The sentence of a problem from a book:
Tickets to a lottery cost \$1. There are two possible prizes: a \$10 payoff with probability
1/50, and a \$1,000,000 payoff with probability 1/2,000,000. What is the expected monetary
value of a lottery ticket? When (if ever) is it rational to buy a ticket? Be precise—show an
equation involving utilities. You may assume current wealth of \$k and that $U(S_k) = 0$. You
may also assume that $U(S_{k+10}) = 10 × U(S_{k+1})$, but you may not make any assumptions
about $U(S_{k+1,000,000}).
Of course, the expected value of the lottery is
$$\frac1{50}\$10 + \frac1{2000000}\$1000000 = \$0.70$$
Now, for the second part.
I would say that it is rational to buy a ticket if
$$\frac{1}{50}U(S_{k+10}) + \frac{1}{2000000}U(S_{k+1000000}) + (1-\frac{1}{50} – \frac{1}{2000000}) U(S_{k-1})>U(S_{k})$$
By that I mean that the expected utility after buying the ticket (the LHS) should be larger than the utility after deciding not to buy the ticket.
But the answer from the book is quite different:
Although \$0.70 < \$1, it is not necessarily irrational to buy the ticket. First we will consider
just the utilities of the monetary outcomes, ignoring the utility of actually playing the lottery
game. Using $U(S_{k+n})$ to represent the utility to the agent of having n dollars more than the
current state, and assuming that utility is linear for small values of money (i.e., $U(S_{k+n}) ≈
n(U(S_{k+1}) − U(S_k))$ for −10 ≤ n ≤ 10), the utility of the lottery is:
$$U(L) =\frac1{50}U(S_{k+10}) + \frac1{2, 000, 000}U(S+{k+1,000,000})$$
$$≈ \frac15U(S_{k+1}) + \frac1{2, 000, 000}U(Sk+1,000,000)$$
This is more than $U(S_{k+1})$ when $U(S_{k+1,000,000}) > 1, 600, 000U(\$1).$
Why is my judgement wrong?
EDIT:
I would say that it is rational to buy a ticket if
$$\frac{1}{50}U(S_{k+10 {\color{red}{-1}} }) + \frac{1}{2000000}U(S_{k+1000000{\color{red}{-1}}}) + (1-\frac{1}{50} – \frac{1}{2000000}) U(S_{k-1})>U(S_{k})$$
I forgot that, after buying the ticket, I have k-1 dollars.
Best Answer
You have an off-by-one error in some of your utilities: playing and winning the lottery means you have $k+9$ or $k+999999$ dollars if you started with $k$ dollars. Alternately, if you start with $k+1$ dollars, then your outcomes are $k+10$ or $k+1000000$ dollars. The book is comparing the utility of $k+1$ dollars with the utility of the lottery. That is, suppose someone gives you a dollar, and you have the option to either play the lottery or keep the dollar. Then the computation the book does is straightforward from the assumption it says you are allowed to make (modulo a bunch of notational shifts).
It would be easiest to just assume $k=0$ and then write everything like this:
$U(L) = \frac{1}{50}U(10) + \frac{1}{2000000}U(1000000) \approx \frac{1}{5} U(1) + \frac{1}{2000000}U(1000000).$