von Neumann–Morgenstern axioms

completeness

$$\begin{array}{c}\forall A,B:A⪰B\vee B⪰A\end{array}$$

for any two lotteries A and B, either A is at least as good as B, or the other way around (or both). the agent always has an opinion.

transitivity

$$\begin{array}{c}A⪰B,B⪰C\Rightarrow A⪰C\end{array}$$

if A beats B and B beats C, then A beats C. without it, an agent with cyclic preferences can be money-pumped: it pays to swap C→B→A→C forever.

continuity

$$\begin{array}{c}A\succ B\succ C\Rightarrow \exists p\in [0,1]:pA+(1-p)C\sim B\end{array}$$

if B sits strictly between A and C, then there is a probability at which a gamble between the best and the worst feels exactly as good as the middle for sure. no outcome is infinitely good or infinitely bad.

independence

$$\begin{array}{c}A⪰B\Rightarrow pA+(1-p)C⪰pB+(1-p)C\end{array}$$

mixing a third option into both sides with the same probability does not flip the preference. C is irrelevant; only the part where the lotteries differ matters. this is the axiom humans most often violate (see the Allais paradox¹).

the theorem

if preferences satisfy all four, there is a utility function $u$ such that

$$\begin{array}{c}A⪰B⟺\mathbb{E}[u(A)]\ge \mathbb{E}[u(B)]\end{array}$$

a rational agent acts as if it maximizes expected utility. $u$ is unique up to positive affine transformation $au+b,a>0$.²

legend

• $A,B,C$ - lotteries (probability distributions over outcomes).
• $⪰$ - weakly preferred (at least as good as).
• $\succ$ - strictly preferred.
• $\sim$ - indifferent.
• $p$ - probability, $p\in [0,1]$.
• $pA+(1-p)C$ - compound lottery: $A$ with probability $p$, else $C$.
• $\forall$, $\exists$ - for all, there exists.

1. Allais paradox
2. Von Neumann–Morgenstern utility theorem