1 Interdependent (common) values. Each bidder receives private signal X [0,w ]. • i ∈ i (w = is possible) i ∞ (X1,X2,...,Xn) are jointly distributed accord- • Auctions 3: ingtocommonlyknownF (f>0). Interdependent Values & • Vi = vi(X1,X2,...,Xn). v (x1,x2,...,xn) E V X = x for all j . i ≡ i | j j h Ni Typically assumed that functional forms vi i=1 are commonly known. { } Linkage Principle v (0, 0,...,0) = 0 and E[V ] < . • i i ∞ 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)). − − − Spectrum licenses: Each of the incumbents knows Affiliation: for all x , x , • • 0 00 ∈ X its own set of customers, interested in knowing f(x x )f(x x ) f(x )f(x ). characteristics of the rest. 0 ∨ 00 0 ∧ 00 ≥ 0 00 If f is twice continuously differentiable and strictly Oil field: each firm runs a test, obtains a noisy positive, affiliation is equivalent to • signal of capacity, ζi. The estimate of the ex- 2 1 ∂ ln f pected value is Vi = ζi,or,V i = aζi +(1 0, N − ∂x ∂x ≥ 1 i j a) N 1 j=i ζj;(notalPl firms may run tests, ..., − 6 1 also equivalent to require that ln f is a supermod- other compP onents to value); Vi = Xi + N ζi. ular function (or that f is log-supermodular). P Job market? Employers bid for employees. Some • Examples: employers can have better info. Art painting: experts, dealers, collectors. Some • bidders are more informative than the others. 2 Brief analysis 3 Second-price auction Common values / Private values / Affiliated val- • ues / Interdependent values. Define Winner’s curse. v(x, y)=E [V1 X1 = x, Y1 = y] . • | Second-price auction: Pivotal bidding–I bid what Equilibrium strategy • I get if I just marginally win. II β (x)=v (x, x). First-price auction: “Usual” analysis–differential • Indeed, equation, .... 1 β− (b) Π(b, x)= (v(x, y) β(y)) g(y x)dy 0 − | English auction: See below. Z 1 • β− (b) = (v(x, y) v(y, y)) g(y x)dy. Z0 − | Revenue ranking: English > SPA > FPA. Π is maximized by choosing β 1(b)=x ,thatis, b = • − β(x). (!) Interdependency and affiliation are important for the first part. 5 English auction (general case) 4Example Astrategyβ EA( x ) is a collection of strategies 1. Suppose S1,S2,andT are uniformly and inde- i i pendently distributed on [0, 1]. There are two N N 1 2 β (x ),β − (x ,z ),...,β (x ,z3,...,z ) . bidders, Xi = Si + T . The object has a common { i i i i N i i N } value 1 Here zk is the “revealed” type of the player who have V = (X1 + X2) . 2 exited when there were k players still active. k 2. In this example, in the first price auction: βi is an intended exit price if no other player exited before. I 2 I 7 β (x)= x, E[R ]= . 3 9 How to construct? (Note that revealing need not to happen in equilibrium) 1 3. In the second-price auction v(x, y)= 2(x + y) and so Remember, i, Vi(x1,...,xn)isincreasinginx i (st) II I 5 ∀ β (x)=x, E[R ]= . and in x i (wk), plus SC. (+other requirements) 6 − Consider some p (suppose all bidders are active). 6Linkagepr inciple Define A W (z, x)=E [P (z) X1 = x, Y1 <z] | ?Exists zi(p) for each i that with xi >zi, i is active, expected price paid by the winning bidder when she with xi <zi is not. receive signal x but bids z. Proposition: (Linkage principle): Everyone can make these inferences. Plug into V s. Let A and B be two auction forms in which the highest For i, Vi(xi,x i) Vi(xi,z i) Vi(zi,z i) S p. bidder wins and (she only) pays positive amount. Sup- − ≥ − ≥ − pose that symmetric and increasing equilibrium exists in both forms. Suppose also that In equilibrium, Vi(zi,z i)=p . − 1. for all x, W A (x, x) W B (x, x). 2 ≥ 2 Equilibrium: Solution to V(z)=p .Onces omeone 2. W A(0, 0) = W B(0, 0) = 0. exits, fixhisz ,continue. 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. M units of the same object are offered for sale. Each bidder has a set of (marginal values) V i = i i i i (V1 ,V2 ,...VM), the objects are substitutes, Vk i ≥ Vk+1. Auctions 4: Extreme cases: unit-demand, the same value for all objects. Multiunit Auctions & Types of auctions: • The discriminatory (“pay-your-bid”); • Cremer-McLean Mechanism Uniform-price; • Vickrey; • Multi-unit English; • 7 Vickrey Auction Let (bi ,bi ,...,bi ) be the vector of bids submit- • 1 2 n ted by i. Ausubel; • Winners: M highest bids. • Dutch, descending uniform-price, • 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 Issues: Existence and description of equilibria, price and win. series if sequential, efficiency, optimality, non-homogenous goods, complementarities,... 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 = V i . • k k 8 Interdependent valuations 8.2 Single-crossing condition MSC (single-crossing) For any admissible k,fora llx and any pair of players i, j k(x), 8.1 Notation { } ⊂ I k ki j ∂V (x) ∂Vj (x) i > . ∂xi ∂xi K objects; given k =(k1,...,kN), denote k k1 kN V = V1 ,...,VN . ³ ´ 8.3 Efficiency: VCG mechanism (general- Winners circle at s, k(s), is the set of bidders with I ized Vickrey auction) the highest value among Vk . Allocation rule: Efficient. k is admissible if 1 k K and • ≤ i ≤ N 0 (k 1) <K. Payments: Vickrey price that player j pays for ≤ i − • iX=1 kth unit won: 9 Cremer & McLean Mechanism Multiple units. Single-crossing and non-independent • values. k k k Efficient, Extract all the surplus. pj = Vj (sj ,x j)= • − Discrete support: i = 0, ∆, 2∆,...,(t 1)∆ , (M k +1)th highest X { i − } − discrete single-crossing is assumed (no need if the val- ues are private). m k m=1..M among Vi (sj ,x j) i=j . { − } 6 Π(x) is the joint probability of x, Πi =(π(x i xi)). − | Theorem: In the above conditions and if Π has a full These are generically different across units and win- rank, there exists a mechanism in which truth-telling ners (unlike with private values). is an efficient ex post equilibrium and in which the seller extracts full surplus from the bidders. Proof: Consider VCG mechanism (Q∗, M∗). Define, Ui ∗ (xi)= π(x i xi)[Qi∗ (x)Vi(x) Mi ∗ (x)] . x i − | − P− This is the expected surplus of buyer i in VCG mech- anism. Define, ui∗ =(Ui ∗ (xi))x i. i∈X There exists ci =(ci(x i))x i i,sucht hatΠ ici = − − ∈X− ui∗ . Equivalently, π(x i xi)ci(x i)=U i ∗ (xi). x i − | − P− CM Then, CM mechanism (Q∗, M )isdefined by CM Mi (x)=M i ∗ (x)+ ci(x i). − Remarks: Private values (correlated), equiv. second price • auction with additional payments. Negative payoffs sometimes, not ex post IR, pay- • offs arbitrarily large if the distribution converges to the independent one. .