Lecture notes for Decision Theory

Let X be a nonempty finite set, ∆(X) = { p : X → [0, 1] | Px∈X p(x) = 1}, and let % denote a binary relation on ∆(X). As usual,  and ∼ denote the asymmetric and symmetric parts of % . In our nomenclature elements of X are outcomes (or consequences or prizes), elements of ∆(X) are lotteries, and % is the preference relation. Axioms: A1 the relation % is complete and transitive. A2 (continuity) ∀p, q, r ∈ ∆(X) with p  q  r, there are α, β ∈ (0, 1) such that αp+(1−α)r  q  βp + (1 − β)r. A3 (independence or substitution) ∀p, q, r ∈ ∆(X), and α ∈ (0, 1] : p  q ⇒ αp+(1−α)r  αq+(1−α)r. (Note that for all p, q ∈ ∆(X), αp+(1−α)q ∈ ∆(X), and [αp+(1−α)q](x) = αp(x)+ (1−α)q(x). Obviously P x∈X [αp(x) + (1− α)q(x)] = 1.) Theorem (for lotteries). A binary relation % on ∆(X) satisfies A1, A2, and A3 iff there exists a (utility) function u : X → R such that, p % q ⇔ P x∈X p(x)u(x) ≥ P x∈X q(x)u(x). Moreover, a function v : X → R satisfies p % q ⇔ Px∈X p(x)u(x) ≥ Px∈X q(x)u(x) iff v(x) = αu(x) + β on X for α, β ∈ R , α > 0.