# Utility of a lottery

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.

## 2 Answers

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:

$U(L) = \frac{1}{50}U(10) + \frac{1}{2000000}U(1000000) \approx \frac{1}{5} U(1) + \frac{1}{2000000}U(1000000).$

• Thank you for your observation, I corrected the question. But having no prior information about the linearity of the utility function, I am not allowed to consider that I am already in $S_{k-1}$ once I chose to play, so the final estimated utility would be $U(S_{k-1}) + \frac{1}{50} U(S_{k+9}) + \frac{1}{2000000} U(S_{k+999999})$, right? Because maybe the possibility of going into state $S_{k-1}$ is 0. Generally, the equation that I used is the correct one, right? – user42768 Jun 9 '16 at 20:21

paying $1 dollar for a ticket is a certainty.$\frac{1}{50}U(S_{k+10}) + \frac{1}{2000000}U(S_{k+1000000}) + U(S_{k-1})>U(S_{k})$And,$U(S_{k}) = 0$and we have been told that we can assume$U(S_{k+10}) = 10 U(S_{k+1})$It doesn't seem a stretch to say,$U(S_{k-1}) = - U(S_{k+1})$Playing the lottery is rational when$\frac{1}{5}U(S_{k+1}) + \frac{1}{2000000}U(S_{k+1000000}) - U(S_{k+1})>0$What the book has done is said$U(L) = \frac{1}{5}U(S_{k+1}) + \frac{1}{2000000}U(S_{k+1000000})$And playing the lottery would be rational when$U(L) > U(S_{k+1})$• Thank you for your response. But, considering, as you said, that, after deciding to play, I have k-1 dollars, isn't the utility at which I may arrive$U(k-1) + \frac1{50}(U(k+9)-U(k-1)) + \frac{1}{2000000}(U(k+999999)-U(k-1))$, where I used U(k) for$U(S_k)$? – user42768 Jun 9 '16 at 20:37 • Something like that. – Doug M Jun 9 '16 at 20:45 • I think it is easiest to read if you let k = 1;$\frac1{50} U(10) + \frac1{2000000} U(1000000)>U(1)$Or set your initial condition to$k+1$dollars. The linear assumption in a neighborhood of k gives you and out. But, you are correct, it should be$U(999,999)\$ for the jackpot. – Doug M Jun 9 '16 at 20:55
• So the first inequality in your response is not correct? As Jacob Hansen answered, the rationality of the decision must be considered in the situation in which you are given 1 dollar and you have the possibility of buying a lottery ticket with it or not (at least that is how it is done in the book). – user42768 Jun 9 '16 at 20:56