In Prospect Theory (2010; Cambridge UP), Peter P. Wakker has an exercise assignment 3.3.6 without solution in the book and I'm really unsure about this one. The exercise states on pages 76-77:

Assignment 3.3.6. This assignment demonstrates that risk aversion implies arbitrage. More generally, it shows that every deviation from risk neutrality implies arbitrage. You may assume that risk neutrality for all 50-50 prospects implies complete risk neutrality. Show that arbitrage is possible whenever there is no risk neutrality, for instance as soon as there is strict risk aversion. A difficulty in this assignment is that we have defined risk neutrality for decision under risk, and arbitrage has been defined for decision under uncertainty. You, therefore, have to understand §2.1-2.3 to be able to do this assignment.

This is in the context of classical expected utility theory (not prospect theory or RDU).

I can set up a scenario in which there are consecutive prospects with the same expected value that the decision maker is willing to swap. For example, assume that the decision maker is risk-averse and the utility function is $U(x)=\sqrt{x}$. Consider only 50-50 prospects and write only the respective gains. Let $x_1$=(\$90, \$10), $x_2$=(\$80, \$20), $x_3$=(\$70, \$30), $x_4$=(\$60, \$40), $x_5$=(\$50, \$50). The expected value is \$50 for each, but we get $EU(x_1)\approx 6.32$, $EU(x_2)\approx 6.71$, $EU(x_3)\approx 6.92$, $EU(x_4)\approx 7.04$, ...

That doesn't show arbitrage, though, it merely reflects the decision maker's willingness to swap prospects with higher risks for those in which the risk is spread more evenly.

How do I show arbitrage?

What puzzles me about this exercise is that there seem to be many no-arbitrage theorems for expected utility and that expected utility is generally accepted as a rationality principle (as opposed to only allowing expected value). But, if I understand correctly, arbitrage in finance is the same as Dutch books, and these are generally taken as arguments for the irrationality of some decision making (e.g. when preferences are cyclic). The bottomline is that I really don't understand this exercise.

N.B.: This is not homework and I'm not an economist. I'm a philosopher who wants to understand Wakker's book.



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