Suppose you are playing a game which contains $N$ rounds ($N$ is large enough). At each round you pick one from $K$ different strategies. The probability you win at a round depends on the strategy you pick and the number of rounds you have already taken. Denote the probability you win at $n$th round by picking $k$th strategy as $P_n(k), k = 1 ... K, n = 1 ... N$. The probability satisfies the following conditions:

- $P_{n_1}(k_1)$ and $P_{n_2}(k_2)$ are independent when $k_1 \neq k_2$.
- $|P_n(k) - P_{n+1}(k)| < \delta$. $\delta$ is known and small. Let's say $\delta = 0.01$.
- $P_n(k)$ is independent to all the previous strategies you take at $1 ... n - 1$ th round.

Now suppose you don't know the values of $P_n(k)$. How do you pick strategies to maximize the total number of rounds you win?