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I'm trying to understand how Q-learning deals with games where the optimal policy is a mixed strategy. The Bellman equation says that you should choose $max_a(Q(s,a))$ but this implies a single unique action for each $s$. Is Q-learning just not appropriate if you believe that the problem has a mixed strategy?

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One possibility is to use softmax and choose each action a randomly with probabiliy $p = \frac{\exp(Q(s,a))}{\sum_a \exp(Q(s,a))}$. I don't thinks it is still Q-learning though.

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