Expected Utility Maximization

Question

This is from Markowitz's Risk-Return Analysis: The Theory and Practice of Rational Investing (Volume One) Chapter 1.

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

$D_0$ < $D_1$ < $D_2$

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 $u_1$ ∈ (0,1). For a typical lottery, |$D_0$| is quite small as compared to $D_1$ and $D_2$. With $k$ ≤ 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.

My 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 $k$ ≤ 0, it only makes sense that $D_0$ must be negative, but again, how could the price of a lottery ticket be negative? What am I not understanding here? If $D_0$ were a positive return with a given probability, then it wouldn't be possible for $k$ to be less than zero, in which case, how would one define a "fair" or a profitable lottery? I'm so confused. Surely I'm reading this wrong.

Answer 1

What you're not understanding is the fact that we're calculating your expected winnings. So what your author means is that if you buy a lottery ticket at a price $|D_0|$ (which is positive), then if you don't win the lottery you'll have earned $D_0$ (which is negative), your overall wealth decreased by $|D_0|$.

So it seems like we're modelling the scenario where a lottery ticket costs $|D_0|$ and with probability $p_1$, I gain some amount money $M_1>0$. In this case, I've paid for the ticket and won $M_1$, so my gains have been $D_1:=M_1+D_0$ (note again that $D_0$ is negative). Similarly, with probability $p_2$, I gain some greater amount of money $M_2>0$, in which case I'll have gained $D_2:=M_2+D_0$ - I've still paid for my ticket.

Thus, with probability $p_0:=1-(p_1+p_2)$, I'll have entered the lottery and not won, and thus lost the money that I paid for the ticket. Therefore, my expected winnings are as written above. The important part is to keep in mind which quantity we're actually keeping track of - in this case, it's your wealth after the lottery is over.

Answer 2

The $D$s are the payment to the player by the casino. So $D_0<0$ represents the fact that the payment to the player is negative, i.e., the player pays the casino $\vert D_0\vert$. I think that answers 1 and 3.

Regarding 2 - the probabilities represent the events "not winning", "winning the net amount $D_1$ and winning the net amount $D_2$. So "not winning" has a probability of $p_0$ in which case the player only plays the lottery ticket.