Tohmé (2006).

  5. For precise characterizations and properties of these rules in Cartesian domains, see Barberà et al. (1993), Le Breton and Sen (1995, 1999). The positive results in these papers must be qualified when some potential alternatives cannot be chosen and the range of the function is not a Cartesian product. See Barberà et al. (1998), Barberà et al. (1997b) and Barberà et al. (2005).

  6. In his classical book, Fishburn (1973 page 178) proposed a classification of different conditions that one may impose on social choice functions, and distinguished, among others, between intraprofile and interprofile conditions, depending on whether the requirements on the outcomes of a function could be expressed in reference to one profile at a time, or needed to identify several ones that were somewhat connected. The conditions we present are on domains of definition, rather than on a function’s outcome, and the classification does not directly apply, but there is a parallel. Single peakedness can be checked profile by profile, while the conditions we are about to present refer to combinations of profiles.

  7. A presentation of some of the properties that follow, directed to a computer science audience, is contained in Barberà et al. (2013).

  8. See Le Breton and Zaporozhets (2009) for a related domain condition.

  9. For details about this “almost necessity” result, see Barberà et al. (2010). The argument is not quite a necessity implication, and in that sense is reminiscent of previous definitions of “necessity” in social choice, like the one used in Sen and Pattanaik (1969).

  10. See for example Gibbard (1973), Myerson (1979), Dasgupta et al. (1979) and Harris and Townsend (1981).

  11. A recent attempt in that direction may be found in Barberà et al. (2018, 2019).

References

  • Arrow K (1963) Social choice and individual values, 2nd edn. Wiley, New York (1st edition 1951)

    Google Scholar 

  • Arrow K, Sen A, Suzumura K (eds) (2002) Handbook of social choice and welfare, vol 1. North Holland-Elsevier, Amsterdam

    Google Scholar 

  • Arrow K, Sen A, Suzumura K (eds) (2010) Handbook of social choice and welfare, vol 2. North Holland-Elsevier, Amsterdam

    Google Scholar 

  • Aswal N, Chatterji S, Sen A (2003) Dictatorial domains. Econ Theory 22:45–62

    Google Scholar 

  • Austen-Smith D, Banks J (1996) Information aggregation, rationality, and the Condorcet jury theorem. Am Polit Sci Rev 90:34–45

    Google Scholar 

  • Austen-Smith D, Feddersen T (2005) Deliberation and voting rules. In: Austen-Smith D, Duggan J (eds) Social choice and strategic decisions: essays in honor of Jeffrey S. Banks. Springer, Berlin

    Google Scholar 

  • Austen-Smith D, Feddersen T (2006) Deliberation, preference uncertainty and voting rules. Am Polit Sci Rev 100(2):209–221

    Google Scholar 

  • Baigent N (2010) Topological theories of social choice, chapter 18. In: Arrow K, Sen A, Suzumura K (eds) Handbook of social choice and welfare, vol 2. North Holland, Amsterdam

    Google Scholar 

  • Ballester MA, Haeringer G (2011) A characterization of the single-peaked domain. Soc Choice Welf 36:305–322

    Google Scholar 

  • Barberà S (1980) Pivotal voters: a new proof of Arrow’s theorem. Econ Lett 6:13–16

    Google Scholar 

  • Barberà S (1983) Strategy-proofness and pivotal voters: a direct proof of the Gibbard–Satterthwaite theorem. Int Econ Rev 24:413–417

    Google Scholar 

  • Barberà S (2007) Indifferences and domain restrictions. Analyse und Kritik 29:146–162

    Google Scholar 

  • Barberà S, Moreno B (2011) Top monotonicity: a common root for single peakedness, single crossing and the median voter result. Games Econ Behav 73:345–359

    Google Scholar 

  • Barberà S, Gul F, Stacchetti E (1993) Generalized median voter schemes and committees. J Econ Theory 61:262–289

    Google Scholar 

  • Barberà S, Jackson MO, Neme A (1997a) Strategy-proof allotment rules. Games Econ Behav 18:1–21

    Google Scholar 

  • Barberà S, Massó J, Neme A (1997b) Voting under constraints. J Econ Theory 76:298–321

    Google Scholar 

  • Barberà S, Massó J, Serizawa S (1998) Strategy-proof voting on compact ranges. Games Econ Behav 25:272–291

    Google Scholar 

  • Barberà S, Massó J, Neme A (2005) Voting by committees under constraints. J Econ Theory 122:185–205

    Google Scholar 

  • Barberà S, Berga D, Moreno B (2010) Individual versus group strategy-proofness: when do they coincide? J Econ Theory 145:1648–1674

    Google Scholar 

  • Barberà S, Berga D, Moreno B (2012) Two necessary conditions for strategy-proofness: on what domains are they also sufficient? Games Econ Behav 75:490–509

    Google Scholar 

  • Barberà S, Berga D, Moreno B (2013) Some new domain restrictions in social choice, and their consequences. In: Torra V et al (eds) MDAI 2013, LNAI 8234. Springer, Berlin, pp 11–24

    Google Scholar 

  • Barberà S, Berga D, Moreno B (2016) Group strategy-proofness in private good economies. Am Econ Rev 106:1073–1099

    Google Scholar 

  • Barberà S, Berga D, Moreno B (2018) Restricted environments and incentive compatibility in interdependent values models. Available at SSRN: https://ssrn.com/abstract=3135163 or https://doi.org/10.2139/ssrn.3135163, and BGSE WP 1024. Accessed July 2018

  • Barberà S, Berga D, Moreno B (2019) Domains admitting ex post incentive compatible and respectful mechanisms: a characterization for the two alternatives case. In: Trockel W (ed) Social design—essays in memory of Leonid Hurwicz. Springer, Berlin

    Google Scholar 

  • Berga D (1998) Strategy-proofness and single-plateaued preferences. Math Soc Sci 35:105–120

    Google Scholar 

  • Bergson A (1938) A reformulation of certain aspects of welfare economics. Q J Econ 52:310–334

    Google Scholar 

  • Black D (1948) On the rationale of group decision making. J Polit Econ 56:23–34

    Google Scholar 

  • Black D (1958) The theory of committees and elections. Cambridge University Press, Cambridge

    Google Scholar 

  • Blau JH (1957) The existence of social welfare functions. Econometrica 25:302–313

    Google Scholar 

  • Border K (1983) Social welfare functions for economic environments with and without the pareto principle. J Econ Theory 29:205–216

    Google Scholar 

  • Bordes G, Le Breton M (1989) Arrovian theorems with private alternatives domains and selfish individuals. J Econ Theory 47:257–281

    Google Scholar 

  • Bordes G, Le Breton M (1990) Arrovian theorems for economic domains: assignments, matchings, and pairings. Soc Choice Welf 7:193–208

    Google Scholar 

  • Bordes G, Campbell DE, Le Breton M (1995) Arrow’s theorem for economic domains and Edgeworth hyperboxes. Int Econ Rev 36:441–454

    Google Scholar 

  • Chatterji S, Sanver R, Sen A (2013) On domains that admit well-behaved strategy-proof social choice functions. J Econ Theory 148:1050–1073

    Google Scholar 

  • Chattherji S, Massó J (2018) On strategy proofness and the salience of single peakedness. Int Econ Rev 59:163–189

    Google Scholar 

  • Chichilnisky G (1980) Social choice and the topology of spaces of preferences. Adv Math 37:165–176

    Google Scholar 

  • Chichilnisky G (1982) Social aggregation rules and continuity. Q J Econ 97:337–352

    Google Scholar 

  • Chichilnisky G, Heal G (1983) Necessary and sufficient conditions for a resolution of the social choice paradox. J Econ Theory 31:68–87

    Google Scholar 

  • Dasgupta P, Hammond P, Maskin E (1979) The implementation of social choice rules: some results on incentive compatibility. Rev Econ Stud 46:185–216

    Google Scholar 

  • Davis OA, Groot DE, Hinich M (1972) Social preference orderings and majority rule. Econometrica 40:147–157

    Google Scholar 

  • Demange G (1982) Single-peaked orders on a tree. Math Soc Sci 3:389–396

    Google Scholar 

  • Ehlers L, Storcken T (2008) Arrow’s possibility theorem for one-dimensional single-peaked preferences. Games Econ Behav 64:533–547

    Google Scholar 

  • Fishburn PC (1973) The theory of social choice. Princeton University Press, Princeton

    Google Scholar 

  • Fishburn PC (1997) Acyclic sets of linear orders. Soc Choice Welf 14:113–124

    Google Scholar 

  • Fishburn PC (2002) Acyclic sets of linear orders. A progress report. Soc Choice Welf 19:431–447

    Google Scholar 

  • Gaertner W (2001) Domain conditions in social choice theory. Cambridge University Press, Cambridge

    Google Scholar 

  • Gaertner W (2002) Domain restrictions, chapter 3. In: Arrow K, Sen A, Suzumura K (eds) Handbook of social choice and welfare, vol 1. North Holland, Amsterdam, pp 131–170

    Google Scholar 

  • Gaertner W (2005) De jure naturae et gentium: Samuel von Pufendorf’s contribution to social choice theory and economics. Soc Choice Welf 25:231–241

    Google Scholar 

  • Gaertner W (2018) Kenneth Arrow’s impossibility theorem stretching to other fields. Public Choice 179:125–131

    Google Scholar 

  • Gans JS, Smart M (1996) Majority voting with single-crossing preferences. J Public Econ 59:219–237

    Google Scholar 

  • Gibbard A (1973) Manipulation of voting schemes: a general result. Econometrica 41:587–601

    Google Scholar 

  • Harris M, Townsend R (1981) Resource allocation under asymmetric information. Econometrica 49:33–64

    Google Scholar 

  • Igersheim H (2017) The death of welfare economics. History of a controversy. The center for the history of political economy working paper, no. 2017-03

  • Kalai E, Muller E, Satterthwaite M (1979) Social welfare functions when preferences are convex and continuous: impossibility results. Public Choice 34:87–97

    Google Scholar 

  • Kramer GH (1973) On a class of equilibrium conditions for majority rules. Econometrica 41:285–297

    Google Scholar 

  • Le Breton M, Sen A (1995) Strategyproofness and decomposability: weak orderings. G.R.E.Q.A.M. 95a38, Université Aix-Marseille III

  • Le Breton M, Sen A (1999) Separable preferences, strategyproofness and decomposability. Econometrica 67:605–628

    Google Scholar 

  • Le Breton M, Weymark JW (2010) Arrovian social choice theory on economic domains, chapter 17. In: Arrow K, Sen A, Suzumura K (eds) Handbook of social choice and welfare. Elsevier, Amsterdam

    Google Scholar 

  • Le Breton M, Zaporozhets V (2009) On the equivalence of coalitional and individual strategy-proofness properties. Soc Choice Welf 33:287–309

    Google Scholar 

  • List C (2012) The theory of judgment aggregation: an introductory review. Synthese 187:179–207

    Google Scholar 

  • List C, Polak B (2010) Introduction to judgment aggregation. J Econ Theory 145:441–466

    Google Scholar 

  • Maskin E (1976) Social welfare functions for economics. Darwin College, Cambridge University and Department of Economics, Harvard University, Cambridge (unpublished manuscript)

    Google Scholar 

  • Maskin E (1979) Fonctions de Préférence Collective Définies sur des Domaines de Préférences Individuelles Soumis à des Contraintes. Cahiers du Séminaire d’Econométrie 20:153–182

    Google Scholar 

  • Massó J, Moreno de Barreda I (2011) On strategy-proofness and symmetric single peakedness. Games Econ Behav 72:467–484

    Google Scholar 

  • McKelvey RD (1976) Intransitivities in multidimensional voting models and some implications for agenda control. J Econ Theory 12:472–482

    Google Scholar 

  • McKelvey RD (1996) Social choice. The Division of Humanities and Social Sciences, California Institute of Technology, Pasadena

    Google Scholar 

  • Moulin H (1980) On strategy-proofness and single peakedness. Public Choice 35:437–455

    Google Scholar 

  • Moulin H (1984) Generalized condorcet-winners for single peaked and single plateau preferences. Soc Choice Welf 1:127–147

    Google Scholar 

  • Muller E (1982) Graphs and anonymous social welfare functions. Int Econ Rev 23:609–622

    Google Scholar 

  • Myerson R (1979) Incentive compatibility and the bargaining problem. Econometrica 47:61–73

    Google Scholar 

  • Nehring K, Puppe C (2007) The structure of strategy-proof social choice—part I: general characterization and possibility results on median spaces. J Econ Theory 135:269–305

    Google Scholar 

  • Penn EM, Patty JW, Gailmard S (2011) Manipulation and single-peakedness: a general result. Am J Polit Sci 55:436–449

    Google Scholar 

  • Plott CR (1967) A notion of equilibrium and its possibility under majority rule. Am Econ Rev 57:787–806

    Google Scholar 

  • Redekop J (1991) Social welfare functions on restricted domains. J Econ Theory 53:396–427

    Google Scholar 

  • Roth A (2008) What have we learned from market design. Hahn Lect Econ J 118:285–310

    Google Scholar 

  • Samuelson PA (1947) Foundations of economic analysis. Harvard University Press, Cambridge

    Google Scholar 

  • Saporiti A, Tohmé F (2006) Single-crossing, strategic voting and the median choice rule. Soc Choice Welf 26:363–383

    Google Scholar 

  • Satterthwaite M (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 

  • Schmitz N (1977) A further note on Arrow’s impossibility theorem. J Math Econ 4:189–196

    Google Scholar 

  • Schofield N (1978) Instability of simple dynamic games. Rev Econ Stud 45:575–594

    Google Scholar 

  • Sen A (1966) A possibilty theorem on majority decisions. Econometrica 34:491–499

    Google Scholar 

  • Sen A, Pattanaik PK (1969) Necessary and sufficient conditions for rational choice under majority decision. J Econ Theory 1:178–202

    Google Scholar 

  • Sprumont Y (1991) The division problem with single-peaked preferences: a characterization of the uniform allocation rule. Econometrica 59:509–519

    Google Scholar 

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Acknowledgements

We are thankful to the Journal Editor, Mark Fleurbaey, and to two extremely helpful referees, for comments and suggestions. We are also grateful to the editors of this special issue for allowing us to participate in this homage to Kenneth Arrow. S. Barberà acknowledges financial support through Grants ECO2014-53052-P and SGR2014-515, and Severo Ochoa Programme for Centres of Excellence in R&D (SEV-2015-0563). D. Berga and B. Moreno acknowledge the financial support from the Spanish Ministry of Science, Industry and Competitiveness through Grants ECO2016-76255-P and ECO2017-86245-P, respectively, and thank the MOMA network.

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Barberà, S., Berga, D. & Moreno, B. Arrow on domain conditions: a fruitful road to travel. Soc Choice Welf 54, 237–258 (2020). https://doi.org/10.1007/s00355-019-01196-4

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