EE 228 a - Lecture 25 - Spring 2006 Combinatorial Auction

This lecture gives a combinatorial market mechanism that is always efficient, budget-balanced, ex post individual rational and ‘almost’ dominant strategy incentive compatible in the complete information case. In the incomplete information case, it is budget-balanced, ex post individual rational and asymptotically efficient and Bayesian incentive compatible. I. THE COMBINATORIAL SELLERS’ BID DOUBLE AUCTION (C-SEBIDA) There are L items l1, . . . , lL, m buyers and n sellers. Buyer i has (true) reservation value v i per unit for a bundle of items Ri ⊆ {l1, . . . , lL}, and submits a buy bid of bi per unit and demands up to δi units of the bundle Ri. Thus, the buyers have quasi-linear utility functions of the form ui(x;ω,Ri) = vi(x) + ω where ω is money and vi(x) = { x · vi, for x ≤ δi, δi · vi, for x > δi. Seller j has (true) per unit cost cj and offers to sell up to σj units of lj at a unit price of aj . Denote Lj = {lj}. The sellers also have quasi-linear utility functions of the form u j(x;ω,Lj) = −cj(x) + ω where ω is money and cj(x) = { x · cj , for x ≤ σj , ∞, for x > σj . The mechanism receives all these bids, and matches some buy and sell bids. The possible matches are described by integers xi, yi : 0 ≤ xi ≤ δi is the number of units of bundle Ri allocated to buyer i and 0 ≤ yj ≤ σj is the number of units of item lj sold by seller j. The mechanism determines the allocation (x∗, y∗) as the solution of the surplus maximization problem MIP: max x,y ∑