Combinatorial Auctions: A Survey by de Vries and Vohra

Transcription

1 Combinatorial Auctions: A Survey by de Vries and Vohra Ashwin Ganesan EE228, Fall 2003 September 30, 2003

2 1 Combinatorial Auctions Problem N is the set of bidders, M is the set of objects b j (S) is the amount agent j N is willing to pay for S M Let b(s) = max j N b j (S) CAP1: max S M b(s)x S such that S i x S 1 i M, x S = 0, 1 S M. Here, x S = 1 means the highest bid on S is accepted.

3 2 Decentralized Methods The integer programming problem is NP-hard in general. One way to reduce the computational burden is set up a fictitious market to determine allocation and prices in a decentralized way. Auctioneer announces prices for objects. Bidders announce which set of objects they will purchase at the posted prices. If two or more bidders compete for same object, auctioneer adjusts the price vector.

4 3 Lagrangean Relaxation The Set Packing Problem formulation: Let c j = highest bid on j-th subset. Z = max j V c jx j s.t. j V a ijx j 1 i M x j = 0, 1 j V. Linear programming relaxation: Z LP = max j V c jx j s.t. j V a ijx j 1 i M 0 x j 1 j V. Z Z LP

5 4 Lagrangean Relaxation Linear programming relaxation: Z LP = max j V c j x j s.t. j V a ij x j 1 i M, 0 x j 1 j V. Consider the problem: Z(λ) = max j V c j x j + i M λ i (1 j V a ij x j ) s.t. 0 x j 1 j V.

6 5 Easy to compute Z(λ) for each λ: Lagrangean Relaxation Z(λ) = max j V c j x j + i M λ i (1 j V a ij x j ) s.t. 0 x j 1 j V. Note that c j x j + i M λ i (1 j V a ij x j ) = j V (c j i M λ i a ij )x j + i M λ i. j V Theorem: Z LP = min λ 0 Z(λ).

7 6 Subgradient Algorithm Use the subgradient algorithm to find: min Z(λ). λ 0 Suppose the value of λ at iteration t is λ t. If x t is the optimal solution associated with Z(λ t ), then where θ t is step size. λ t+1 = λ t + θ t (Ax t 1), Notice that λ t+1 i > λ t i for any i such that j a ijx t j term increased on any constraint being violated. > 1. Penalty

8 7 Market Interpretation The market interpretation of the above algorithm: The auctioneer chooses a price vector λ for individual objects, and bidders submit bids. If c j, the highest bid on j-th bundle exceeds i M a ijλ i, this bundle is tentatively assigned to the bidder. After λ is announced, bidders can simply state which objects are acceptable to them at the announced prices. If conflict in assignment, auctioneer uses subgradient algorithm to adjust prices and repeats the process.

9 8 Cutting Plane Methods Can solve an integer programming (IP) problem by solving a sequence of linear programming (LP) problems. A generic cutting plane algorithm: Solve the LP relaxation. Let x be the optimal solution. If x is integer, stop; x is an optimal solution. If not, add a linear inequality constraint to the LP relaxation that all integer solutions to the IP problem satisfy, but x does not; go to first step. Performance of the algorithm depends on choice of inequality used to cut x May take an exponential number of iterations.

10 9 An Example There are 6 objects, with highest bids on subsets as follows: b({1, 2}) = b({2, 3}) = b({3, 4}) = b({4, 5}) = b({1, 5, 6}) = 2, b({6}) = 1. LP relaxation: max 2x x x x x x 6 s.t. x 12 + x 156 1, x 12 + x 23 1 x 23 + x 34 1, x 34 + x 45 1 x 45 + x 156 1, x x 6 1, x i 0 Optimal fractional solution is to set all variables equal to half. Add inequality: x 12 + x 23 + x 34 + x 45 + x

11 10 Incentive Issues So far, problems focused on choosing allocation to maximize seller s revenue. But no guarantee that bidders submit their true valuations. For example, suppose there are 3 bidders and two objects {x, y}, with true values of subsets to bidders given by v 1 (x, y) = 100, v 1 (x) = v 1 (y) = 0 v 2 (x, y) = 0, v 2 (x) = v 2 (y) = 75 v 3 (x, y) = 0, v 3 (x) = v 3 (y) = 40 If bidders bid truthfully, auctioneer should award x to 2 and y to 3, to maximize revenue. Bidder 2, under the assumption that bidder 3 bids truthfully, has the incentive to shade his bid down to 65.

12 11 Incentive Issues To discuss incentive issues, we first need a model of bidder preferences - simplest model is a list {v j (S)} S M. Choose an auction design that does three things: Induces bidders to reveal actual valuations (incentive compatibility) No bidder is made worse off (in expectation) by participating in the auction. Subject to the above, seller maximizes expected revenue. We can restrict to auctions with a single round. Optimal auction design is not known. Under the additional restriction of efficiency, the optimal auction is the VCG scheme.

13 12 Efficiency An auction is efficient if the allocation of objects to bidders chosen by seller solves the following: s.t. max j N S i j N S M S M v j (S)y(S, j) y(s, j) 1 i M y(s, j) 1 j N y(s, j) = 0, 1 S M, j N. This is just CAP2 with b i replaced by v i.

14 13 Vickrey-Clarke-Grove Scheme The VCG scheme is an optimal auction that is also efficient. Bidder j reports v j. Seller chooses the allocation that solves for y : V = max j N S M vj (S)y(S, j) s.t. y(s, j) 1 i M S M S i j N y(s, j) 1 j N y(s, j) = 0, 1 S M, j N.

15 14 Vickrey-Clarke-Grove Scheme For each k, solve for y k : V k = max j N\k S M vj (S)y(S, j) s.t. S i j N\k y(s, j) 1 i M S M y(s, j) 1 j N\k y(s, j) = 0, 1 S M, j N\k. y k is the optimal allocation when bidder k is excluded from the auction. The payment made by bidder k is V k [V S M vk y (S, k)]. The VCG scheme is impractical to implement. Computing approximately optimal solutions for y and y k need not preserve incentive compatibility.

16 15 Computational Experiments Done using the branch and bound exact method solver of CPLEX. They provide plots of log of running time versus number of bids (for a fixed number of items) for CAP1. Problems belong to four classes of distributions Random: For each bid, pick the number of items randomly from 1,...,m. Randomly choose that many items without replacement. Pick the prices randomly from [0, 1]. When the problems become large, many singletons in optimal allocation. Weighted Random: As above, but price is a real number between 0 and number of items in the bid. Modelling issue: how to model how bidders values different subsets of objects?

17 16 References Bertsimas and Tsitsiklis, Introduction to Linear Optimization, (for IP and LP) R. Myerson, Optimal Auction Design, Mathematics of Operations Research, 6, p , (direct revelation principle) Krishna and Perry, Efficient Mechanism Design, manuscript, 1998, available online. (on VCG scheme)