Exercise 22.30. * (1) Show that in Theorem 22.7, a necessary condition for a pure strategy symmetric equilibrium, with pA = pB = p∗, to exist is that the matrix X G g=1 λghg(0)D2Ug (p∗) + X G g=1 λg ∂hg(0) ∂σ (DUg (p∗)) · (DUg (p∗))T is negative semidefinite, where D2Ug denotes the Jacobian matrix of Ug. (2) Derive a sufficient condition for such a symmetric equilibrium to exist. [Hint: distinguish between local and global maxima]. (3) Show that without any assumptions on Ug (·)’s (beyond concavity), the sufficient condition for a symmetric equilibrium can only be satisfied if all Hg’s are uniform.