A release-checked medium problem for training Games, Discrete Random Variables.
Question
You’re at the bank on a lucky day. The banker will randomly choose a positive integer, and so will you. Both of you can set a probability distribution over the positive integers that you select from. If you both select the same integer, $n$, you win $n^2$ dollars; otherwise, you get nothing. Neither party knows the other's distribution. Assuming optimal strategies from both sides, what is your expected payout? Express your answer in the form $\dfrac{a}{\pi^b}$ for integers $a$ and $b$, and find the product $ab$.
Practice focus
This Probability problem is tagged Games, Discrete Random Variables. State the random variables and conditioning information explicitly, then check the result against boundary cases before opening hints or a solution.