Difference between revisions of "Von Neumann-Morgenstern preferences"
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* <strong>Transitivity:</strong> if <math>\mu \succeq \nu</math> and <math>\nu \succeq \rho</math>, then <math>\mu \succeq \rho</math>. | * <strong>Transitivity:</strong> if <math>\mu \succeq \nu</math> and <math>\nu \succeq \rho</math>, then <math>\mu \succeq \rho</math>. | ||
* <strong>Continuity:</strong> if <math>\mu \succeq \nu \succeq \rho</math>, then there exists <math>p \in [0,1]</math> such that <math>p \mu + (1-p) \rho \sim \nu</math>. | * <strong>Continuity:</strong> if <math>\mu \succeq \nu \succeq \rho</math>, then there exists <math>p \in [0,1]</math> such that <math>p \mu + (1-p) \rho \sim \nu</math>. | ||
| − | * <strong>Independence:</strong> <math>\mu \succeq \nu</math> if and only if <math>p \mu + (1-p) \rho \succeq p \nu + (1-p) \rho</math>. | + | * <strong>Independence of irrelevant alternatives (IIA):</strong> <math>\mu \succeq \nu</math> if and only if <math>p \mu + (1-p) \rho \succeq p \nu + (1-p) \rho</math>. |
The von Neumann-Morgenstern theorem states that any von Neumann-Morgenstern preference is equivalently descrbied by an expected score (called utility). In other words, there exists a utility function <math>u : X \rightarrow \mathbb R</math> such that <math>\mu \succeq \nu</math> if and only if <math>\mathbb E_{x \leftarrow \mu} [u(x)] \geq \mathbb E_{x \leftarrow \nu} [u(x)]</math>. | The von Neumann-Morgenstern theorem states that any von Neumann-Morgenstern preference is equivalently descrbied by an expected score (called utility). In other words, there exists a utility function <math>u : X \rightarrow \mathbb R</math> such that <math>\mu \succeq \nu</math> if and only if <math>\mathbb E_{x \leftarrow \mu} [u(x)] \geq \mathbb E_{x \leftarrow \nu} [u(x)]</math>. | ||
It was also shown that this utility function is nearly unique. It is defined up to a positive affine transformation, i.e., <math>v</math> is a utility function for the von Neumann-Morgenster preference if and only if there exists <math>a \in \mathbb R</math> and <math>b > 0</math> such that <math>v = a+bu</math>. | It was also shown that this utility function is nearly unique. It is defined up to a positive affine transformation, i.e., <math>v</math> is a utility function for the von Neumann-Morgenster preference if and only if there exists <math>a \in \mathbb R</math> and <math>b > 0</math> such that <math>v = a+bu</math>. | ||
Latest revision as of 23:36, 22 February 2020
A Von Neumann-Morgenstern preference VonneumannMorgensternBook44 is a consistent order relation over probabilistic outcomes. The Von Neumann-Morgenstern theorem states that any Von Neumann-Morgenstern preference is equivalently described by expected scores.
Formal theorem
Consider a set [math]X[/math] of outcomes. We denote [math]\Delta(X)[/math] the set of probability distributions over [math]X[/math]. For countable sets [math]X[/math], a probabilistic outcome [math]\mu \in \Delta(X)[/math] is thus a function that assigns a probability [math]\mu(x)[/math] to any outcome [math]x \in X[/math], and that satisfies [math]\sum_{x \in X} \mu(x) = 1[/math].
A preference is a total order relation over [math]\Delta(X)[/math]. To be a von Neumann-Morgenstern preference, it needs to satisfy the following very natural axioms:
- Completeness: [math]\mu \succ \nu[/math], [math]\nu \prec \mu[/math] or [math]\mu \sim \nu[/math].
- Transitivity: if [math]\mu \succeq \nu[/math] and [math]\nu \succeq \rho[/math], then [math]\mu \succeq \rho[/math].
- Continuity: if [math]\mu \succeq \nu \succeq \rho[/math], then there exists [math]p \in [0,1][/math] such that [math]p \mu + (1-p) \rho \sim \nu[/math].
- Independence of irrelevant alternatives (IIA): [math]\mu \succeq \nu[/math] if and only if [math]p \mu + (1-p) \rho \succeq p \nu + (1-p) \rho[/math].
The von Neumann-Morgenstern theorem states that any von Neumann-Morgenstern preference is equivalently descrbied by an expected score (called utility). In other words, there exists a utility function [math]u : X \rightarrow \mathbb R[/math] such that [math]\mu \succeq \nu[/math] if and only if [math]\mathbb E_{x \leftarrow \mu} [u(x)] \geq \mathbb E_{x \leftarrow \nu} [u(x)][/math].
It was also shown that this utility function is nearly unique. It is defined up to a positive affine transformation, i.e., [math]v[/math] is a utility function for the von Neumann-Morgenster preference if and only if there exists [math]a \in \mathbb R[/math] and [math]b \gt 0[/math] such that [math]v = a+bu[/math].
