[Question] Expected utility and repeated choices

[Marco Discendenti]

Maybe this is a well known kind of problem but I am a novice and it looks puzzling to me.

Here is a lottery: I have these two choices:

• (a) get 0.5$ for sure

• (b) win 1$ with probability $\frac{2}{3}$ or nothing with probability $\frac{1}{3}$

My utility function is $U\left(x\right)=\sqrt{x}$ .

What should I choose?

Let’s compute the expected utilities:

• expected utility for one single game is for (b) $\frac{2}{3}\cdot \sqrt{1}=\frac{2}{3}\approx 0.67$ while for (a) is $\sqrt{0.5}\approx 0.7$ so I have maximized expected utility with choice (a)

• if I compute expected utility for two games I get a different prescription:

  • utility for chosing (a) two times is $U\left(1\right)=\sqrt{1}=1$

  • the expected utility for chosing (b) two times is

$$P\left(\text{2 wins}\right)U\left(2\right)+P\left(\text{1 win}\right)U\left(1\right)={\left(\frac{2}{3}\right)}^2\cdot \sqrt{2}+2\cdot \frac{2}{3}\cdot \frac{1}{3}\cdot \sqrt{1}$$

This last computation is equal to $\frac{4}{9}(\sqrt{2}+1)\approx 1.07$ which is greater than the utility of double (a) (i.e. 1) so in order to maximize expected utility I should actually prefer to play (b) two times rather than playing (a) two times.

So we have this apparent inconsistency:

• for one single game it’s better to choose (a)

• for two games it’s better to choose (b) both times

This result is puzzling to me because I would expect that utility maximization for one single game should be enough in order to take the decision regardless of what I am allowed to do in future choices. It seems instead that the mere possibility that I could play this same lottery another time changes the convenience of the choices about what to play in the first game. If this is the case then utility theory seems almost useless: I would be forced to put in my computation the whole list of my possible future choices!

Am I missing something or is this an actual problem?