Can a game with negative expectation still have a positive utility?

Question

Intuitively, I think not. But I can't clearly prove why. Specifically, I've been thinking about lottery games, where the expectation is obviously negative. But can the utility of hitting the jackpot ever be enough to justify playing?

Answer 1

Imagine the following game: You toss a fair coin, if heads you get a buck, if tails you pay $2$ bucks. Clearly the expectation is $-0.5\ $bucks . We now define the following utility function: $u(x)=(x+2)^3$. Notice if you don't play the expected utility is $8$, on the other hand the expected utility if you play is $\frac{0+3^3}{2}=\frac{27}{2}=13.5$. So this gives us a positive utility.

Answer 2

A common assumption about a utility function is that it is concave down: $U(tx + (1-t)y) \geq tU(x) + (1-t)U(y)$ for any $x,y$ and $0 \leq t \leq 1$. One can prove that any concave down function also satisfies the condition that, if $p_i$, $1 \leq i \leq n$ are nonnegative numbers with $\sum_i p_i = 1$, then for any $x_i$ we have:

$$U(\sum_ip_ix_i) \geq \sum_i p_i U(x_i)$$

In other words, if $E$ is the expectation of the game and $E_U$ the expected utility:

$$U(E) \geq E_U$$

It is also generally assumed that $U$ is increasing, and $U(0) = 0$. Hence if $E < 0$ one has $E_U \leq U(E) < U(0) = 0$, so the game has negative expected utility.

If one does not assume that utility is concave down this result does not hold, as shown in Jorge's answer.