Suppose, for example, that a decision maker can choose any probabilities ${p}_{0}$, ${p}_{1}$, ${p}_{2}$ that he or she wants for specified dollar outcomes

${D}_{0}$

and that they have a given expected value

${p}_{0}{D}_{0}+{p}_{1}{D}_{1}+{p}_{2}{D}_{2}=k$

For example, if ${D}_{0}<0$ were the price of a lottery ticket with possible prizes ${D}_{1}$ and ${D}_{2}$, then $k=0$ would define a “fair” lottery, while $k<0$ would afford the lottery organizer a profit. We may arbitrarily let the utilities of ${D}_{0}$ and ${D}_{2}$ be ${u}_{0}=0$ and ${u}_{2}=1$; then the utility of ${D}_{1}$ is ${u}_{1}\in (0,1)$. For a typical lottery, |${D}_{0}$| is quite small as compared to ${D}_{1}$ and ${D}_{2}$. With $k\le 0$, this implies that feasible ${p}_{1}$ and ${p}_{2}$ are small, with ${p}_{1}+{p}_{2}$ well under $0.5$, and therefore with ${p}_{0}$ well over $0.5$.

Questions:

1. If ${D}_{0}$ is the price of a lottery ticket, how could it possibly be less than zero?

2. Why include the price of a lottery ticket in an EV calculation? The prizes ${D}_{1}$ and ${D}_{2}$ have a probability associated with them, that makes sense when calculating expected value. But the price of a lottery ticket? What does it mean for a ticket price to have a probability "well over $0.5$"

3. For $k\le 0$, it only makes sense that ${D}_{0}$ must be negative, but again, how could the price of a lottery ticket be negative?