# Congruence relations on a choice space

- 67 Downloads

## Abstract

A choice space is a finite set of alternatives endowed with a map associating to each menu a nonempty subset of selected items. A congruence on a choice space is an equivalence relation that preserves its structure. Intuitively, two alternatives are congruent if the agent is indifferent between them, and, in addition, her choice is influenced by them in exactly the same way. We give an axiomatic characterization of the notion of congruence in terms of three natural conditions: binary fungibility, common destiny, and repetition irrelevance. Further, we show that any congruence satisfies the following desirable properties: (hereditariness) it induces a well-defined choice on the quotient set of equivalence classes; (reflectivity) the primitive behavior can be always retrieved from the quotient choice, regardless of any feature of rationality; (consistency) all basic axioms of choice consistency are preserved back and forth by passing to the quotient. We also prove that the family of all congruences on a choice space forms a lattice under set-inclusion, having equality as a minimum, and a unique maximum, called revealed indiscernibility. The latter relation can be seen as a limit form of revealed similarity as the agent’s rationality increases.

## Notes

### Acknowledgements

The authors are very grateful to two anonymous referees for their valuable comments, which determined a substantial improvement in the overall presentation of the topic. The second author also wishes to thank Alessio E. Biondo for some fruitful discussions.

## References

- Alcantud JCR (2002) Revealed indifference and models of choice behavior. J Math Psychol 46:418–430CrossRefGoogle Scholar
- Aleskerov F, Bouyssou D, Monjardet B (2007) Utility maximization, choice and preference. Springer, BerlinGoogle Scholar
- Apesteguía J, Ballester MA (2013) Choice by sequential procedures. Games Econ Behav 77:90–99CrossRefGoogle Scholar
- Arrow KJ (1959) Rational choice functions and orderings. Economica 26:121–127CrossRefGoogle Scholar
- Au PH, Kawai K (2011) Sequentially rationalizable choice with transitive rationales. Games Econ Behav 73(2):608–614CrossRefGoogle Scholar
- Bandyopadhyay T, Sengupta K (1991) Revealed preference axioms for rational choice. Econ J 101(405):202–213CrossRefGoogle Scholar
- Bandyopadhyay T, Sengupta K (1993) Characterization of generalized weak orders and revealed preference. Econ Theor 3:571–576CrossRefGoogle Scholar
- Blyth T, Janowitz M (1972) Residuation theory. Pergamon Press, OxfordGoogle Scholar
- Cantone D, Giarlotta A, Greco S, Watson S (2016) \((m, n)\)-rationalizable choices. J Math Psychol 73:12–27CrossRefGoogle Scholar
- Cantone D, Giarlotta A, Watson S (2017a) On the preservation of formulae by congruence relations in a choice space. University of Catania, MimeoGoogle Scholar
- Cantone D, Giarlotta A, Watson S (2017b) The satisfiability for Boolean set theory with a choice correspondence. Proceedings of the eight international symposium on games, automata, logics and formal verification (GandALF 2017), 20–22 September, Rome, ItalyGoogle Scholar
- Chambers CP, Echenique F (2016) Revealed preference theory (Econometric Society Monographs). Cambridge University Press, CambridgeCrossRefGoogle Scholar
- Cherepanov V, Feddersen T, Sandroni A (2013) Rationalization. Theor Econ 8:775–800CrossRefGoogle Scholar
- Chernoff H (1954) Rational selection of decision functions. Econometrica 22:422–443CrossRefGoogle Scholar
- Debreu G (1960) Review of “Individual choice behavior: a theoretical analysis” by Luce R. D American Economic Review 50:186–188Google Scholar
- Doignon JP, Falmagne JC (1999) Knowledge spaces. Springer, BerlinCrossRefGoogle Scholar
- Eliaz K, Ok EA (2006) Indifference or indecisiveness? Choice-theoretic foundations of incomplete preferences. Games Econ Behav 56:61–86CrossRefGoogle Scholar
- Falmagne JC, Doignon JP (2011) Learning spaces. Springer, BerlinCrossRefGoogle Scholar
- Fishburn PC (1970) Intransitive indifference with unequal indifference intervals. J Math Psychol 7:144–149CrossRefGoogle Scholar
- Fishburn PC (1985) Interval orders and interval graphs. Wiley, NewYorkCrossRefGoogle Scholar
- Giarlotta A (2014) A genesis of interval orders and semiorders: transitive NaP-preferences. Order 31:239–258CrossRefGoogle Scholar
- Giarlotta A (2015) Normalized and strict NaP-preferences. J Math Psychol 66:34–40CrossRefGoogle Scholar
- Giarlotta A, Greco S (2013) Necessary and possible preference structures. J Math Econ 42(1):163–172CrossRefGoogle Scholar
- Giarlotta A, Watson S (2014) The pseudo-transitivity of preference relations: Strict and weak \((m, n)\)-Ferrers properties. J Math Psychol 58:45–54CrossRefGoogle Scholar
- Giarlotta A, Watson S (2016) Universal semiorders. J Math Psychol 73:80–93CrossRefGoogle Scholar
- Giarlotta A, Watson S (2017a) Well-graded families of NaP-preferences. J Math Psychol 77:21–28CrossRefGoogle Scholar
- Giarlotta A, Watson S (2017b) Necessary and possible indifferences. J Math Psychol 81:98–109CrossRefGoogle Scholar
- Giarlotta A, Watson S (2018) Strict \((m, 1)\)-Ferrers properties. J Math Psychol 82:84–96CrossRefGoogle Scholar
- Hansson B (1968) Choice spaces and preference relations. Synthese 18:443–458CrossRefGoogle Scholar
- Herzberger H (1973) Ordinal preference and rational choice. Econometrica 41(2):187–237CrossRefGoogle Scholar
- Houthakker HS (1950) Revealed preference and the utility function. Economica 17:159–174CrossRefGoogle Scholar
- Jamison DT, Lau LJ (1973) Semiorders and the theory of choice. Econometrica 41:901–912CrossRefGoogle Scholar
- Jamison DT, Lau LJ (1975) Semiorders and the theory of choice: a correction. Econometrica 43:975–977CrossRefGoogle Scholar
- Johnson MR, Dean RA (2001) Locally complete path independent choice functions and their lattices. Math Soc Sci 42:53–87CrossRefGoogle Scholar
- Kalai G, Rubinstein A, Spiegler R (2002) Rationalizing choice functions by multiple rationales. Econometrica 70(6):2481–2488CrossRefGoogle Scholar
- Krause D, Coelho A (2005) Identity, indiscernibility, and philosophical claims. Axiomathes 15(2):191–210CrossRefGoogle Scholar
- Leibniz GW (1966) Logical papers. Translated and edited by G. H. R. Parkinson. Clarendon Press, OxfordGoogle Scholar
- Lleras JS, Masatlioglu Y, Nakajima D, Ozbay EY (2017) When more is less: limited consideration. J Econ Theory 170:70–85CrossRefGoogle Scholar
- Luce RD (1956) Semiorders and a theory of utility discrimination. Econometrica 24:178–191CrossRefGoogle Scholar
- Luce RD (1959) Individual choice behavior: a theoretical analysis. Wiley, New YorkGoogle Scholar
- Luce DR, Raiffa H (1957) Games and decisions: introduction and critical survey. Wiley, New YorkGoogle Scholar
- Mandler M (2009) Indifference and incompleteness distinguished by rational trade. Games Econ Behav 67:300–314CrossRefGoogle Scholar
- Manzini P, Mariotti M (2007) Sequentially rationalizable choice. Am Econ Rev 97:1824–1839CrossRefGoogle Scholar
- Masatlioglu Y, Nakajima D, Ozbay EY (2012) Revealed attention. Am Econ Rev 102(5):2183–2205CrossRefGoogle Scholar
- McFadden DL (1974) Conditional logit analysis of qualitative choice behavior. In: Zarembka P (ed) Frontiers in econometrics. Academic Press, New York, pp 105–142Google Scholar
- Moulin H (1985) Choice functions over a finite set: a summary. Soc Choice Welf 2:147–160CrossRefGoogle Scholar
- Pawlak Z (1982) Rough sets. Int J Inform Comput Sci 11:341–356CrossRefGoogle Scholar
- Pawlak Z (1991) Rough sets. Theoretical aspects of reasoning about data. Kluwer Academic Publisher, DordrechtGoogle Scholar
- Pirlot M, Vincke P (1997) Semiorders: properties, representations, applications. Kluver, DordrechtCrossRefGoogle Scholar
- Plott CR (1973) Path independence, rationality, and social choice. Econometrica 41(6):1075–1091CrossRefGoogle Scholar
- Ribeiro M, Riella G (2017) Regular preorders and behavioral indifference. Theory Decis 82(1):1–12CrossRefGoogle Scholar
- Richter MK (1966) Revealed preference theory. Econometrica 34:635–645CrossRefGoogle Scholar
- Samuelson P (1938) A note on the pure theory of consumer’s behavior. Economica 5:61–71CrossRefGoogle Scholar
- Schwartz T (1976) Choice functions, “rationality” conditions, and variations on the weak axiom of revealed preference. J Econ Theory 13:414–427CrossRefGoogle Scholar
- Sen A (1969) Quasi-transitivity, rational choice and collective decisions. Rev Econ Stud 36:381–393CrossRefGoogle Scholar
- Sen A (1971) Choice functions and revealed preferences. Rev Econ Stud 38:307–317CrossRefGoogle Scholar
- Sen A (1986) Social choice theory. In: Arrow KJ, Intriligator MD (Eds.), Handbook of mathematical economics, vol. III, Elsevier Science Publishers, North-Holland, pp. 1073–1181Google Scholar
- Sen A (1993) Internal consistency of choice. Econometrica 61:495–521CrossRefGoogle Scholar
- Suzumura K (1976) Rational choice and revealed preference. Rev Econ Stud 43:149–158CrossRefGoogle Scholar
- Suzumura K (1983) Rational choice, collective decisions, and social welfare. Cambridge University Press, CambridgeCrossRefGoogle Scholar
- Tyson CJ (2013) Behavioral implications of shortlisting procedures. Soc Choice Welf 41:941–963CrossRefGoogle Scholar
- Ward M (1942) The closure operators of a lattice. Ann Math 43:191–196CrossRefGoogle Scholar