Utility Representation of Lotteries
So I figured I'd post the proof for the third condition now that I know it.


  We want to show that if $U$ and $V$ are linear representations of $\succeq$ on  $\mathscr{L}$, then $\exists \ \beta &gt; 0, \gamma \in \mathbb{R}$ such that $V = \beta U + \gamma$


Case 1: All the lotteries are all equally preferred.

If $\forall \ L_1, L_2 \in \mathscr{L}$ we also have $L_1 \sim L_2$, then $U$ is constant.

$V = \beta U$ holds $\forall \beta &gt; 0, \gamma  &gt; 0$

Case 2: All the lotteries are not equally preferred.

$\forall L \in \mathscr{L}$ where $L \neq L_b, L_w$
$$L_b \succeq L \succeq L_w$$
$$L_b \succ L_w$$

For a lottery $L \in \mathscr{L}$,

$$\lambda_L \equiv \frac{U(L) - U(L_w)}{U(L_b) - U(L_w)}$$

for $\lambda \in [0,1]$

$$1 - \lambda_L \equiv \frac{U(L_b) - U(L)}{U(L_b) - U(L_w)}$$

So now we can write $U(L), \ \forall L \in \mathscr{L}$ as

$$U(L) = \lambda_L U(L_b) + (1-\lambda_L)U(L_w)$$
$$\implies L \sim \lambda_L L_b + (1-\lambda_L)L_w$$

For $V$ to also represent $\succeq$,

$$V(L) = \lambda_L V(L_b) + (1-\lambda_L)V(L_w)$$

$$V(L) = \bigg[\frac{U(L) - U(L_w)}{U(L_b) - U(L_w)}\bigg] V(L_b) + \bigg[\frac{U(L_b) - U(L)}{U(L_b) - U(L_w)}\bigg] V(L_w)$$

$$V(L) = \frac{U(L)V(L_b) - U(L_w)V(L_b)}{U(L_b) - U(L_w)} + \frac{U(L_b)V(L_w) - U(L)V(L_w)}{U(L_b) - U(L_w)}$$

$$V(L) = \frac{V(L_b) - V(L_w)}{U(L_b) - U(L_w)} \cdot U(L) + \frac{U(L_b)V(L_w) - U(L)V(L_b)}{U(L_b) - U(L_w)}$$

Define $\frac{V(L_b) - V(L_w)}{U(L_b) - U(L_w)} \equiv \beta$ and $\frac{U(L_b)V(L_w) - U(L)V(L_b)}{U(L_b) - U(L_w)} \equiv \gamma$

Thus $V(L) = \beta U(L) + \gamma$, $\forall L$

$\square$