Abstract
A weak (strict) preference relation is continuous if it has a closed (open) graph; it is hemicontinuous if its upper and lower contour sets are closed (open). If preferences are complete these four conditions are equivalent. Without completeness continuity in each case is stronger than hemicontinuity. This paper provides general characterizations of continuity in terms of hemicontinuity for weak preferences that are modeled as (possibly incomplete) preorders and for strict preferences that are modeled as strict partial orders. Some behavioral implications associated with the two approaches are also discussed.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aliprantis CD, Border KC (2006) Infinite dimensional analysis, 3rd edn. Springer, Berlin
Bergstrom TC, Parks RP, Rader T (1976) Preferences which have open graphs. J Math Econ 3: 265–268
Bewley TF (1986/2002) Knightian decision theory. Part I. Cowles foundation discussion paper no. 807, 1986. Reprinted in (2002) Decis Econ Finance 25:79–110
Bridges DS, Mehta GB (1995) Representations of preference orderings. Springer, Berlin
Dubra J (2011) Continuity and completeness under risk. Math Soc Sci 61: 80–81
Dubra J, Maccheroni F, Ok EA (2004) Expected utility theory without the completeness axiom. J Econ Theory 115: 118–133
Gerasimou G (2010) Consumer theory with bounded rational preferences. J Math Econ 46: 708–714
Ghirardato P, Maccheroni F, Marinacci M, Siniscalchi M (2003) A subjective spin on roulette wheels. Econometrica 71: 1897–1908
Harary F (1961) A parity relation partitions its field distinctly. Am Math Mon 68: 215–217
Manzini P, Mariotti M (2008) On the representation of incomplete preferences over risky alternatives. Theory Decis 65: 303–323
Nachbin L (1950) Topologia e Ordem. Printed by the University of Chicago Press. English edition (1965) Topology and order. Princeton, Van Nostrand
Schmeidler D (1969) Competitive equilibria in markets with a continuum of traders and incomplete preferences. Econometrica 37: 578–585
Schmeidler D (1971) A condition for the completeness of partial preference relations. Econometrica 39: 403–404
Shafer WJ (1974) The nontransitive consumer. Econometrica 42: 913–919
Stecher JD (2008) Existence of approximate social welfare. Soc Choice Welf 30: 43–56
Steen LA, Seebach JA (1995) Counterexamples in Topology. Dover, Mineola
Ward LE (1954) Partially ordered topological spaces. Proc Am Math Soc 5: 144–161
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicated to Konstantinos Gerasimou.
Rights and permissions
About this article
Cite this article
Gerasimou, G. On continuity of incomplete preferences. Soc Choice Welf 41, 157–167 (2013). https://doi.org/10.1007/s00355-012-0673-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00355-012-0673-3