# Preferences over lotteries

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  and 0.5 ⊕ 0.5 with the lottery 0.5 ⊕ 0.5.)*

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

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 $L_2=0.5\oplus0.5$ 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}