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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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  • $\begingroup$ What is a mixed strategy? Define this concept formally. $\endgroup$ – nbro May 20 '19 at 20:37
  • $\begingroup$ It's a basic concept in game theory. wikiwand.com/en/Strategy_(game_theory)#/… Essentially it's a strategy of sometimes choosing one action and sometimes choosing another in the same situation. $\endgroup$ – Thomas Johnson May 21 '19 at 20:03
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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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