In economics, convex preferences refer to a property of an individual's ordering of various outcomes which roughly corresponds to the idea that "averages are better than the extremes". The concept roughly corresponds to the concept of diminishing marginal utility without requiring utility functions.
Comparable to the greaterthanorequalto ordering relation \geq for real numbers, the notation \succeq below can be translated as: 'is at least as good as' (in preference satisfaction). Use x, y, and z to denote three consumption bundles (combinations of various quantities of various goods). Formally, a preference relation P on the consumption set X is convex if for any

x, y, z \in X where y \succeq x and z \succeq x ,
it is the case that

\theta y + (1\theta) z \succeq x for any \theta \in [0,1] .
That is, the preference ordering P is convex if for any two goods bundles that are each viewed as being at least as good as a third bundle, a weighted average of the two bundles is also viewed as being at least as good as the third bundle.
Moreover, P is strictly convex if for any

x, y, z \in X where y \succeq x , z \succeq x , and y \neq z,
it is also true that

\theta y + (1\theta) z \succ x for any \theta \in (0,1);
here \succ can be translated as 'is better than' (in preference satisfaction). Thus the preference ordering P is strictly convex if for any two distinct goods bundles that are each viewed as being at least as good as a third bundle, a weighted average of the two bundles (including a positive amount of each bundle) is viewed as being better than the third bundle.
A set of convexshaped indifference curves displays convex preferences: Given a convex indifference curve containing the set of all bundles (of two or more goods) that are all viewed as equally desired, the set of all goods bundles that are viewed as being at least as desired as those on the indifference curve is a convex set.
Convex preferences with their associated convex indifference mapping arise from quasiconcave utility functions, although these are not necessary for the analysis of preferences.
References

Hal R. Varian; Intermediate Microeconomics A Modern Approach. New York: W. W. Norton & Company. ISBN 0393927024

MasColell, Andreu; Whinston, Michael; & Green, Jerry (1995). Microeconomic Theory. Oxford: Oxford University Press. ISBN 9780195073409
See also
This article was sourced from Creative Commons AttributionShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov and content contributors is made possible from the U.S. Congress, EGovernment Act of 2002.
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a nonprofit organization.