nach oben

Social Choice and Welfare

Erschienen in:

11.06.2018 | Original Paper

On single-peaked domains and min–max rules

verfasst von: Gopakumar Achuthankutty, Souvik Roy

Erschienen in: Social Choice and Welfare | Ausgabe 4/2018

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config

KI-gestützte Suche

Aus

Abstract

We consider social choice problems where the admissible set of preferences of each agent is single-peaked. First, we show that if all the agents have the same admissible set of preferences, then every unanimous and strategy-proof social choice function (SCF) is tops-only. Next, we consider situations where different agents have different admissible sets of single-peaked preferences. We show by means of an example that unanimous and strategy-proof SCFs need not be tops-only in this situation, and consequently provide a sufficient condition on the admissible sets of preferences of the agents so that unanimity and strategy-proofness guarantee tops-onlyness. Finally, we characterize all domains on which (i) every unanimous and strategy-proof SCF 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 in Public Choice 35(4):437–455. https://doi.org/10.1007/BF00128122, 1980; Weymark in SERIEs 2(4):529–550. https://doi.org/10.1007/s13209-011-0064-5, 2011), minimally rich single-peaked domains (Peters et al. in J Math Econ 52:123–127. https://doi.org/10.1016/j.jmateco.2014.03.008. http://www.sciencedirect.com/science/article/pii/S0304406814000470, 2014), maximal regular single-crossing domains (Saporiti in Theor Econ 4(2):127–163, 2009, J Econ Theory 154:216–228. https://doi.org/10.1016/j.jet.2014.09.006. http://www.sciencedirect.com/science/article/pii/S0022053114001276, 2009), and distance based single-peaked domains.

Vorheriger Artikel A paradox of expert rights in abstract argumentation

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Nur mit Berechtigung zugänglich

Barberà et al. (1993) and Ching (1997) provide equivalent presentations of this class of SCFs.

A rich literature has developed around the single-peaked restriction by considering various generalizations and extensions (see Barberà et al. 1993; Demange 1982; Schummer and Vohra 2002; Nehring and Puppe 2007a, and Nehring and Puppe 2007b).

See, for instance, the domain restriction considered in models of voting (Tullock 1967; Arrow 1969), taxation and redistribution (Epple and Romer 1991), determining the levels of income redistribution (Hamada 1973; Slesnick 1988), and measuring tax reforms in the presence of horizontal inequity (Hettich 1979). Recently, Puppe (2018) shows that under mild conditions these domains form subsets of the maximal single-peaked domain.

Chatterji and Sen (2011) provide a sufficient condition on a domain so that every unanimous and strategy-proof SCF on it is tops-only. However, an arbitrary single-peaked domain does not satisfy their condition.

The top-connectedness property is well studied in the literature (see Barberà and Peleg 1990; Aswal et al. 2003; Chatterji and Sen 2011; Chatterji et al. 2014; Chatterji and Zeng 2018; Puppe 2018 and Roy and Storcken 2016).

A domain is regular if every alternative appears as the top-ranked alternative of some preference in the domain.

Single-crossing domains appear in models of taxation and redistribution (Roberts 1977; Meltzer and Richard 1981), local public goods and stratification (Westhoff 1977; Epple and Platt 1998; Epple et al. 2001), coalition formation (Demange 1994; Kung 2006), selecting constitutional and voting rules (Barberà and Jackson 2004), and designing policies in the market for higher education (Epple et al. 2006).

Saporiti (2014) provides a different but equivalent functional form of these SCFs which he calls augmented representative voter schemes.

A single-peaked preference is called left (or right) single-peaked if every alternative to the left (or right) of the peak is preferred to every alternative to its right (or left).

Such preferences appear in directional theories of issue voting (Stokes 1963; Rabinowitz 1978; Rabinowitz et al. 1982; Rabinowitz and Macdonald 1989).

For details, see Weymark (2011).

A median rule is a min–max rule that is anonymous. More formally, a min–max rule with respect to parameters \((\beta _S)_{S \subseteq N}\) is a median rule if \(\beta _S=\beta _{\bar{S}}\) for all \(S,\bar{S} \subseteq N\) with \(|S|=|\bar{S}|\). For details see Moulin (1980).

Since \(f(P'_{i},P_{N {\setminus } i}) = y'\), if \(r_{1}(P'_i) \ne y'\), then by strategy-proofness, \(f(P''_{i},P_{N {\setminus } i}) = y'\) for some \(P''_{i} \in \mathcal {S}_{i}\) with \(r_{1}(P''_{i}) = y'\). Similarly, if \(r_{1}(P_{j}) < y'\) for some \(j \in N {\setminus } i\), then by strategy-proofness, \(f(P'_{i},P'_{j},P_{N {\setminus } \{i,j\}}) = y'\) for some \(P'_{j} \in \mathcal {S}_{j}\) with \(r_{1}(P'_{j}) = y'\).

Here \(\mathcal {D}_{i}\) satisfies the unique seconds property defined in Aswal et al. (2003) and the SCF f considered here is similar to the one used in the proof of Theorem 5.1 in Aswal et al. (2003).

Arribillaga RP, Massó J (2016) Comparing generalized median voter schemes according to their manipulability. Theor Econ 11(2):547–586. https://doi.org/10.3982/TE1910 CrossRef

Arrow KJ (1969) Tullock and an existence theorem. Public Choice 6:105–111. http://www.jstor.org/stable/30022601 CrossRef

Aswal N, Chatterji S, Sen A (2003) Dictatorial domains. Econ Theory 22(1):45–62. https://doi.org/10.1007/s00199-002-0285-8 CrossRef

Barberà S, Jackson MO (2004) Choosing how to choose: Self-stable majority rules and constitutions. Q J Econ 119(3):1011–1048. http://www.jstor.org/stable/25098708 CrossRef

Barberà S, Peleg B (1990) Strategy-proof voting schemes with continuous preferences. Soc Choice Welf 7(1):31–38. https://doi.org/10.1007/BF01832918 CrossRef

Barberà S, Gul F, Stacchetti E (1993) Generalized median voter schemes and committees. J Econ Theory 61(2):262–289. https://doi.org/10.1006/jeth.1993.1069. http://www.sciencedirect.com/science/article/pii/S0022053183710690 CrossRef

Barberà S, Massò J, Neme A (1999) Maximal domains of preferences preserving strategy-proofness for generalized median voter schemes. Soc Choice Welf 16(2):321–336. https://doi.org/10.1007/s003550050146 CrossRef

Barberà S, Berga D, Moreno B (2010) Individual versus group strategy-proofness: when do they coincide? J Econ Theory 145(5):1648–1674. https://doi.org/10.1016/j.jet.2010.03.006. http://www.sciencedirect.com/science/article/pii/S0022053110000566 CrossRef

Black D (1948) On the rationale of group decision-making. J Political Econ 56(1):23–34. http://www.jstor.org/stable/1825026 CrossRef

Chatterji S, Sen A (2011) Tops-only domains. Econ Theory 46(2):255–282. https://doi.org/10.1007/s00199-009-0509-2 CrossRef

Chatterji S, Zeng H (2018) On random social choice functions with the tops-only property. Games Econ Behav 109:413–435. https://doi.org/10.1016/j.geb.2017.11.011. http://www.sciencedirect.com/science/article/pii/S0899825617302427 CrossRef

Chatterji S, Sen A, Zeng H (2014) Random dictatorship domains. Games Econ Behav 86:212–236. https://doi.org/10.1016/j.geb.2014.03.017. http://www.sciencedirect.com/science/article/pii/S0899825614000700 CrossRef

Ching S (1997) Strategy-proofness and “median voters”. Int J Game Theory 26(4):473–490. https://doi.org/10.1007/BF01813886 CrossRef

Demange G (1982) Single-peaked orders on a tree. Math Soc Sci 3(4):389–396. https://doi.org/10.1016/0165-4896(82)90020-8 CrossRef

Demange G (1994) Intermediate preferences and stable coalition structures. J Math Econ 23(1):45–58. https://doi.org/10.1016/0304-4068(94)90035-3 CrossRef

Elkind E, Faliszewski P, Skowron P (2014) A characterization of the single-peaked single-crossing domain. In: AAAI, vol 14, pp 654–660

Epple D, Platt GJ (1998) Equilibrium and local redistribution in an urban economy when households differ in both preferences and incomes. J Urban Econ 43(1):23–51. https://doi.org/10.1006/juec.1996.2030. http://www.sciencedirect.com/science/article/pii/S0094119096920306 CrossRef

Epple D, Romer T (1991) Mobility and redistribution. J Political Econ 99(4):828–858. http://www.jstor.org/stable/2937782 CrossRef

Epple D, Romer T, Sieg H (2001) Interjurisdictional sorting and majority rule: an empirical analysis. Econometrica 69(6):1437–1465. https://doi.org/10.1111/1468-0262.00253 CrossRef

Epple D, Romano R, Sieg H (2006) Admission, tuition, and financial aid policies in the market for higher education. Econometrica 74(4):885–928. https://doi.org/10.1111/j.1468-0262.2006.00690.x CrossRef

Gibbard A (1973) Manipulation of voting schemes: a general result. Econometrica 41(4):587–601. http://www.jstor.org/stable/1914083 CrossRef

Hamada K (1973) A simple majority rule on the distribution of income. J Econ Theory 6(3):243–264. https://doi.org/10.1016/0022-0531(73)90049-5 CrossRef

Hettich W (1979) A theory of partial tax reform. Can J Econ Revue canadienne d’Economique 12(4):692–712. http://www.jstor.org/stable/134874 CrossRef

Hotelling H (1929) Stability in competition. Econ J 39:41–57

Kung FC (2006) An algorithm for stable and equitable coalition structures with public goods. J Public Econ Theory 8(3):345–355. https://doi.org/10.1111/j.1467-9779.2006.00267.x CrossRef

Meltzer AH, Richard SF (1981) A rational theory of the size of government. J Political Econ 89(5):914–927. http://www.jstor.org/stable/1830813 CrossRef

Moulin H (1980) On strategy-proofness and single peakedness. Public Choice 35(4):437–455. https://doi.org/10.1007/BF00128122 CrossRef

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(1):269–305. https://doi.org/10.1016/j.jet.2006.04.008. http://www.sciencedirect.com/science/article/pii/S0022053106001050 CrossRef

Nehring K, Puppe C (2007) Efficient and strategy-proof voting rules: a characterization. Games Econ Behav 59(1):132–153. https://doi.org/10.1016/j.geb.2006.05.002. http://www.sciencedirect.com/science/article/pii/S0899825606000674 CrossRef

Peters H, Roy S, Sen A, Storcken T (2014) Probabilistic strategy-proof rules over single-peaked domains. J Math Econ 52:123–127. https://doi.org/10.1016/j.jmateco.2014.03.008. http://www.sciencedirect.com/science/article/pii/S0304406814000470 CrossRef

Puppe C (2018) The single-peaked domain revisited: a simple global characterization. J Econ Theory 176:55–80. https://doi.org/10.1016/j.jet.2018.03.003. http://www.sciencedirect.com/science/article/pii/S0022053118300863 CrossRef

Rabinowitz G (1978) On the nature of political issues: insights from a spatial analysis. Am J Political Sci 22(4):793–817. http://www.jstor.org/stable/2110591 CrossRef

Rabinowitz G, Macdonald SE (1989) A directional theory of issue voting. Am Political Sci Rev 83(1):93–121. https://doi.org/10.2307/1956436 CrossRef

Rabinowitz G, Prothro JW, Jacoby W (1982) Salience as a factor in the impact of issues on candidate evaluation. J Politics 44(1):41–63. https://doi.org/10.2307/2130283 CrossRef

Roberts KW (1977) Voting over income tax schedules. J Public Econ 8(3):329–340. https://doi.org/10.1016/0047-2727(77)90005-6 CrossRef

Romer T, Rosenthal H (1979) Bureaucrats versus voters: on the political economy of resource allocation by direct democracy. Q J Econ 93(4):563–587. http://www.jstor.org/stable/1884470 CrossRef

Roy S, Storcken T (2016) Unanimity, pareto optimality and strategy-proofness on connected domains. Working Paper

Saporiti A (2009) Strategy-proofness and single-crossing. Theor Econ 4(2):127–163

Saporiti A (2014) Securely implementable social choice rules with partially honest agents. J Econ Theory 154:216–228. https://doi.org/10.1016/j.jet.2014.09.006. http://www.sciencedirect.com/science/article/pii/S0022053114001276 CrossRef

Satterthwaite MA (1975) Strategy-proofness and arrow’s conditions: existence and correspondence theorems for voting procedures and social welfare functions. J Econ Theory 10(2):187–217. https://doi.org/10.1016/0022-0531(75)90050-2 CrossRef

Schummer J, Vohra RV (2002) Strategy-proof location on a network. J Econ Theory 104(2):405–428. https://doi.org/10.1006/jeth.2001.2807. http://www.sciencedirect.com/science/article/pii/S0022053101928078 CrossRef

Slesnick D (1988) The political economy of redistribution policy. Unpublished University of Texas Working Paper

Stokes DE (1963) Spatial models of party competition. Am Political Sci Rev 57(2):368–377. https://doi.org/10.2307/1952828 CrossRef

Tullock G (1967) The general irrelevance of the general impossibility theorem. Q J Econ 81(2):256–270. http://www.jstor.org/stable/1879585 CrossRef

Westhoff F (1977) Existence of equilibria in economies with a local public good. J Econ Theory 14(1):84–112. https://doi.org/10.1016/0022-0531(77)90086-2 CrossRef

Weymark JA (2011) A unified approach to strategy-proofness for single-peaked preferences. SERIEs 2(4):529–550. https://doi.org/10.1007/s13209-011-0064-5 CrossRef

Titel: On single-peaked domains and min–max rules
verfasst von: Gopakumar Achuthankutty
Souvik Roy
Publikationsdatum: 11.06.2018
Verlag: Springer Berlin Heidelberg
Erschienen in: Social Choice and Welfare / Ausgabe 4/2018
Print ISSN: 0176-1714
Elektronische ISSN: 1432-217X
DOI: https://doi.org/10.1007/s00355-018-1137-1

Springer Professional

Abstract

Bitte loggen Sie sich ein, um Zugang zu Ihrer Lizenz zu erhalten.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Wirtschaft"

Weitere Artikel der Ausgabe 4/2018

The Electoral College, battleground states, and rule-utilitarian voting

Composite indices, alternative weights, and comparison robustness

A paradox of expert rights in abstract argumentation

Does decentralization of decisions increase the stability of large groups?

On some oligarchy results when social preference is fuzzy

The flow network method