# VNM Theorem

TagLast edit: 22 Sep 2020 18:47 UTC by

The VNM theorem is one of the classic results of Bayesian decision theory. It establishes that, under four assumptions known as the VNM axioms, a preference relation must be representable by maximum-expectation decision making over some real-valued utility function. (In other words, rational decision making is best-average-case decision making.)

Starting with some set of outcomes, gambles (or lotteries) are defined recursively. An outcome is a gamble, and for any finite set of gambles, a probability distribution over those gambles is a gamble.

Preferences are then expressed over gambles via a preference relation. if is preferred to , this is written . We also have indifference, written . If is either preferred to or indifferent with , this can be written .

The four VNM axioms are:

1. Completeness. For any gambles and , either , , or .

2. Transitivity. If and , then .

3. Continuity. If , then there exists a probability such that . In other words, there is a probability which hits any point between two gambles.

4. Independence. For any and , we have if and only if . In other words, substituting for in any gamble can’t make that gamble worth less.

In contrast to Utility Functions, this tag focuses specifically on posts which discuss the VNM theorem itself.

# Money pump­ing: the ax­io­matic approach

5 Nov 2009 11:23 UTC
26 points

# Why you must max­i­mize ex­pected utility

13 Dec 2012 1:11 UTC
48 points

# VNM ex­pected util­ity the­ory: uses, abuses, and interpretation

17 Apr 2010 20:23 UTC
35 points

# VNM agents and lot­ter­ies in­volv­ing an in­finite num­ber of pos­si­ble outcomes

21 Feb 2013 21:58 UTC
26 points

# Con­se­quen­tial­ists: One-Way Pat­tern Traps

16 Jan 2023 20:48 UTC
47 points

# Con­se­quences of ar­bi­trage: ex­pected cash

13 Nov 2009 10:32 UTC
16 points

# Gen­er­al­iz­ing Foun­da­tions of De­ci­sion Theory

4 Mar 2017 16:46 UTC
12 points

# Ver­ify­ing vNM-ra­tio­nal­ity re­quires an ontology

13 Mar 2019 0:03 UTC
24 points

# Con­ti­nu­ity ax­iom of vNM

30 Jul 2014 16:27 UTC
5 points

# Com­pu­ta­tional effi­ciency rea­sons not to model VNM-ra­tio­nal prefer­ence re­la­tions with util­ity functions

25 Jul 2018 2:11 UTC
16 points

# An At­tempt at Prefer­ence Uncer­tainty Us­ing VNM

16 Jul 2013 5:20 UTC
15 points