Bayesian Optimal No-Deficit Mechanism Design
- 694 Downloads
One of the most fundamental problems in mechanism design is that of designing the auction that gives the optimal profit to the auctioneer. For the case that the probability distributions on the valuations of the bidders are known and independent, Myerson  reduces the problem to that of maximizing the common welfare by considering the virtual valuations in place of the bidders’ actual valuations. The Myerson auction maximizes the seller’s profit over the class of all mechanisms that are truthful and individually rational for all the bidders; however, the mechanism does not satisfy ex post individual rationality for the seller. In other words, there are examples in which for certain sets of bidder valuations, the mechanism incurs a loss.
We consider the problem of merging the worst case no-deficit (or ex post seller individual rationality) condition with this average case Bayesian expected profit maximization problem. When restricting our attention to ex post incentive compatible mechanisms for this problem, we find that the Myerson mechanism is the optimal no-deficit mechanism for supermodular costs, that Myerson merged with a simple thresholding mechanism is optimal for all-or-nothing costs, and that neither mechanism is optimal for general submodular costs. Addressing the computational side of the problem, we note that for supermodular costs the Myerson mechanism is NP-hard to compute. Furthermore, we show that for all-or-nothing costs the optimal thresholding mechanism is NP-hard to compute. Finally, we consider relaxing the ex post incentive compatibility constraint and show that there is a Bayesian incentive compatible mechanism that achieves the same expected profit as Myerson, but never incurs a loss.
KeywordsCombinatorial Auction Threshold Mechanism Optimal Mechanism Post Incentive Optimal Auction
Unable to display preview. Download preview PDF.
- 1.Archer, A., Papadimitriou, C., Tawar, K., Tardos, E.: An Approximate Truthful Mechanism for Combinatorial Auctions with Single Parameter Agents. In: Proc. 14th Symp. on Discrete Alg. ACM/SIAM (2003)Google Scholar
- 2.Archer, A., Tardos, E.: Truthful mechanisms for one-parameter agents. In: Proc. of the 42nd IEEE Symposium on Foundations of Computer Science (2001)Google Scholar
- 5.Feigenbaum, J., Papadimitriou, C., Shenker, S.: Sharing the Cost of Multicast Transmissions. In: Proc. of 32nd Symposium Theory of Computing, pp. 218–226. ACM Press, New York (2000)Google Scholar
- 6.Fiat, A., Goldberg, A., Hartline, J., Karlin, A.: Competitive Generalized Auctions. In: Proc. 34th ACM Symposium on the Theory of Computing. ACM Press, New York (2002)Google Scholar
- 7.Goldberg, A.V., Hartline, J.D.: Competitiveness via Concensus. In: Proc. 14th Symp. on Discrete Alg. ACM/SIAM (2003)Google Scholar
- 8.Goldberg, A.V., Hartline, J.D., Wright, A.: Competitive Auctions and Digital Goods. In: Proc. 12th Symp. on Discrete Alg., pp. 735–744. ACM/SIAM (2001)Google Scholar
- 9.Hartline, J.: Stanford cs364b: Topics in algorithmic game theory. Course Notes (2006)Google Scholar
- 11.Krishna, V.: Auction theory. Academic Press, San Diego (2002)Google Scholar
- 13.Mehta, A., Shenker, S., Vazirani, V.: Profit-Maximizing Multicast Pricing Via Approximate Fixed Points. In: Proc. of 4th ACM Conference on Electronic Commerce. ACM Press, New York (2003)Google Scholar
- 16.Ronen, A.: On Approximating Optimal Auctions. In: Proc. of Third ACM Conference on Electronic Commerce. ACM Press, New York (2001)Google Scholar