Consider the following calculation of Nash equilibria from Wikipedia:
Player B plays H Player B plays T
Player A plays H −1, +1 +1, −1
Player A plays T +1, −1 −1, +1
To compute the mixed strategy Nash equilibrium, assign A the probability p
of playing H and (1−p) of playing T, and assign B the probability q of
playing H and (1−q) of playing T.
E[payoff for A playing H] = (−1)q + (+1)(1−q) = 1−2q
E[payoff for A playing T] = (+1)q + (−1)(1−q) = 2q−1
E[payoff for A playing H] = E[payoff for A playing T] ⇒ 1−2q = 2q−1 ⇒ q = 1/2
We have calculated the strategy for player 2, but we haven't used their payoffs, only those for player 1! How is this possible?