Q1.

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 bestresponse whenever player 2 chooses q

above