St Petersburg paradox

Question

I am wondering what it means to say: "The gamble is worth $x$ dollars."

Does it mean the expected outcome\value is $x$ dollars?

In the photo (last sentence of paragraph) it says that "the gamble is worth a much more realistic 2 dollars."

(https://i.stack.imgur.com/nMV7o.jpg)

Answer 1

I don´t read that the word "dollar" is mentioned. If you look at the table there are no dollar signs at the utility row. The utility has no unit. But you can transform the dollars into utility, et vice versa.

The utility function is $U(x)=\log_2(x)+2$, where $x$ is the winning in dollars. For instance, if the winning is 4 the corresponding utility is $U(4)=\log_2(4)+2=2+2=4$.

The utility of the winnings for sequence $y$ is $U(y)=y+1$. The corresponding probability is $p(y)=\left(\frac12\right)^y$.

Therefore the expected utility is $\mathbb E(U(y))=\sum\limits_{y=1}^{\infty }p(y)\cdot u(y)=\sum\limits_{y=1}^{\infty }\left(\frac12\right)^y \cdot (y+1)=3$

But the exected value of the winnings (in dollars) is $\mathbb E(w)=\sum\limits_{x=1}^{\infty }p(x)\cdot w(x)=\sum\limits_{x=1}^{\infty }\left(\frac12\right)^x \cdot 2^{x-1}=\infty$

The expected value of the utilities does converge. One main reason is that the marginal utility function $(U'(x))$ is decreasing. But the expected value of the winnings does not converge.

it says that "the gamble is worth a much more realistic 2 dollars.

You will get that number if you insert the value of the expected utility ($\mathbb E(U)=3$) into the equation $U(x)=\log_2(x)+2$. Then solve the equation for $x$.