<p>The von Neumann-Morgenstern (vNM) utility function takes the form 
\begin{equation}
U(p,x)=\sum_{i=1}^np_iu(x_i)
\end{equation}
where $x=(x_1,\dots,x_n)$ with $x_i$ being the (monetary) payoff associated with outcome $i$ and $p=(p_1,\dots,p_n)$ with $p_i$ being the probability that $i$ occurs. </p>

<p>In behavioral economics, generalizations of the vNM utility usually happen in either (or both) of the following two channels:</p>

<ol>
<li><p>Non-linear probability weighting: instead of using $p_i$'s, which can be considered objective probabilities, a model may use <em>decision weights</em> $w_i$'s which map the objective probability distribution into a new distribution. </p></li>
<li><p>Reference dependent utility function: instead of $u(x_i)$, which depends only on the payoff in outcome $i$, a model may consider a utility function $u(x_i,r)$ that depends on both $x_i$ and some reference payoff $r$. </p></li>
</ol>

<p>Modifications made through either of these will give rise to a non-expected utility function, which is supposed to improve the model's descriptive accuracy of people's decision under risk. </p>

<h1>Example of 1: Rank-Dependent Utility</h1>

<p>Suppose $x_1&lt;x_2&lt;\cdots&lt;x_n$, and calculate decision weights ($w_i$'s) as follows: 
\begin{equation}
w_i=\pi\left(\sum_{j=i}^np_j\right)-\pi\left(\sum_{j=i+1}^np_j\right),
\end{equation}
where $\pi(\cdot)$ is called a probability weighting function (PWF). 
A common PWF is the Tversky-Kahneman PWF: 
\begin{equation}
\pi(t)=\frac{s^\gamma}{(s^\gamma+(1-s)^\gamma)^{1/\gamma}}.
\end{equation}
Then the rank-dependent utility with T-K PWF is 
\begin{equation}
RDU(p,x;\gamma)=\sum_{i=1}^nw_iu(x_i).
\end{equation}
Note that $w_i=p_i$ if $\gamma=1$, so that vNM EU is a special case of RDU.</p>

<h1>Example of 2: Disappointment Aversion</h1>

<p>\begin{equation}
DAU(p,x;\overline U)=\sum_{i=1}^np_i[u(x_i)-D(u(x_i)-\overline U)],
\end{equation}
where $D(\cdot)$ is a function that captures how disappointment ($u(x_i)&lt;\overline U$) or elation ($u(x_i)&gt;\overline U$) affects an individual's evaluation of a prospect. Here, the reference utility level $\overline U$ can be something like the certainty equivalent level of utility. EU can also be derived as a special case from DAU by requiring that $D(\cdot)$ be a constant function.</p>

<h1>Example of 1+2: Prospect Theory</h1>

<p>Prospect theory (PT) incorporates both non-linear probability weighting and reference dependence. Again, assume $x_1&lt;\cdots&lt;x_n$. PT distinguishes between gains ($x_i&gt;r$) and losses ($x_i&lt;r$). For gains, decision weights are given by 
\begin{equation}
w_i=\pi^+(p_i+\cdots+p_n)-\pi^+(p_{i+1}+\cdots+p_n),
\end{equation}
while for losses the decision weights are 
\begin{equation}
w_i=\pi^-(p_i+\cdots+p_n)-\pi^-(p_{i+1}+\cdots+p_n),
\end{equation}
where $\pi^+(\cdot)$ and $\pi^-(\cdot)$ are both PWFs. For instance, one can have both PWFs be the Tversky-Kahneman form, but with different parameters. In addition, the utility function also takes a reference dependent form: 
\begin{equation}
u(x_i,r)=
\begin{cases}
(x_i-r)^\alpha &amp;\text{if }x_i\ge r\\
-\lambda(r-x_i)^\beta &amp;\text{if }x_i&lt;r
\end{cases}
\end{equation} 
where $\alpha,\beta\in(0,1)$ are parameters of risk aversion (differentiated between gains and losses) and $\lambda&gt;0$ is the coefficient of loss aversion. 
Thus, the non-expected utility function under PT is 
\begin{equation}
V(p,x;\text{parameters})=\sum_{i=1}^nw_iu(x_i,r)
\end{equation}</p>

CC BY-SA 3.0
2017-11-26T13:08:54.187Z
Thank you :). I have a thought: Can we always formulate a von neumann utility function with additional variables that is equivalent to the utility functions you mention?
user56834
CC BY-SA 3.0
2017-11-26T13:14:04.477Z
@Programmer2134 I'm afraid I don't understand what you mean
Herr K.
CC BY-SA 3.0
2017-11-26T16:55:33.117Z
Sorry. The functions that you mention are functions of $x$ and $p$ that are non linear in $p$ (i.e. non von Neumann). My question is, is it possible to generate a third variable and then formulate a von Neumann utility function that takes variables $x$ and $p$, but that takes that third variable, such that it is equivalent to the non-von neumann function?
user56834
CC BY-SA 3.0
2017-11-26T18:15:05.810Z
@Programmer2134: Well, RDU is sort of a one-parameter-extension. Also note that with PT, you can set $\pi^+(p)=\pi^-(p)=p$ and $\alpha=\beta$ to get a non-EU with loss aversion. Are these the sort of extension you have in mind?
Herr K.
CC BY-SA 4.0
2021-12-03T14:14:07.090Z
Does prospect theory apply only to decision under risk?
Papayapap
CC BY-SA 4.0
2021-12-04T03:23:59.153Z
@Papayapap: Prospect theory has also been used to analyze decision under (Knightian) uncertainty, if that's what you mean. See for example, [Tversky and Kahneman (1992)](http://cemi.ehess.fr/docannexe/file/2780/tversjy_kahneman_advances.pdf).
Herr K.
CC BY-SA 3.0
2017-11-26T09:32:49.717Z
2017-11-26T18:35:45.750Z
2017-11-26T18:35:45.750Z
Herr K.
<p>A utility function that does not incorporate the probabilities associated with the outcomes cannot be expressed as an expected value of the $p\cdot u(x_1) + (1-p) \cdot u(x_2)$ form. Therefore all such utilities are non-vNM.</p>

<p>The most familiar case may be Leontief preferences. Assuming that you get outcome $x_1$ with probability $p$ or outcome $x_2$ with probability $1-p$, your utility would be $\min(x_1,x_2)$. This cannot be expressed as expected value. It has a nice interpretation though: A person with such preferences is an absolute pessimist. No matter the actual probabilites, he is sure he will get the worst possible outcome.</p>

<p>Similarly an extreme optimist would have $U(x_1,x_2) = \max(x_1,x_2)$.</p>

CC BY-SA 3.0
2017-11-24T23:33:28.540Z
2017-11-24T23:33:28.540Z
Giskard
<p>Von-Neumann Morgenstern preferences are preferences over lotteries that can be represented as the expectation of a "deterministic" utility function over outcomes. </p>

<p>"non-von-Neumann-Morgenstern" preferences are therefore preferences that cannot be described this way, so that we cannot apply expected utility maximization. </p>

<p>I know of at least one non-von-Neumann Morgenstern preference relation, namely so called "EZ preferences".</p>

<p><strong>What are some other non-von-Neumann-Morgenstern preferences that are actually used in economics (or some other field)?</strong></p>

CC BY-SA 3.0
2017-11-26T19:01:28.277Z
Actually, von Neumann and Morgenstern only considered preferences over lotteries with known "objective" probabilities. So even subjective expected utility theory is outside the domain of vNM expected utility theory.
Michael Greinecker
<microeconomics><decision-theory><expected-utility>
What are some non-von-Neumann-Morgenstern preferences used in economics?
user56834
# What are some non-von-Neumann-Morgenstern preferences used in economics?

(Asked by: username_0 on 2017-11-24)

Von-Neumann Morgenstern preferences are preferences over lotteries that can be represented as the expectation of a "deterministic" utility function over outcomes.

"non-von-Neumann-Morgenstern" preferences are therefore preferences that cannot be described this way, so that we cannot apply expected utility maximization.

I know of at least one non-von-Neumann Morgenstern preference relation, namely so called "EZ preferences".

**What are some other non-von-Neumann-Morgenstern preferences that are actually used in economics (or some other field)?**

**Tags:** microeconomics, decision-theory, expected-utility

> **Comments:**
> **username_1**: Actually, von Neumann and Morgenstern only considered preferences over lotteries with known "objective" probabilities. So even subjective expected utility theory is outside the domain of vNM expected utility theory.  

---

## 2 answers

### Answer by username_2 on 2017-11-26. Score: 4

The von Neumann-Morgenstern (vNM) utility function takes the form
\begin{equation}
U(p,x)=\sum\_{i=1}^np\_iu(x\_i)
\end{equation}
where $x=(x\_1,\dots,x\_n)$ with $x\_i$ being the (monetary) payoff associated with outcome $i$ and $p=(p\_1,\dots,p\_n)$ with $p\_i$ being the probability that $i$ occurs.

In behavioral economics, generalizations of the vNM utility usually happen in either (or both) of the following two channels:

1. Non-linear probability weighting: instead of using $p\_i$'s, which can be considered objective probabilities, a model may use *decision weights* $w\_i$'s which map the objective probability distribution into a new distribution.
2. Reference dependent utility function: instead of $u(x\_i)$, which depends only on the payoff in outcome $i$, a model may consider a utility function $u(x\_i,r)$ that depends on both $x\_i$ and some reference payoff $r$.

Modifications made through either of these will give rise to a non-expected utility function, which is supposed to improve the model's descriptive accuracy of people's decision under risk.

# Example of 1: Rank-Dependent Utility

Suppose $x\_1

# Example of 2: Disappointment Aversion

\begin{equation}
DAU(p,x;\overline U)=\sum\_{i=1}^np\_i[u(x\_i)-D(u(x\_i)-\overline U)],
\end{equation}
where $D(\cdot)$ is a function that captures how disappointment ($u(x\_i)<\overline U$) or elation ($u(x\_i)>\overline U$) affects an individual's evaluation of a prospect. Here, the reference utility level $\overline U$ can be something like the certainty equivalent level of utility. EU can also be derived as a special case from DAU by requiring that $D(\cdot)$ be a constant function.

# Example of 1+2: Prospect Theory

Prospect theory (PT) incorporates both non-linear probability weighting and reference dependence. Again, assume $x\_1<\cdotsr$) and losses ($x\_i0$ is the coefficient of loss aversion.
Thus, the non-expected utility function under PT is
\begin{equation}
V(p,x;\text{parameters})=\sum\_{i=1}^nw\_iu(x\_i,r)
\end{equation}

> **Comments:**
> **username_0**: Thank you :). I have a thought: Can we always formulate a von neumann utility function with additional variables that is equivalent to the utility functions you mention?  
> **username_2**: @Programmer2134 I'm afraid I don't understand what you mean  
> **username_0**: Sorry. The functions that you mention are functions of $x$ and $p$ that are non linear in $p$ (i.e. non von Neumann). My question is, is it possible to generate a third variable and then formulate a von Neumann utility function that takes variables $x$ and $p$, but that takes that third variable, such that it is equivalent to the non-von neumann function?  
> **username_2**: @Programmer2134: Well, RDU is sort of a one-parameter-extension. Also note that with PT, you can set $\pi^+(p)=\pi^-(p)=p$ and $\alpha=\beta$ to get a non-EU with loss aversion. Are these the sort of extension you have in mind?  
> **username_3**: Does prospect theory apply only to decision under risk?  
> **username_2**: @username_3: Prospect theory has also been used to analyze decision under (Knightian) uncertainty, if that's what you mean. See for example, [Tversky and Kahneman (1992)](http://cemi.ehess.fr/docannexe/file/2780/tversjy_kahneman_advances.pdf).  

---

### Answer by username_4 on 2017-11-24. Score: 1

A utility function that does not incorporate the probabilities associated with the outcomes cannot be expressed as an expected value of the $p\cdot u(x\_1) + (1-p) \cdot u(x\_2)$ form. Therefore all such utilities are non-vNM.

The most familiar case may be Leontief preferences. Assuming that you get outcome $x\_1$ with probability $p$ or outcome $x\_2$ with probability $1-p$, your utility would be $\min(x\_1,x\_2)$. This cannot be expressed as expected value. It has a nice interpretation though: A person with such preferences is an absolute pessimist. No matter the actual probabilites, he is sure he will get the worst possible outcome.

Similarly an extreme optimist would have $U(x\_1,x\_2) = \max(x\_1,x\_2)$.


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