Auctions 3: Interdependent Values & Linkage Principle 1 Interdependent (common) values. • Each bidder receives private signal Xi ∈ [0,wi]. (wi = ∞ is possible) • (X1,X2,...,Xn) are jointly distributed accord-ingtocommonlyknown F (f> 0). • Vi = vi(X1,X2,...,Xn). vi(x1,x2,...,xn) ≡ E h Vi | Xj = xj for all j i . Typically assumed that functional forms {vi}N i=1 are commonly known. • vi(0, 0,...,0) = 0 and E[Vi] < ∞. • Note the difference: The actual ‘value’ impact of information vs strategic effect of knowing the other’s values.• Symmetric case: vi(xi, x−i)= v(xi, x−i)= v(xi,π(x−i)). • Affiliation: for all x0 , x00 ∈ X , f(x 0 ∨ x 00)f(x 0 ∧ x 00) ≥ f(x 0)f(x 00). If f is twice continuously differentiable and strictly positive, affiliation is equivalent to ∂2 ln f ∂xi∂xj ≥ 0, also equivalent to require that ln f is a supermod-ular function (or that f is log-supermodular). Examples: • Art painting: experts, dealers, collectors. Some bidders are more informative than the others. • Spectrum licenses: Each of the incumbents kno ws its own set of customers, interested in knowing characteristics of the rest. • Oil field: each firm runs a test, obtains a noisy signal of capacity, ζi. The estimate of the ex-pected value is Vi = 1 N P ζi,or, Vi = aζi +(1 − a) 1 N−1 P j6=i ζj;(notall firms may run tests, ..., other components to value); Vi = Xi + 1 N P ζi. • Job market? Employers bid for employees. Some employers can have better info.2 Brief analysis • Common values / Private values / Affiliated val-ues / Interdependent values. • Winner’s curse. • Second-price auction: Pivotal bidding–I bid what I get if I just marginally win. • First-price auction: “Usual” analysis–differential equation, .... • English auction: See below. • Revenue ranking: English > SPA > FPA. (!) Interdependency and affiliation a re important for the first pa rt. 3 Second-price auction Define v(x, y)= E [V1 | X1 = x, Y1 = y] . Equilibrium strategy βII(x)= v(x, x). Indeed, Π(b, x)= Z β−1(b) 0 (v(x, y) − β(y)) g(y|x)dy = Z β−1(b) 0 (v(x, y) − v(y, y)) g(y|x)dy. Π is maximized by choosing β−1(b)= x,thatis, b = β(x).4Example 1. Suppose S1,S2,and T are uniformly and inde-pendently distributed on [0, 1]. There are two bidders, Xi = Si+ T . The object has a common value V = 1 2 (X1 + X2) . 2. In this example, in the first price auction: βI(x)= 2 3x, E[RI]= 7 9 . 3. In the second-price auction v(x, y)= 1 2(x + y) and so βII(x)= x, E[RI]= 5 6 . 5 English auction (general case) Astrategy βEA i (xi) is a collection of strategies {βN i (xi),βN−1 i (xi,zN),...,β2 i(xi,z3,...,zN)}. Here zk is the “revealed” t ype of the player who have exited when there were k players still active. βk i is an intended exit p rice if no other player exited before. How to construct? (Note that revealing need not to happen in equilibrium) Remember, ∀i, Vi(x1,...,xn)isincreasingin xi (st) and in x−i (wk), plus SC. (+other requirements) Consider some p (suppose all bidders are active).?Exists zi(p) for each i that with xi >zi, i is active, with xi <zi is not. Everyone can make these inferences. Plug into V s. For i, Vi(xi,x−i) ≥ Vi(xi,z−i) ≥ Vi(zi,z−i) S p. In equilibrium, Vi(zi,z−i)= p. Equilibrium: Solution to V(z)= p.Once someone exits, fixhis z,continue. 6Linkage principle Define WA(z, x)= E [P (z) | X1 = x, Y1 <z] expected price paid by the winning bidder when she receive signal x but bids z. Proposition: (Linkage principle): Let A and B be two a uction forms in which the highest bidder wins and (she only) pays positive amount. Sup-pose that symmetric and increasing equilibrium exists in both forms. Suppose also that 1. for all x, WA 2 (x, x) ≥ WB 2 (x, x). 2. WA(0, 0) = WB(0, 0) = 0. Then, the expected revenue in A is at least as large as the expected revenue in B. So,the greater the linkage between a bidder’s own in-formation and how he perceives the others will bid the greater is the expected price paid upon winning.Auctions 4: Multiunit Auctions & Cremer-McLean Mechanism M units of the same object are offered for sale. Each bidder has a set of (marginal values) Vi = (Vi 1 ,Vi 2 ,...Vi M), the objects are substitutes, Vi k ≥ Vi k+1. Extreme cases: unit-demand, the same value fo r all objects. • Types of auctions: • The discriminatory (“pa y-your-bid”); • Uniform-price; • Vickrey; • Multi-unit English;• Ausubel; • Dutch, descending uniform-price, • ... Issues: Existence and description of equilibria, p rice series if sequential, efficiency, optimality, non-homogenous goods, complementarities,... 7 Vickrey Auction • Let (bi 1,bi 2,...,bi n) be the vector of bids submit-ted by i. • Winners: M highest bids. • Payments: If player i wins m objects, then has to pay the sum of m highest non-winning bids from the others. Or, price for each unit is: minimal value to have and win. E.g. to win 3d unit need to bid among (M − 2) highest bids, p =(M − 2)sd highest bid of the others. • Weakly dominant to bid truthfully, bi k = Vi k.8 Interdependent valuations 8.1 Notation K objects; given k =(k1,...,kN), denote Vk = ³ Vk1 1 ,...,VkN N ´ . Winners circle at s, Ik(s), is the set of bidders with the highest value among Vk . k is admissible if 1 ≤ ki ≤ K and 0 ≤ NX i=1 (ki − 1) <K. 8.2 Single-crossing condition MSC (single-crossing ) For any admissible k,for all x and any pair of players {i, j} ⊂ Ik(x), ∂Vki i (x) ∂xi > ∂V kj j (x) ∂xi . 8.3 Efficiency: VCG mechanism (general-ized Vickrey auction) • Allocation rule: Efficient. • Payments: Vickrey price that player j pays for kth unit won:p k j = Vk j (s k j,x−j)= (M − k +1)th highest among {Vm i (s k j,x−j)}m=1..M i6=j . These are generically different across units and win-ners (unlike with private values). 9 Cremer & McLean Mechanism • Multiple units. Single-crossing and non-independent values. • Efficient, Extract all the surplus. Discrete support: X i = {0, ∆, 2∆,...,(ti − 1)∆}, discrete single-crossing is assumed (no need if the val-ues are private). Π(x) is the joint probability of x, Πi =(π(x−i|xi)). Theorem: In the above conditions and if Π has a full rank, there exists a mechanism in which