Lê Nguyên Hoang: /* Formal theorem */

2020-02-22T22:36:52Z

Formal theorem

Lê Nguyên Hoang: Created page with "A Von Neumann-Morgenstern preference [https://pdfs.semanticscholar.org/0375/379194a6f34b818962ea947bff153adf621c.pdf VonneumannMorgensternBook][https://scholar.google.ch/schol..."

2020-02-22T22:36:01Z

Created page with "A Von Neumann-Morgenstern preference [https://pdfs.semanticscholar.org/0375/379194a6f34b818962ea947bff153adf621c.pdf VonneumannMorgensternBook][https://scholar.google.ch/schol..."

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A Von Neumann-Morgenstern preference [https://pdfs.semanticscholar.org/0375/379194a6f34b818962ea947bff153adf621c.pdf VonneumannMorgensternBook][https://scholar.google.ch/scholar?hl=en&as_sdt=0%2C5&q=Theory+of+games+and+economic+behaviors+von+neumann+morgenstern&btnG= 44] 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: <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 > 0</math> such that <math>v = a+bu</math>.

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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>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>.
 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>.

Von Neumann-Morgenstern preferences - Revision history

Lê Nguyên Hoang: /* Formal theorem */

Lê Nguyên Hoang: Created page with "A Von Neumann-Morgenstern preference [https://pdfs.semanticscholar.org/0375/379194a6f34b818962ea947bff153adf621c.pdf VonneumannMorgensternBook][https://scholar.google.ch/schol..."