17  The Von Neumann Morgenstern Theorem

Warning

This section contains more advanced material and can be skipped on a first reading.

The Von Neumann Morgenstern Theorem is one of the most important results in rational choice theory. It shows that if a decision maker’s preferences satisfy four axioms, then the decision maker’s preferences are expected utility representable. Moreover, the utility function that represents the decision maker’s preferences is unique up to linear transformations. This means that if two utility functions represent the same preferences, then they are equal up to a linear transformation.

Recall the four axioms discussed in the previous sections.

Preference (Definition 8.3)

Suppose that \(\mathcal{L}\) is a set of lotteries. \((P, I)\) is a rational preference relation over \(\mathcal{L}\).

Compound Lottery Axiom (Chapter 14)

For any lottery \(L\), the decision maker is indifferent between \(L\) and the simplification of \(L\). Formally, if \(I\) represents the decision maker’s indifference relation, then for all lotteries \(L\), \(L\mathrel{I}s(L)\).

Independence Axiom (Chapter 15)

Suppose that \(\mathcal{L}\) is a set of lotteries and \((P, I)\) is a rational preference over \(\mathcal{L}\). For all \(L, L', L''\in \mathcal{L}\) and \(r\in (0, 1]\), \[L\mathrel{P} L'\quad\mbox{if, and only if,}\quad [L:r,\ L'':(1-r)]\mathrel{P} [L':r,\ L'':(1-r)].\] \[L\mathrel{I} L'\quad\mbox{if, and only if,}\quad [L:r,\ L'':(1-r)]\mathrel{I} [L':r,\ L'':(1-r)].\]

Continuity Axiom (Chapter 16)

Suppose that \(\mathcal{L}\) is a set of lotteries and \((P, I)\) is a rational preference over \(\mathcal{L}\). For all \(L, L', L''\in \mathcal{L}\), if \(L\mathrel{P} L'\mathrel{P} L''\), then there exists an \(r\in (0, 1)\) such that \[L'\mathrel{I} [L:r,\ L'':(1-r)].\]

Theorem 17.1 (Von Neumann Morgenstern Representation Theorem) Suppose that \(\mathcal{L}\) is a set of lotteries. Then, \((P, I)\) satisfies Preference, Compound Lotteries, Independence and Continuity if, and only if, \((P, I)\) is represented by a linear utility function (Definition 13.2).

Moreover, \(u\) is unique up to linear transformations: \(u':\mathcal{L}\rightarrow\mathbb{R}\) also represents \((P, I)\) if, and only if, there are real numbers \(c>0\) and \(d\) such that for all lotteries \(L\in\mathcal{L}\), \[u'(L)=c*u(L)+d.\]