Preference relation

A binary relation that represents how an agent ranks pairs of options. Its properties determine whether choices can be compared, ordered, or made consistently.

Binary relation: A relation that associates elements of one set with elements of another, or with the same set. Preference relations are binary relations over a set of options.
Utility function: A function assigning numerical values to outcomes so that higher values represent greater preference. It can represent a preference relation when the ranking satisfies suitable conditions.
Completeness and transitivity: Two axioms requiring every pair to be comparable and rankings to remain consistent across chains of comparisons. Together they are common rationality conditions, not automatic features of every preference relation.
Consumer theory: The economic analysis of how consumers allocate resources among goods and services. Preference rankings help derive consumer choices under budget constraints.
Completeness (mathematics): A property requiring every pair of elements in a domain to be comparable under a relation. Completeness means an agent can rank either option at least as highly as the other.
Choice function: A rule that selects one or more alternatives from each eligible set of options. It translates rankings into selections from available alternatives.
Cyclic preference: A ranking pattern in which an option is preferred to a second, the second to a third, and the third to the first. It demonstrates how a relation can violate transitivity and frustrate consistent ranking.
Pareto efficiency: A state in which no feasible change can improve someone’s outcome without worsening another’s. Welfare comparisons depend on how individual preferences rank feasible outcomes.
Transitive relation: A relation in which holding from one element to a second and from the second to a third implies it holds from the first to the third. Transitivity rules out preference cycles across three options.
Revealed preference: An inference about an agent's rankings drawn from observed choices among available alternatives. Observed choices can provide evidence about an otherwise unobserved preference relation.

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