Convex preferences

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Convex preferences refer to a property of utility functions commonly represented in an indifference curve as a bulge toward the origin. It roughly corresponds to the "law" of diminishing marginal utility but uses modern theory to represent the concept. Comparable to the greater-than-or-equal-to ordering relation $\geq$ for real numbers, the notation $\succeq$ below can be translated as: 'is as at least as good as' (in preference satisfaction). Formally, if $\succeq$ is a preference relation on the consumption set X, then $\succeq$ is convex if for any $x, y, z \in X$ where $y \succeq x$ and $z \succeq x$ , then it is the case that $\theta y + (1-\theta) z \succeq x$ for any $\theta \in [0,1]$ .

$\succeq$ is strictly convex if for any $x, y, z \in X$ where $y \succeq x$ and $z \succeq x$ , and $y \neq z$ then it is also true that $\theta y + (1-\theta) z \succ x$ for any $\theta \in (0,1)$ . It can be translated as: 'is better than relation' (in preference satisfaction).