can a player always have a pure best response strategy?

Question

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?

Answer 1

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.