<p>That's true, but it should be constructed the other way round: The assignment of numbers <span class="math-container">$u_i$</span> to outcomes is not <em>given</em> by the Bernoulli utility function but <em>defines</em> it. The terminology of Bernoulli and Von Neumann-Morgenstern utility functions is common, but not universally used in the context of EUT.</p>

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2020-05-14T09:12:04.080Z
2020-05-14T09:12:04.080Z
VARulle
<p>This is the definition of an expected utility function that I'm working with:</p>

<p>The utility function <span class="math-container">$U:\mathcal{L}\to\mathbb{R}$</span> has an expected utility form if there is an assignment of numbers <span class="math-container">$(u_{1},...,u_{N})$</span> to the <span class="math-container">$N$</span> outcomes <span class="math-container">$\{\omega_{1},...,\omega_{N}\}$</span>, such that the following is satisfied for any simple lottery <span class="math-container">$L=(\omega_{1},...,\omega_{N};p_{1},...,p_{N})\in\mathcal{L}$</span>:
<span class="math-container">\begin{equation}\label{eq:vnm}
    U(L)=\sum_{i=1}^{N}p_{i}u_{i}.
\end{equation}</span>
<strong>My question</strong>: Is it true that the assignment of numbers <span class="math-container">$(u_{1},...,u_{N})$</span> are given by the Bernoulli function <span class="math-container">$u:\Omega\to\mathbb{R}$</span>, where <span class="math-container">$\Omega$</span> denotes the sample space. And is it true that <span class="math-container">$U(L):\mathcal{L}\to\mathbb{R}$</span> is called the Von Neumann-Morgenstern expected utility function, where <span class="math-container">$\mathcal{L}$</span> denotes the set of simple lotteries?</p>

<microeconomics>
Help understanding definition of expected utility function
Miski123
# Help understanding definition of expected utility function

(Asked by: username_0 on 2020-05-14)

This is the definition of an expected utility function that I'm working with:

The utility function $U:\mathcal{L}\to\mathbb{R}$ has an expected utility form if there is an assignment of numbers $(u\_{1},...,u\_{N})$ to the $N$ outcomes $\{\omega\_{1},...,\omega\_{N}\}$, such that the following is satisfied for any simple lottery $L=(\omega\_{1},...,\omega\_{N};p\_{1},...,p\_{N})\in\mathcal{L}$:
\begin{equation}\label{eq:vnm}
U(L)=\sum\_{i=1}^{N}p\_{i}u\_{i}.
\end{equation}
**My question**: Is it true that the assignment of numbers $(u\_{1},...,u\_{N})$ are given by the Bernoulli function $u:\Omega\to\mathbb{R}$, where $\Omega$ denotes the sample space. And is it true that $U(L):\mathcal{L}\to\mathbb{R}$ is called the Von Neumann-Morgenstern expected utility function, where $\mathcal{L}$ denotes the set of simple lotteries?

**Tags:** microeconomics

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## 1 answers

### Answer by username_1 on 2020-05-14. Score: 1 | Accepted answer

That's true, but it should be constructed the other way round: The assignment of numbers $u\_i$ to outcomes is not *given* by the Bernoulli utility function but *defines* it. The terminology of Bernoulli and Von Neumann-Morgenstern utility functions is common, but not universally used in the context of EUT.


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