maxa1A1mina2A2u(a1,a2)=mina2A2maxa1A1u(a1,a2). We say that this equation has a solution (a1,a2) if a1 is an arg max of the left hand side and a2 is an arg min of the right hand side. a) Show that if (a1,a2) is a solution of (1), then u(a1,a2)=maxa1A1mina2A2u(a1,a2)=mina2A2maxa1A1u(a1,a2). b) In class, we called the solution of (1) as an optimal strategy for the game. Using part a), show that if (a1,a2) is an optimal strategy, then (a1,a2) is a saddle point for the utility function, i.e., u(a1,a2)u(a1,a2),a1A1,u(a1,a2)u(a1,a2),a2A2. c) Now suppose that (a1,a2) is saddle point of the utility function u (i.e., it satisfies the equations given in part b). Show that the game has a value and (a1,a2) is an optimal strategy (i.e., it satisfies (1)). Hint: Try to use the definitions of maxmin and minmax value of the game. d) An immediate implication of part a) (you don't need to prove this in your solution) is that if (1) has two solutions (a1,a2) and (b1,b2) such that a1=b1 and a2=b2, then u(a1,a2)=u(b1,b2). Show that it is also the case that both of these terms are also equal to u(a1,b2) and also to u(b1,a2). Hint: Use part b) to show that u(b1,b2)u(b1,a2)u(a1,a2)andu(a1,a2)u(a1,b2)u(b1,b2)