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)?

  • $\begingroup$ 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. $\endgroup$ – Michael Greinecker Nov 26 '17 at 19:01

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 $x_1<x_2<\cdots<x_n$, 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.

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 $x_1<\cdots<x_n$. PT distinguishes between gains ($x_i>r$) and losses ($x_i<r$). 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 &\text{if }x_i\ge r\\ -\lambda(r-x_i)^\beta &\text{if }x_i<r \end{cases} \end{equation} where $\alpha,\beta\in(0,1)$ are parameters of risk aversion (differentiated between gains and losses) and $\lambda>0$ 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}

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  • $\begingroup$ 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? $\endgroup$ – user56834 Nov 26 '17 at 13:08
  • $\begingroup$ @Programmer2134 I'm afraid I don't understand what you mean $\endgroup$ – Herr K. Nov 26 '17 at 13:14
  • $\begingroup$ 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? $\endgroup$ – user56834 Nov 26 '17 at 16:55
  • $\begingroup$ @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? $\endgroup$ – Herr K. Nov 26 '17 at 18:15

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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