Guessing with Sets (noch nicht übersetzt)

Two players play a game. At the start of the game each player secretly chooses an integer; the first player from $1,...,n$ and the second player from $1,...,m$. Then they take alternate turns, starting with the first player. The player, whose turn it is, displays a set of numbers and the other player tells whether their secret number is in the set or not. The player to correctly guess a set with a single number is the winner and the game ends.

Let $p(m,n)$ be the winning probability of the first player assuming both players play optimally. For example $p(1, n) = 1$ and $p(m, 1) = 1/m$.

You are also given $p(7,5) \approx 0.51428571$.

Find $\displaystyle \sum_{i=0}^{20}\sum_{j=0}^{20} p(7^i, 5^j)$ and give your answer rounded to 8 digits after the decimal point.