I have a question with mixed strategies. The question is as follows, if we're in a strategy profile that is a Nash equilibrium and a player is playing a mixed strategy, can the pure strategies that are part of such a mixed strategy give different payoffs?
My first impression is no. If, for example, we consider a game such that the mixed strategy assigns $\mu$ to pure strategy $A$ and $1-\mu$ to pure strategy $B$ and if $A$ gives higher payouts than $B$, then another mixed strategy that gives probability 1 to playing $A$ and 0 playing $B$ would give higher payoffs for said player, contrary to it being a Nash equilibrium.
I don't know if I'm right or if I have a mistake, I would really appreciate your comments or if you have a better way of arguing it or have a counterexample.
Update: For example, in matching pennies, we are in an equilibrium when we play the strategy (0.5,0.5) for each player and yet the payoffs of the associated pennies are different. I don't know if this serves to exemplify that it is possible.