That Alice has to have a randomised strategy is obvious as otherwise Bob will win evey play (now it may turn out that the random starategy is to always select the nickel but we will let the sums tell us that).
So start by assuming that A chooses the nickel with probability p_1, and that B guesses nickel with probbaility p_2. Work out the expected value of the game to A under these assumptions. This is the objective that A will seek to maximise under the assumption that B will simultaneously try to minimise it.
All of this is subject to the constraints that 0<=p_1<=1, 0<=p_2<=1.