1. This question is about some basic properties of projection operators. Remember from class that if W is a subspace of R" which has the orthonormal basis {1'51\" . . ,ii'P}, then the (orthogonal) projection onto W is given by multiplication by the matrix P = U UT, where U = [t'il . . . 113,] is the n x p matrix whose columns are the basis vectors of W. Also remember that UTU = I, and that any vector :i." can be written in exactly one way in the form =t+5 oemsewi. By the way, be careful of the terminology here. The matrix P of an orthogonal projection is not generally an orthogonal matrix. Remember that an orthogonal matrix S is one such that S\"1 = ST. In particular, it must be invertible. The only projection which is invertible is I. (Don't blame me; I didn't invent the terminology!) (a) Show that P2 = P and PT = P. (b) Dene Q = I P. Show that Q2 = Q, QT = Q, and P0 = QP = 0, the zero matrix. (c) Explain why Col(P) = Coi(U). (For this, remember that the column space of a matrix A is the same as the range of the linear transformation whose matrix is A. That is, the vectors in Col(A) are precisely the vectors of the form A5.) (d) Show that if a e Col(P) and 5' 6 (301(0), then a. i; = o. (Hint: Remember that a. 5' = 5T3.) (e) Explain why :i.' = Pf + inf. This means that 001(0) = Col(P)-". (Remark: It is true that if P is any n x 11. matrix such that P2 = P and PT = P, then P is the matrix of an orthogonal projection. If all you have is the P2 = P, then P is still a projection, but it doesn't have to be an orthogonal projection. That is, instead mapping vectors onto a subsPace in a direction perpendicular to the subspace, it will instead map in some oblique direction.) (1') Let A be any n x p matrix with linearly independent columns. The p x p matrix ATA must then be invertible. (You don't need to explain why this is true, but you should at least think about it and try to gure out a reason why it's true.) Dene R = A(ATA)'1AT. Show that 32 = R and RT = R. (Remark: In light of the previous remark, then, R is an orthogonal projection. In fact, it's the projection onto the column space of A. The difference here is that while the columns of A are a basis for the column space, they aren't necessarily an orthonormal basis.) (3) Show that S = I - 2P is an orthogonal matrix. (Not an orthogonal projection!). Markets with asymmetric information can be modeled as games of incomplete Information, resulting in Bayesian Nash equilibrium outcomes with inefficient trade outcomes. Harsanyi's purification theorem suggests that mixed-strategy equilibria in games of complete information can be thought of as representing pure-strategy Bayesian Nash equilibria of games with heterogeneous players. Exercises 12.1 Chicken Revisited: Consider the game of chicken in Section 12.2.1 with the parameters R = 8. H = 16, and L = 0 as described there. A preacher, who knows some game theory, decides to use this model to claim that moving to a society in which all parents are lenient will have detrimental effects on the behavior of teenagers. Does equilibrium analysis support this claim? What if R =8, H = 0, and L = 16? 12.2 Cournot Revisited: Consider the Cournot duopoly model in which two firms, I and 2, simultaneously choose the quantities they supply, q, and 92. The price each will face is determined by the market demand function p(91. 92) = a - b(91 + 92). Each firm has a probability & of having a marginal unit cost of cr and a probability 1 - u of having a marginal unit cost of CH. These probabilities are common knowledge, but the true type is revealed only to each firm individually. Solve for the Bayesian Nash equilibrium. 12.3 Armed Conflict: Consider the following strategic situation: Two rival armies plan to seize a disputed territory. Each army's general can choose either to attack (A) or to not attack (N). In addition, each army is either strong ($) or weak (W) with equal probability, and the realizations for each army are independent. Furthermore the type of each army is known only to that army's general. An army can capture the territory if either (i) it attacks and its rival does not or (ii) it and its rival attack, but it is strong and the rival is weak. If both attack and are of equal strength then neither captures the territory. As for payoffs, the territory is worth m if captured and each army has a cost of fighting equal to s if it is strong and w if it is weak, where s