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Let's we have a tictactoe design using RL against a random player. We can describe the system by enhancing and giving rewards to good actions. But what if the Rl model is played with itself? What changes in this case than playing with a random player?

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It would not change much. Instead of using a random policy, you would just replace the opponent with the policy you are training (assuming it is a zero-sum game, where both agents have the same actions and aim to achieve the same objective) $\pi_{\theta}(a\mid s)$. You probably want to update the policy of the opponent less often than the policy you are training for stability.

In the case of tictactoe, chess or Go, a typical reward would be 1 if we win and -1 if we lose, that would be given at the very end of the episode.

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