# Expected utility and St. Petersburg paradox

Can someone explain to me how they get the $10.94$ at the Expected utility theory section of the solutions to the St. Petersburg paradox?

My problem is that they use a formula to calculate the Expected value but I dont know how they get $10.94$ if the person has $1000000$ as wealth. And how to simulate this example with a Utility function $\mathbb {U} = \ln(x)$ in Excel.

The line begins with :: For example, with $\log$ utility a millionaire should be willing to pay up to $10.94$.

• Exclamation points!!!!! – Asaf Karagila Feb 10 '14 at 12:30
• This is a possible duplicate with math.stackexchange.com/questions/566414/…. Can you explain precisely what makes your question different from the above? Or why the answer to the above is not enough for your to answer your question? – Martin Van der Linden Feb 10 '14 at 13:03

So the answer to your question is in Maximum amount willing to gamble given utility function $U(W)=\ln(W)$ and $W=1000000$ in the game referred to in St. Petersberg's Paradox?

The value of $F$ is found by finding the solution to the equation

$\displaystyle \sum_{h=0}^{\infty}\ln(1000000-F+2^h)\cdot\left(\frac{1}{2}\right)^{h+1}=\ln(1000000)$

So you must compute the value of $\displaystyle \sum_{h=0}^{\infty}\ln(1000000-F+2^h)\cdot\left(\frac{1}{2}\right)^{h+1}$ for different values of $F$ and try to get closer and closer to the value of $F$ that will make $\displaystyle \sum_{h=0}^{\infty}\ln(1000000-F+2^h)\cdot\left(\frac{1}{2}\right)^{h+1} = \ln(1000000)$.

This can be done with a spreadsheet software like OpenCalc or excel. If you are stuck with how to do this precisely, you should start by learning the basics of these spreadsheet software. Google is your friend if your are lost (type "excel basics tutorial" or something?) or think about other SE site such as https://superuser.com/.

• You are welcome. As you seem to be new to SE websites, you might want to know that you can mark an answer as "accepted" by clicking the "v" below the arrows for upvote/downvote. This indicates that you are ok with the answer and do not need further help with the question. Do it ONLY if the answer is satisfactory to you. If it is not explain why and what else you need help with. – Martin Van der Linden Feb 10 '14 at 17:19
• But which value should I take for F. – user2970322 Feb 10 '14 at 17:21
• I do not know how to explain it better than in my answer " compute the value of the sum for different values of F and try to get closer and closer to the value of F that will make the equation hold". Then chose the value of $F$ which makes you closest from satisfying the equation with equality. The level of accuracy is up to your preferences and the time you are ready to put in repeating iterations. – Martin Van der Linden Feb 10 '14 at 18:46