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I have a general simple question. Does a player always have a pure best response strategy in response to mixed strategies of other players in all kinds of games?

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  • $\begingroup$ It need not be unique. $\endgroup$ – David G. Stork Feb 16 '18 at 21:23
  • $\begingroup$ Define pure best. $\endgroup$ – orlp Feb 16 '18 at 21:23
  • $\begingroup$ @orlp pure strategy best response and not a mixed one; $\endgroup$ – Reza_va Feb 16 '18 at 21:28
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In general, there are (infinite) games where no best response (either pure or mixed) exists. See f.i. the standard example of a Bertrand duopoly with symmetric costs and continuous prices.

If a best response exists, it may not be unique. However, if the best response is a mixed strategy $\sigma$, it is necessary that the player is indifferent among all the pure strategy in the support of $\sigma$. Therefore, if a best response exists, there must be at least one that is also a pure strategy.

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