VNM Theorem

TagLast edit: 22 Sep 2020 18:47 UTC by abramdemski

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

Stuart_Armstrong5 Nov 2009 11:23 UTC
26 points
93 comments5 min readLW link

Why you must max­i­mize ex­pected utility

Benya13 Dec 2012 1:11 UTC
48 points
77 comments21 min readLW link

VNM ex­pected util­ity the­ory: uses, abuses, and interpretation

Academian17 Apr 2010 20:23 UTC
35 points
51 comments10 min readLW link

VNM agents and lot­ter­ies in­volv­ing an in­finite num­ber of pos­si­ble outcomes

AlexMennen21 Feb 2013 21:58 UTC
26 points
8 comments3 min readLW link

Con­se­quen­tial­ists: One-Way Pat­tern Traps

David Udell16 Jan 2023 20:48 UTC
47 points
3 comments14 min readLW link

Con­se­quences of ar­bi­trage: ex­pected cash

Stuart_Armstrong13 Nov 2009 10:32 UTC
16 points
28 comments3 min readLW link

Gen­er­al­iz­ing Foun­da­tions of De­ci­sion Theory

abramdemski4 Mar 2017 16:46 UTC
12 points
0 comments10 min readLW link

Ver­ify­ing vNM-ra­tio­nal­ity re­quires an ontology

jeyoor13 Mar 2019 0:03 UTC
24 points
5 comments1 min readLW link

Con­ti­nu­ity ax­iom of vNM

Stuart_Armstrong30 Jul 2014 16:27 UTC
5 points
13 comments1 min readLW link

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

AlexMennen25 Jul 2018 2:11 UTC
16 points
5 comments3 min readLW link

An At­tempt at Prefer­ence Uncer­tainty Us­ing VNM

[deleted]16 Jul 2013 5:20 UTC
15 points
33 comments6 min readLW link

Don’t Get Dis­tracted by the Boilerplate

johnswentworth26 Jul 2018 2:15 UTC
52 points
16 comments2 min readLW link

Geo­met­ric Ra­tion­al­ity is Not VNM Rational

Scott Garrabrant27 Nov 2022 19:36 UTC
123 points
22 comments3 min readLW link
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