Auctions with Budget Constraints

In a combinatorial auction k different items are sold to n bidders, where the objective of the seller is to maximize the revenue. The main difficulty to find an optimal allocation is due to the fact that the valuation function of each bidder for bundles of items is not necessarily an additive function over the items. An auction with budget constraints is a common special case where bidders generally have additive valuations, yet they have a limit on their maximal valuation. Auctions with budget constraints were analyzed by Lehmann, Lehmann and Nisan [11], as part of a wider class of auctions, where they have shown that maximizing the revenue is NP-hard, and presented a greedy 2-approximation algorithm. In this paper we present exact and approximate algorithms for auctions with budget constraints. We present a randomized algorithm with an approximation ratio of \(\frac{e}{e-1}\cong\) 1.582, which can be derandomized. We analyze the special case where all bidders have the same budget constraint, and show an algorithm whose approximation ratio is between 1.3837 and 1.3951. We also present an FPTAS for the case of a constant number of bidders.

[1]  Raymond E. Miller,et al.  Complexity of Computer Computations , 1972 .

[2]  Richard M. Karp,et al.  Reducibility Among Combinatorial Problems , 1972, 50 Years of Integer Programming.

[3]  Ellis Horowitz,et al.  Exact and Approximate Algorithms for Scheduling Nonidentical Processors , 1976, JACM.

[4]  Jan Karel Lenstra,et al.  Approximation algorithms for scheduling unrelated parallel machines , 1987, 28th Annual Symposium on Foundations of Computer Science (sfcs 1987).

[5]  Ronald L. Rivest,et al.  Introduction to Algorithms , 1990 .

[6]  Ronald M. Harstad,et al.  Computationally Manageable Combinational Auctions , 1998 .

[7]  Yoav Shoham,et al.  Taming the Computational Complexity of Combinatorial Auctions: Optimal and Approximate Approaches , 1999, IJCAI.

[8]  Noam Nisan,et al.  Bidding and allocation in combinatorial auctions , 2000, EC '00.

[9]  Daniel Lehmann,et al.  Combinatorial auctions with decreasing marginal utilities , 2001, EC '01.

[10]  Ronald L. Rivest,et al.  Introduction to Algorithms, Second Edition , 2001 .

[11]  Moshe Tennenholtz,et al.  Tractable combinatorial auctions and b-matching , 2002, Artif. Intell..

[12]  Bidding clubs in first-price auctions , 2002, AAAI/IAAI.

[13]  Noam Nisan,et al.  Truthful approximation mechanisms for restricted combinatorial auctions , 2008, Games Econ. Behav..

[14]  Anna R. Karlin,et al.  Truthful and Competitive Double Auctions , 2002, ESA.

[15]  Tuomas Sandholm,et al.  Algorithm for optimal winner determination in combinatorial auctions , 2002, Artif. Intell..

[16]  Amos Fiat,et al.  Competitive generalized auctions , 2002, STOC '02.

[17]  Andrew V. Goldberg,et al.  Competitiveness via consensus , 2003, SODA '03.

[18]  Noam Nisan,et al.  Incentive compatible multi unit combinatorial auctions , 2003, TARK '03.

[19]  Éva Tardos,et al.  An approximate truthful mechanism for combinatorial auctions with single parameter agents , 2003, SODA '03.