Suppose the prize space (in dollars) is $\mathbb{Z}$ = {1, 2, 3, 4, 5, 6, 7, 8} and consider choices by an agent whose preferences (over lotteries) satisfy the von Neumann-Morgenstern axioms.

A risk averse von Neumann-Morgenstern agent (with increasing utility) has to compare the following two gambles

p = ($\frac{1}{8}$, $\frac{1}{8}$, $\frac{1}{8}$,$\frac{1}{8}$,$\frac{1}{8}$,$\frac{1}{8}$,$\frac{1}{8}$,$\frac{1}{8}$)

q=($\frac{2}{8}$,$0$, $\frac{1}{8}$,$\frac{1}{8}$,$\frac{1}{8}$,$\frac{1}{8}$,$0$,$\frac{2}{8}$)

Question is: Can you say unambiguously which one she would prefer?

I am thinking that if I can show one gamble second order stochastically dominates the other I can find the answer but I could not do that. Any help will be appreciated thanks!

  • $\begingroup$ Are you familiar with any of the definitions of risk aversion? How about applying one or the other? $\endgroup$
    – Giskard
    Oct 22 '21 at 21:46
  • $\begingroup$ Can you give me a hint? $\endgroup$
    – Mrnobody
    Oct 22 '21 at 22:51
  • $\begingroup$ Sure. Look up the definition in your textbook. $\endgroup$
    – Giskard
    Oct 23 '21 at 4:34
  • $\begingroup$ I looked up the definition but still it didn't help enough. I know a risk averse individual has concave utility function but I couldn't adapt it here. $\endgroup$
    – Mrnobody
    Oct 23 '21 at 17:01
  • $\begingroup$ The expected value is the same. What about variance? How is variance related to a risk-averse utility? $\endgroup$
    – user141240
    Oct 25 '21 at 6:24

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