EconCSLib.Foundation.Utility.VNMAxioms #
The four axioms for expected utility theory, stated as predicates on a preference relation over lotteries.
Main definitions #
VNM.Completeness— every pair of lotteries is comparableVNM.Transitivity— preference is transitiveVNM.Independence— mixing with a common lottery preserves preferenceVNM.Continuity— intermediate mixtures exist [MSZ Axiom 2.12]strict,indiff— shared derived strict preference and indifference
Main results #
VNM.continuity_independent— ∃ preference violating only Continuity [MSZ Ex 2.5]VNM.completeness_independent— ∃ preference violating only CompletenessVNM.transitivity_independent— ∃ preference violating only TransitivityVNM.independence_independent— ∃ preference violating only Independence
References #
- [MSZ] Chapter 2, Axioms 2.12–2.17, Exercise 2.5
Lottery-specific axioms #
strict, indiff, Completeness, Transitivity are defined in
Foundation.Preference for general binary relations. Here we add the two
lottery-specific axioms that reference Lottery.mix.
Independence: mixing both sides with a common lottery preserves preference.
L₁ ≿ L₂ ↔ [α L₁, (1-α) N] ≿ [α L₂, (1-α) N] for α > 0.
Equations
- VNM.Independence pref = ∀ (L₁ L₂ N : ↑(Lottery 𝕜 O)) (α : 𝕜) (hα₀ : 0 < α) (hα₁ : α ≤ 1), pref L₁ L₂ ↔ pref (Lottery.mix α ⋯ hα₁ L₁ N) (Lottery.mix α ⋯ hα₁ L₂ N)
Instances For
Continuity (Archimedean / MSZ Axiom 2.12): for L₁ ≿ L₂ ≿ L₃,
there exists θ ∈ [0,1] such that L₂ ∼ [θ L₁, (1-θ) L₃].
Equations
- VNM.Continuity pref = ∀ (L₁ L₂ L₃ : ↑(Lottery 𝕜 O)), pref L₁ L₂ → pref L₂ L₃ → ∃ (θ : 𝕜) (hθ₀ : 0 ≤ θ) (hθ₁ : θ ≤ 1), indiff pref L₂ (Lottery.mix θ hθ₀ hθ₁ L₁ L₃)
Instances For
Consequences of the axioms #
The Sure-Thing Principle [MSZ Ex 2.12]: The common lottery in a mixture
does not affect strict preference. If Independence holds, then
[α L₁, (1-α) L₃] ≻ [α L₂, (1-α) L₃] iff [α L₁, (1-α) L₄] ≻ [α L₂, (1-α) L₄]
for any lotteries L₁, L₂, L₃, L₄ and α ∈ [0,1].
Exercise 2.5: Independence of the vNM Axioms [MSZ Ex 2.5] #
For each axiom, we construct a preference relation on Lottery ℚ (Fin 3) that
violates that axiom while satisfying the other three.
Counterexample 1: ¬Completeness #
Use the trivial preference L₁ ≿ L₂ ↔ L₁ = L₂. Only identical lotteries are
comparable, so completeness fails. The other three axioms hold trivially or by
injectivity of mixing.
Counterexample 2: ¬Continuity #
Lexicographic preference on probability vectors: compare L(0) first, then L(1).
This is a total order satisfying independence, but no mixture of pure outcomes
A₀ and A₂ is indifferent to pure A₁.
Counterexample 3: ¬Transitivity #
Define L₁ ≿ L₂ iff L₁(0) ≥ L₂(0) OR L₁(1) ≥ L₂(1). This is complete
(for any pair, at least one coordinate comparison holds) and satisfies independence
(linear mixing preserves each coordinate comparison). But it is NOT transitive:
the "or" allows preference chains that don't compose.
Counterexample 4: ¬Independence #
Use a threshold preference: L₁ ≿ L₂ iff L₁(0) ≥ 1/2 or L₂(0) < 1/2.
This partitions lotteries into "high" (L(0) ≥ 1/2) and "low" (L(0) < 1/2);
high is preferred to low, and within each class everything is indifferent.
Mixing can move a lottery across the threshold, violating independence.
Continuity holds because θ=0 or θ=1 always gives indifference.
Main theorem #
Exercise 2.5 [MSZ]: The four vNM axioms are independent. For each axiom, there exists a preference relation on lotteries that violates that axiom while satisfying the other three.