Do the following exercise:
An agent, i, entertains preferences over the alternatives in U, which are represented by a transitive and complete preference relation, ≳i. Transitivity requests that a ≳i b and b ≳i c imply a ≳i c for all a, b, c ∈ U, and completeness requires that any pair of alternatives a, b ∈ U is comparable, i.e., it holds that either a ≳i b or b ≳i a or both. A utility function u: U → ℝ is said to represent a preference relation on U if, for all a, b ∈ U, u(a) ≥ u(b) if and only if a ≳ b . Show that, when U is finite, a preference relation can be represented by a utility function if and only if it is it is transitive and complete.
Updated December 04 2020 by FST Course Production Staff