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:
- If $D_0$ is the price of a lottery ticket, how could it possibly be less than zero?
- 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"
- 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.
Best Answer
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.