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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:

  1. $P_{n_1}(k_1)$ and $P_{n_2}(k_2)$ are independent when $k_1 \neq k_2$.
  2. $|P_n(k) - P_{n+1}(k)| < \delta$. $\delta$ is known and small. Let's say $\delta = 0.01$.
  3. $P_n(k)$ 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?

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  • $\begingroup$ In this form your problem is unsolvable, its much too general. As an example, since $N$ shall be large assumption 2. can be fulfilled if f.i. $P_n(k) = 0$ for $n \leq 900$ and $P_n(k) = 1$ for $n \geq 1000$. And then, what is the meaning of 1. and 3.? I don't know independent probabilities. Maybe you mean independent realisations at each stage. $\endgroup$ – Dieter Kadelka 2 days ago
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    $\begingroup$ I understand what 3) means, though it's already implicit in the notation. 1) is incomprehensible. The question basically comes down to: "What's my optimal strategy in a game where I don't know the payoffs?" Whatever the answer, can you really expect it to still be optimal if all $P_n(k)$ are replaced with $1-P_n(k)$ ? $\endgroup$ – Steven Landsburg 2 days ago

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