theorem
staged
Independence Of The VNM Axioms
The vNM axioms are logically independent: for each axiom, there is a preference relation over lotteries that violates that axiom while satisfying the others.
Proof
Sketch
Construct separate finite lottery preference relations. For completeness, use a preference that compares only identical lotteries. For transitivity, create a cycle. For continuity, use a lexicographic-style preference. For independence, use a preference rule whose ranking changes under common mixing. Each example keeps the remaining axioms valid.
References
- [MSZ, Chapter 2, Exercise 2.5] Maschler, Solan, and Zamir, Game Theory. The vNM axioms are independent.