VNM 效用函数

Definition The utility function $u : G \to R$ has the expected utility property if, for every $g \in G$ ,

u (g) = \sum_{i = 1}^{n} p_{i} u (a_{i})

where $(p_{1} a_{1}, \dots, p_{n} a_{n})$ is the simple gamble induced by $g$ .

Theorem Existence of a VNM Utility Function on $G$
Let preference $≿$ over gambles in $G$ satisfy axioms G1 to G6. The there exists a utility function $u : G \to R$ representing $≿$ on $G$ , such that $u$ has the expected utility property.

Theorem VNM Utility Functions are Unique up to Positive Affine Transformations
Suppose that the VNM utility function $u (\cdot)$ represents $≿$ . Then the VNM utility function, $v (\cdot)$ , represents those same preferences if and only if for some scalar $α$ and some scalar $β > 0$ ,

v (g) = α + β u (g)

for all gambles $g$ .