Abstract
This paper contributes to the axiomatization of additive rules for ranking sets of objects under the psychological principle of categorization. Firstly we proceed with the case where the elements in the sets are categorized into at most three groups, namely good (with value 1), neutral (with value 0), and bad (with value −1). Secondly, we solve the case where there are only good and neutral elements. In both instances the evaluation of the sets is purely additive. Lastly, we show that dropping one of the axioms in our general characterization produces an axiomatization of the more general class of evaluations where good and bad elements are weighted differently. Areas of research in Economics such as committee selection problems, hedonic games and matching are among the ranking sets models where our results could potentially be applied.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Alcantud JCR, Arlegi R (2008) A characterization of additively representable rankings of sets. Theory Dec 64(2–3): 147–171
Allport GW (1954) The nature of prejudice. Addison Wesley, Reading
Arlegi R (2007) Sequentially consistent rules of choice under complete uncertainty. J Econ Theory 135(1): 131–143
Banerjee S, Konishi T, Sönmez T (2001) Core in a simple coalition formation game. Soc Choice Welf 18(1): 135–153
Barberà S, Barrett CR, Pattanaik P (1984) On some axioms for ranking sets of alternatives. J Econ Theory 33: 301–308
Barberà S, Bossert W, Pattanaik P (2004) Ranking sets of objects. In: Barberà S, Hammond P, Seidl C (eds) Handbook of utility theory, vol 2. Kluwer Academic Publishers, Dordrecht, pp 893–977
Barberà S, Maschler M, Shalev J (2001) Voting for voters: a model of electoral evolution. Games Econ Behav 37: 40–78
Barberà S, Sonnenschein H, Zhou L (1991) Voting by committees. Econometrica 59: 595–609
Berga D, Bergantiños G, Massò J, Neme A (2004) Stability and voting by committees with exit. Soc Choice Welf 23: 229–247
Bogomolnaia A, Jackson M (2002) The stability of hedonic coalition structures. Games Econ Behav 38: 201–230
Bogomolnaia A, Moulin H, Stong R (2005) Collective choice under dichotomous preferences. J Econ Theory 122: 165–184
Bogomolnaia A, Moulin H (2004) Random matching under dichotomous preferences. Econometrica 72(1): 257–279
Bossert W, Pattanaik P, Xu Y (2000) Choice under complete uncertainty: axiomatic characterizations of some decision rules. Econ Theory 16: 295–312
Brams S, Fishburn PC (2002) Voting procedures. In: Arrow KJ, Sen AK, Suzumura K (eds) Handbook of social choice and welfare, vol 1. Elsevier Science, Amsterdam, pp 173–236
Brams S, Fishburn PC (1978) Approval voting. Am Polit Sci Rev 72(3): 831–847
Burani N, Zwicker WS (2003) Coalition formation games with separable preferences. Math Soc Sci 45: 27–52
De Finetti B (1931) Sul significato soggettivo della probabilità. Fundam Math 17: 298–329
Dimitrov D, Borm P, Hendrickx R, Sung SC (2006) Simple priorities and core stability in hedonic games. Soc Choice Welf 26: 421–433
Dimitrov D, Sung SC, Xu Y (2007) Procedural group identification. Math Soc Sci 54(2): 137–146
Fishburn PC (1992) Signed orders and power set extensions. J Econ Theory 56: 1–19
Gale D, Shapley LS (1962) College admisions and the stability of marriage. Am Math Mon 69(1): 9–15
Gärdenfors P (1976) Manipulation of social choice functions. J Econ Theory 13: 217–228
Jones P, Sugden R (1982) Evaluating choice. Int Rev Law Econ 2: 47–65
Ju BG (2003) A characterization of strategy-proof voting rules for separable weak orderings. Soc Choice Welf 21: 469–499
Ju BG (2005) An efficiency characterization of plurality social choice on simple preference domains. Econ Theory 26: 115–128
Kelly JS (1977) Strategy-proofness and social choice functions without single-valuedness. Econometrica 45: 439–446
Kraft CH, Pratt JW, Seidenberg A (1959) Intuitive probability on finite sets. Ann Math Statist 30: 408–419
Pattanaik PK (1978) Strategy and group choice. North-Holland, Amsterdam
Pattanaik PK, Xu Y (1990) On ranking opportunity sets in terms of freedom of choice. Rech Econ Louv 56: 383–390
Roth AE, Sotomayor M (1990) Two-sided matching: a study in game-theoretic modeling and analysis. Cambridge University Press, Cambridge
Romero-Medina A (2001) More on preferences and freedom. Soc Choice Welf 18: 179–191
Samet D, Schmeidler D (2003) Between liberalism and democracy. J Econ Theory 110: 213–233
Scott D (1964) Measurement structures and linear inequalities. J Math Psychol 1: 233–247
Sugden R (1998) The metric of opportunity. Econ Philos 14: 307–337
Open Access
This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
José C. R. Alcantud acknowledges financial support from the Spanish Ministerio de Ciencia e Innovación (Project ECO2009-07682), and from Junta de Castilla y León (Project SA024A08 and GR-99 Funding). R. Arlegi (corresponding author) acknowledges financial support from the Spanish Ministerio de Ciencia e Innovación (Projects SEJ2006-11510 and ECO2009-12836), and from Junta de Castilla y León (Project VA040A05). We are grateful for constructive comments and suggestions by an anonymous referee.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Alcantud, J.C.R., Arlegi, R. An axiomatic analysis of ranking sets under simple categorization. SERIEs 3, 227–245 (2012). https://doi.org/10.1007/s13209-011-0061-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13209-011-0061-8