Weak Orders and Utility Functions

A weak order is a complete and transitive order on a set X, whose elements we refer to as preferences. So, x ≽ y is thought of as x being preferred over y. The preference is strict if x ≻ y. A utility function on X is a function u: X → R that describes preferences numerically. We say that u represents the binary relation ≽ on X if x ≽ y ⇔ u(x) ≥ u(y). In other words, the utility function reflects the preference ordering ≽. We will show that if X is finite, then ≽ is a preference relation if and only if it is represented by a utility function. This will utilize the notion of upper and lower contour sets. After presenting a counterexample, we will show how to extend this result to sets such as X = R^n. This will require an additional assumption of continuity, namely that for all x∈X the upper and lower contour sets are closed. We then briefly discuss how weak orders and utility functions arise in expected utility theory and prospect theory, and conclude with some speculation regarding possible future applications in mathematical psychology with regard to stereotype threat theory.