<p>A lottery is essentially a probability distribution. So the variance of a lottery is the variance of the probability distribution it represents, which is given by the formula 
\begin{equation}
\mathrm{Var}(L)=\sum_{i=1}^np_i(x_i-\mathbb E(L))^2,
\end{equation}
where $p_i$ is the probability that outcome $i$ occurs, $x_i$ is the monetary value of outcome $i$, and $\mathbb E(L)$ is the expected value of the lottery. </p>

<p>Take the lottery $L_2=0.5[0]\oplus0.5[4]$ for example. We can easily verify that $\mathbb E(L_2)=2$. By the formula above, then, 
\begin{equation}
\mathrm{Var}(L_2)=0.5(0-2)^2+0.5(4-2)^2=4.
\end{equation}</p>

<p>I'll leave you to finish the rest of the question.</p>

CC BY-SA 4.0
2018-06-26T22:47:40.727Z
2018-06-26T22:47:40.727Z
Herr K.
<p>Can anybody give me a hint how to solve the following problem?</p>

<p>*One way to construct preferences over lotteries with monetary prizes is by
evaluating each lottery L on the basis of two numbers, Ex(L), the expectation
of L and var(L), L’s variance. Such a procedure may or may not be consistent
with vNM assumptions.</p>

<p>Show that u(L) = Ex(L) − (1/4)var(L) induces a preference relation that
is not consistent with the vNM assumptions. (For example, consider
the mixtures of each of the lotteries [1] and 0.5[0] ⊕ 0.5[4] with the
lottery 0.5[0] ⊕ 0.5[2].)*</p>

<p>I am especially not sure about how to compute the variance in this case.</p>

<microeconomics>
Preferences over lotteries
hermanzegerman
# Preferences over lotteries

(Asked by: username_0 on 2018-06-26)

Can anybody give me a hint how to solve the following problem?

\*One way to construct preferences over lotteries with monetary prizes is by
evaluating each lottery L on the basis of two numbers, Ex(L), the expectation
of L and var(L), L’s variance. Such a procedure may or may not be consistent
with vNM assumptions.

Show that u(L) = Ex(L) − (1/4)var(L) induces a preference relation that
is not consistent with the vNM assumptions. (For example, consider
the mixtures of each of the lotteries [1] and 0.5[0] ⊕ 0.5[4] with the
lottery 0.5[0] ⊕ 0.5[2].)\*

I am especially not sure about how to compute the variance in this case.

**Tags:** microeconomics

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## 1 answers

### Answer by username_1 on 2018-06-26. Score: 1 | Accepted answer

A lottery is essentially a probability distribution. So the variance of a lottery is the variance of the probability distribution it represents, which is given by the formula
\begin{equation}
\mathrm{Var}(L)=\sum\_{i=1}^np\_i(x\_i-\mathbb E(L))^2,
\end{equation}
where $p\_i$ is the probability that outcome $i$ occurs, $x\_i$ is the monetary value of outcome $i$, and $\mathbb E(L)$ is the expected value of the lottery.

Take the lottery $L\_2=0.5[0]\oplus0.5[4]$ for example. We can easily verify that $\mathbb E(L\_2)=2$. By the formula above, then,
\begin{equation}
\mathrm{Var}(L\_2)=0.5(0-2)^2+0.5(4-2)^2=4.
\end{equation}

I'll leave you to finish the rest of the question.


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