# Convex preferences

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 greater-than-or-equal-to 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 convex-shaped 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 quasi-concave utility functions, although these are not necessary for the analysis of preferences.