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 (“pay-your-bid”); • Uniform-price; • Vickrey; • Multi-unit English;• Ausubel; • Dutch, descending uniform-price, • ... Issues: Existence and description of equilibria, price series if sequential, efficiency, optimality, non-homogenous goods, complementarities,... 7 Vickrey Auction • Let (bi 1,bi 2,...,bi n) be the vecto r 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 suppo rt: 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 truth-telling is an efficient ex post equilibrium and in which the seller extracts full surplus from the bidders. Proof: Consider VCG mechanism (Q∗ , M∗). Define, U ∗ i (xi)= P x−i π(x−i|xi)[Q ∗ i(x)Vi(x) − M ∗ i (x)] .This is the expected surplus of buyer i in VCG mech-anism. Define, u ∗ i =(U ∗ i (xi))xi∈X i. There exists ci =(ci(x−i))x−i∈X−i,such that Πici = u ∗ i . Equivalently, P x−i π(x−i|xi)ci(x−i)= U ∗ i (xi). Then, CM mechanism (Q∗ , MCM)isdefined by MCM i (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, pa y-offs arbitrarily large if the distribution converges to the independent one.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 suppo rt: 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 truth-telling is an efficient ex post equilibrium and in which the seller extracts full surplus from the bidders. Proof: Consider VCG mechanism (Q∗ , M∗). Define, U ∗ i (xi)= P x−i π(x−i|xi)[Q ∗ i(x)Vi(x) − M ∗ i (x)] .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:• Ausubel; • Dutch, descending uniform-price, • ... Issues: Existence and description of equilibria, price series if sequential, efficiency, optimality, non-homogenous goods, complementarities,... 7 Vickrey Auction • Let (bi 1,bi 2,...,bi n) be the vecto r 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