Generous Banker

You are at the bank and it is your lucky day. The banker is going to pick a random positive integer and so are you. Each of you chooses the probability distribution on the positive integers that you select from, independently. If you and the banker select the same value, say $n$, you receive $n^2$ dollars from the banker. Otherwise, you receive nothing. Neither party knows the distribution the other selected. You want to maximise your expected payout, while the banker wants to minimise it. Assuming optimal strategy from both parties, what is your expected payout from this game? The answer will be in the form $\frac{a}{\pi^b}$ for integers $a$ and $b$. Find $ab$.