Consider a game where N = {1, 2} and S1 = {X, Y } and S2 = {A, B}. Let p mix =

(pX,(1 p)Y ) and q mix = (qA,(1 q)B) be generic mixed strategies of players 1 and 2,

respectively. Let q* be such that U1(X,(q*A,(1 q* )B)) = U1(Y,(q*A,(1 q* )B)).Show that for any p, U1((pX,(1 p)Y ),(q*A,(1 q* )B)) = U1(X,(q*A,(1 q* )B)). Note that this finding justifies why we include each player 1's strategy as his best-response whenever player 2 chooses q above.