Risk seeking utility

Question

I am stuck on a question in an archived course on BerkeleyX's CS188x Artificial Intelligence.

Which of the following would be a utility function for a risk-seeking preference? That is, for which utility(s) would Pacman prefer entering a lottery for a random food pellet, with expected size $s$, over receiving a pellet of size $s$?

The utilities are functions of the radius, $r$, of the food pellet:

• $U(r) =$ constant
• $U(r) = r$
• $U(r) = -r$
• $U(r) = r^2$
• $U(r) = -r^2$
• $U(r) = -\sqrt{r}$
• $U(r) = \sqrt{r}$

Answer 1

• Risk-neutral behavior is characterized by linear utility functions
• Risk-averse behavior is characterized by concave utility functions
• Risk-seeking behavior is characterized by convex utility functions You can check this by solving $\sum_i{p_i r_i^x} > \sum_i{p_i r_i}$ (risk-seeking), which results in $x > 1$. Conversely, $\sum_i{p_i r_i^x} < \sum_i{p_i r_i}$ (risk-averse), gives us $x < 1$.

Thus, $U(r) = r^2$ and $U(r) = -\sqrt{r}$ are risk-seeking (see here and here).

As an example, let's suppose we have two possible choices A or B. A has two possible consequences: reward 10 with 0.6 probability and reward 0 with 0.4 probability. B has only one possible outcome: 6. Note that both have the same expected return (6), but A should be preferred by a risk-seeking agent. By using $U(r) = r^2$, the utility of A becomes $0.6 * 100 + 0.4 * 0 = 60$, while the utility of B becomes $1.0 * 36 = 36$. We can do the same with $U(r) = -\sqrt{r}$: $A \approx 0.6 * -3.16 + 0.4 * 0 \approx -1.9$ and $B \approx 1.0 * -2.4 \approx -2.4$. In both cases, A has higher utility.