Matching-integer game paying the square of the number

A banker and you each secretly pick a positive integer, each using any probability distribution you like on the positive integers. If you pick the same integer $n$, you receive $n^2$ dollars; otherwise nothing. Neither sees the other's distribution, and both play optimally (you maximise, the banker minimises). Your expected payout has the form $\dfrac{a}{\pi^b}$ with integers $a,b$. Find $ab$.

$12$. The banker should make your payoff the same for every $n$, i.e. choose $n$ with probability $\propto 1/n^2$; normalising with $\sum 1/n^2=\pi^2/6$ gives $\tfrac{6}{\pi^2 n^2}$. Then any guess $n$ earns $n^2\cdot\tfrac{6}{\pi^2n^2}=\tfrac6{\pi^2}$, so $a=6$, $b=2$.