# can a player always have a pure best response strategy?

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?

• It need not be unique. – David G. Stork Feb 16 '18 at 21:23
• Define pure best. – orlp Feb 16 '18 at 21:23
• @orlp pure strategy best response and not a mixed one; – Reza_va Feb 16 '18 at 21:28

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.