Von Neumann–Morgenstern Utility Theorem

In decision theory, the von Neumann–Morgenstern (VNM) utility theorem shows that, under certain axioms of rational behavior, a decision-maker faced with risky (probabilistic) outcomes of different choices will behave as if he or she is maximizing the expected value of some function defined over the potential outcomes at some specified point in the future. This function is known as the von Neumann–Morgenstern utility function. The theorem is the basis for expected utility theory.

In 1947, John von Neumann and Oskar Morgenstern proved that any individual whose preferences satisfied four axioms has a utility function; Neumann, John von and Morgenstern, Oskar, ''Theory of Games and Economic Behavior''. Princeton, NJ. Princeton University Press, 1953. such an individual's preferences can be represented on an interval scale and the individual will always prefer actions that maximize expected utility. That is, they proved that an agent is (VNM-)rational ''if and only if'' there exists a real-valued function ''u'' defined by possible outcomes such that every preference of the agent is characterized by maximizing the expected value of ''u'', which can then be defined as the agent's ''VNM-utility'' (it is unique up to adding a constant and multiplying by a positive scalar). No claim is made that the agent has a "conscious desire" to maximize ''u'', only that ''u'' exists.

The expected utility hypothesis is that rationality can be modeled as maximizing an expected value, which given the theorem, can be summarized as "''rationality is VNM-rationality''". However, the axioms themselves have been critiqued on various grounds, resulting in the axioms being given further justification.

VNM-utility is a ''decision utility'' in that it is used to describe ''decision preferences''. It is related but not equivalent to so-called ''E-utilities'' (experience utilities), notions of utility intended to measure happiness such as that of Bentham's Greatest Happiness Principle.

Set-up

In the theorem, an individual agent is faced with options called ''lotteries''. Given some mutually exclusive outcomes, a lottery is a scenario where each outcome will happen with a given probability, all probabilities summing to one. For example, for two outcomes ''A'' and ''B'',

::$L = 0.25A + 0.75B$

denotes a scenario where ''P''(''A'') = 25% is the probability of ''A'' occurring and ''P''(''B'') = 75% (and exactly one of them will occur). More generally, for a lottery with many possible outcomes ''A_i'', we write:

::$L = \sum p_i A_i,$

with the sum of the $p_i$s equalling 1.

The outcomes in a lottery can themselves be lotteries between other outcomes, and the expanded expression is considered an equivalent lottery: 0.5(0.5''A'' + 0.5''B'') + 0.5''C'' = 0.25''A'' + 0.25''B'' + 0.50''C''.

If lottery ''M'' is preferred over lottery ''L'', we write $M \succ L$, or equivalently, $L \prec M$. If the agent is indifferent between ''L'' and ''M'', we write the ''indifference relation'' Kreps, David M. ''Notes on the Theory of Choice''. Westview Press (May 12, 1988), chapters 2 and 5. $L\sim M.$ If ''M'' is either preferred over or viewed with indifference relative to ''L'', we write $L \preceq M.$

The axioms

The four axioms of VNM-rationality are then ''completeness'', ''transitivity'', ''continuity'', and ''independence''.

Completeness assumes that an individual has well defined preferences:

:Axiom 1 (Completeness) For any lotteries ''L,M'', at least one of the following holds:

::$\, L\succeq M$, $\, M\succeq L$

(the individual must express ''some'' preference or indifferenceImplicit in denoting indifference by equality are assertions like if $L\prec M = N$ then $L\prec N$. To make such relations explicit in the axioms, Kreps (1988) chapter 2 denotes indifference by $\,\sim$, so it may be surveyed in brief for intuitive meaning.). Note that this implies reflexivity.

Transitivity assumes that preferences are consistent across any three options:

:Axiom 2 (Transitivity) If $\,L \succeq M$ and $\,M \succeq N$, then $\,L \succeq N$.

Continuity assumes that there is a "tipping point" between being ''better than'' and ''worse than'' a given middle option:

:Axiom 3 (Continuity): If $\,L \preceq M\preceq N$, then there exists a probability \,p\in ,1/math> such that
::$\,pL + (1-p)N\, \sim \,M$

where the notation on the left side refers to a situation in which ''L'' is received with probability ''p'' and ''N'' is received with probability (1–''p'').

Instead of continuity, an alternative axiom can be assumed that does not involve a precise equality, called the Archimedean property. It says that any separation in preference can be maintained under a sufficiently small deviation in probabilities:

:Axiom 3′ (Archimedean property): If $\,L \prec M\prec N$, then there exists a probability $\,\varepsilon\in(0,1)$ such that

::$\,(1-\varepsilon)L + \varepsilon N\, \prec \,M \, \prec \,\varepsilon L + (1-\varepsilon)N.$

Only one of (3) or (3′) need to be assumed, and the other will be implied by the theorem.

Independence of irrelevant alternatives assumes that a preference holds independently of the possibility of another outcome:

:Axiom 4 (Independence): For any $\,N$ and $\,p\in(0,1]$,
::$\,L\preceq M\qquad \text\qquad pL+(1-p)N \preceq pM+(1-p)N.$
::

Note that the "only if" direction is necessary for the theorem to work. Without that, we have this counterexample: there are only two outcomes $A, B$, and the agent is indifferent on $\$, and strictly prefers all of them over $A$. With the "only if" direction, we can argue that $\frac 12 A + \frac 12 B \succeq \frac 12 B + \frac 12 B$ implies $A \succeq B$, thus excluding this counterexample.

The independence axiom implies the axiom on reduction of compound lotteries:

:Axiom 4′ (Reduction of compound lotteries): For any lotteries $L, L', N, N'$ and any p, q \in ,1/math>,

:: $\text \qquad L\sim qL'+(1-q)N',$
:: $\text \quad pL+(1-p)N \sim pqL'+ p(1-q)N' + (1-p)N.$

To see how Axiom 4 implies Axiom 4', set $M = qL'+(1-q)N'$ in the expression in Axiom 4, and expand.

The theorem

For any VNM-rational agent (i.e. satisfying axioms 1–4), there exists a function ''u'' which assigns to each outcome ''A'' a real number ''u(A)'' such that for any two lotteries,

::$L\prec M \qquad \mathrm \qquad E(u(L)) < E(u(M)),$

where ''E(u(L))'', or more briefly ''Eu''(''L'') is given by

::$Eu(p_1A_1 + \cdots + p_nA_n) = p_1u(A_1) + \cdots + p_nu(A_n).$

As such, ''u'' can be uniquely determined (up to adding a constant and multiplying by a positive scalar) by preferences between ''simple lotteries'', meaning those of the form ''pA'' + (1 − ''p'')''B'' having only two outcomes. Conversely, any agent acting to maximize the expectation of a function ''u'' will obey axioms 1–4. Such a function is called the agent's von Neumann–Morgenstern (VNM) utility.

Proof sketch

The proof is constructive: it shows how the desired function $u$ can be built. Here we outline the construction process for the case in which the number of sure outcomes is finite.

Suppose there are ''n'' sure outcomes, $A_1\dots A_n$. Note that every sure outcome can be seen as a lottery: it is a degenerate lottery in which the outcome is selected with probability 1. Hence, by the Completeness and Transitivity axioms, it is possible to order the outcomes from worst to best:
:$A_1\preceq A_2\preceq \cdots \preceq A_n$

We assume that at least one of the inequalities is strict (otherwise the utility function is trivial—a constant). So $A_1\prec A_n$. We use these two extreme outcomes—the worst and the best—as the scaling unit of our utility function, and define:

:$u(A_1)=0$ and $u(A_n)=1$

For every probability p\in ,1/math>, define a lottery that selects the best outcome with probability $p$ and the worst outcome otherwise:
:$L(p) = p\cdot A_n + (1-p)\cdot A_1$
Note that $L(0)\sim A_1$ and $L(1)\sim A_n$.

By the Continuity axiom, for every sure outcome $A_i$, there is a probability $q_i$ such that:
:$L(q_i) \sim A_i$

and
:$0 = q_1\leq q_2\leq \cdots \leq q_n = 1$

For every $i$, the utility function for outcome $A_i$ is defined as
:$u(A_i)=q_i$

so the utility of every lottery $M=\sum_i p_i A_i$ is the expectation of ''u'':
:$u(M) = u\left(\sum_i p_i A_i \right) = \sum_i p_i u(A_i) = \sum_i p_i q_i$

To see why this utility function makes sense, consider a lottery $M = \sum_i p_i A_i$, which selects outcome $A_i$ with probability $p_i$. But, by our assumption, the decision maker is indifferent between the sure outcome $A_i$ and the lottery $q_i\cdot A_n + (1-q_i)\cdot A_1$. So, by the Reduction axiom, he is indifferent between the lottery $M$ and the following lottery:
:M' = \sum_i p_i _i\cdot A_n + (1-q_i)\cdot A_1
:$M' = \left(\sum_i p_i q_i \right) \cdot A_n + \left(\sum_i p_i(1-q_i)\right)\cdot A_1$
:$M' = u(M)\cdot A_n + (1-u(M))\cdot A_1$
The lottery $M'$ is, in effect, a lottery in which the best outcome is won with probability $u(M)$, and the worst outcome otherwise.

Hence, if $u(M)>u(L)$, a rational decision maker would prefer the lottery $M$ over the lottery $L$, because it gives him a larger chance to win the best outcome.

Hence:
::$L\prec M \;$ if and only if $E(u(L)) < E(u(M)).$

Reaction

Von Neumann and Morgenstern anticipated surprise at the strength of their conclusion. But according to them, the reason their utility function works is that it is constructed precisely to fill the role of something whose expectation is maximized:

"Many economists will feel that we are assuming far too much ... Have we not