Convex preferences

In economics, convex preferences are an individual's ordering of various outcomes, typically with regard to the amounts of various goods consumed, with the property that, roughly speaking, "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 for real numbers, the notation below can be translated as: 'is at least as good as' (in preference satisfaction). Similarly, can be translated as 'is strictly better than' (in preference satisfaction), and Similarly, can be translated as 'is equivalent to' (in preference satisfaction). Use x, y, and z to denote three consumption bundles (combinations of various quantities of various goods). Formally, a preference relation on the consumption set X is called convex if whenever where and , then for every : i.e., for any two bundles that are each viewed as being at least as good as a third bundle, a weighted average of the two bundles is viewed as being at least as good as the third bundle. A preference relation is called strictly convex if whenever where , , and , then for every : i.e., for any two distinct 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 strictly better than the third bundle. Use x and y to denote two consumption bundles. A preference relation is called convex if for any where then for every : That is, if a bundle y is preferred over a bundle x, then any mix of y with x is still preferred over x. A preference relation is called strictly convex if whenever where , and , then for every : That is, for any two bundles that are viewed as being equivalent, a weighted average of the two bundles is better than each of these bundles.

1. If there is only a single commodity type, then any weakly-monotonically-increasing preference relation is convex. This is because, if , then every weighted average of y and ס is also .