Preferences are said to satisfy convexity if, whenever any two consumption vectors \(y\) and \(z\) that are at least as desirable as a consumption vector \(x\) are chosen, every consumption vector obtained by mixing \(y\) and \(z\) in any proportion is also guaranteed to be at least as desirable as \(x\). Preferences satisfying convexity can be characterized by quasi-concave utility functions.

A consumption vector that is at least as desirable as every other consumption vector is called a satiation point. A preference relation satisfies nonsatiation or local nonsatiation if there is no satiation point in the consumption set.

A preference relation satisfies monotonicity if the consumer prefers to consume more of every good. When there exists a utility function representing the preference relation, the preference relation is monotonic if and only if the utility function is monotonically increasing.

A preference relation is said to satisfy continuity if, for every consumption vector, both the strict upper contour set and the strict lower contour set are open sets in the consumption set. Under the continuity assumption, the consumer's preferences are guaranteed to vary continuously. Moreover, preferences represented by a continuous utility function satisfy continuity.

Completeness and transitivity may appear to be natural and reasonable assumptions. In practice, however, there can be various situations in which consumers' preferences fail to be rational. This article examines situations in which consumer preferences fail to satisfy rationality, including the Condorcet paradox and framing effects.

A preference relation is rational if it satisfies both completeness and transitivity. This article explains the independence of these two properties, the definition of rational preferences, preference ordering, and the relationship between rational preferences and utility functions.