Game theory problem

Attached is a game theory problem related to advanced math. Problem 1 (Auctions with Interdependent Valuations). Consider bidder N bidders bidding for a singleindivisible good. Bidder i observes a private signal X; that is informative about his value of the object, withX = (X1, X2, . . . , XN). The vector S = (S1, S2, . . ., SM) includes additional random variables that may beinformative about the value of the object. Bidder i’s value is Vi = ui(S, X). We can assume that ui(., .) is symmetric across i in that there is a function u(., .) on RM+N such that for all i, u; (S, X ) = u(S, Xi, {Xj}j+i).It means that all bidders’ valuations depend on S in the same manner and each bidder’s valuation is asymmetric function of the other bidders’ signals. In class, we have discussed the case where M = 0 and Vi = ui(Xi), which is called the pure independent-private-value model. When Vi = ui(S), it is called common-value model as the valuations all depend on the random vector S.We assume that the bidders are risk-neutral. If bidder i wins the auction and pays price p for the object,his payoff is Vi – p. Denote Y as the highest signal among bidders other than i. It has cdf, Fy(y|x), andpdf, fy (y x ), conditioned on the realization x of Xi. In class, we have discussed the case where the signalsare independent, then Fy (y|x), and pdf, fy (y|x) become Fy (y) and fy (y), respectively. We also define theexpected value conditioned on the winning as v(x, y) := E[Vi|X; = x, Y = y] and the the expected valuer(x) := E[Vi|Xi = x]. These two functions are symmetric across i so the indices i of v and r are dropped.Under this formulation, we can revisit the first price auction and aim to find symmetric Bayesian Nashequilibrium u(x). Here, we assume that u(x) is increasing and and u-I(x) exists. In a first-price auction,bidder i’s optimal bidding strategy solvesmax E[(Vi – a)1{u(Y)<a}lXi = x] = max(a)(v(xly) – a) fy (yl.x )dy.where 1,.; is the indicator function and x is the infimum of the support of Y.(i) Suppose the boundary condition u(x) = v(x, x). Use the first-order condition to show that the first-price equilibrium bidding strategy is given byH(x) = v(x, x) – exp (-fy (tit)FY (tit)it ) d(oly, “).

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