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Mechanism Design

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  1. Mechanism Design Milan Vojnović Lab tutorial, March 2010

  2. Mechanism design is about designing a game so as to achieve a desired goal

  3. b1 b2 bn ... Input: Output: (x, p) other input allocation: payment:

  4. Ex 1: sponsored search

  5. Ex 1: sponsored search (cont’d) a Position 1 Position 2 Position 3 Position 4 advertisers Position 5

  6. Ex 2: online contests

  7. Ex 2 (cont’d)

  8. Ex 3: resource allocation... communication networks, data centres, distributed systems x2 x2 x2 C2 C C C3 C/w P P P C x1 C1 x1 C/w C x1 C2 x1 1 x1 x1 w x2 w C1 C2 C3 x2 x2 C 1

  9. Part of Computer Science

  10. ... this mechanism is strategy proof ... however, it is not ex-post individually rational ... there is a high efficiency loss ... U(x) – px ... ... maximizes virtual surplus...

  11. Some developments ... 1961 Vickery’s auction ... 1981 Myerson’s optimal auction design ... 1997 Overture’s auction; network resource allocation (Kelly) 1999 Algorithmic mechanism design (Nisan & Ronen) 2001 Competitive auctions and digital goods (Goldberg et al) 2002 Generalized Second Price Auction ... 2007 Algorithmic game theory (Nisan et al)

  12. Active research area • Algorithmic problems • Efficient and user-friendly mechanisms • Prior-free and online learning • Alternative solutions concepts • Computational / communication complexity • The use of models to better understand and inform design • Realistic models of rational agents

  13. The importance of irrelevant alternatives Economist.com Economist.com SUBSCRIPTIONS SUBSCRIPTIONS • Welcome to • The Economist Subscription Centre • Pick the type of subscription you want to buy • or renew •  Economist.com subscription – US $59.00 • One-year subscription to Economist.com • Includes online access to all articles from • The Economist since 1997. •  Print subscription – US $125.00 • One-year subscription to the print edition of the Economist. • Print & web subscription – US $125.00 • One-year subscription to the print edition • of The Economist and online access to all • articles from The Economist since 1997. Welcome to The Economist Subscription Centre Pick the type of subscription you want to buy or renew  Economist.com subscription – US $59.00 One-year subscription to Economist.com Includes online access to all articles from The Economist since 1997.  Print & web subscription – US $125.00 One-year subscription to the print edition of The Economist and online access to all articles from The Economist since 1997. OPINION WORLD BUSINESS FINANCE & ECONOMICS SCIENCE & TECHNOLOGY PEOPLE BOOKS & ARTS MARKETS & DATA DIVERSIONS OPINION WORLD BUSINESS FINANCE & ECONOMICS SCIENCE & TECHNOLOGY PEOPLE BOOKS & ARTS MARKETS & DATA DIVERSIONS Source: Ariely D. (2008)

  14. This tutorial agenda • Design objectives • Vickery & Myerson auctions • Prior-free auctions • Auctions for resource allocation

  15. Standard goals Max seller’s profit “optimal auction design” Max social welfare “efficient”

  16. Examples of other goals min makespan, max flow, max weighted flow machines ... v2 vn v1 processing speed jobs

  17. Standard constraints • Incentive-compatibility = it is to the agents’ best interests to report true types Also known as implementation theory, the theory of incentives, or strategy-proof mechanisms • Individual rationality = ensure the agents’ profits are non-negativeAlso known as voluntary participation

  18. Two kinds of games • Incomplete information • Complete information • Types are private information • Types are public information • Types drawn from a distribution F • F is public information

  19. Vickery auctionfor allocation of a single item • Allocation to the buyer with highest bid • Payment equal to the second highest bid

  20. Incentive compatibility equal profit equal profit win only by overbidding dominated by truthful win only if truthful equal profit = 0 lose in either case

  21. Vickery auction is a truthful efficient auction But how do I maximize my profit?

  22. ... setting a reserve price should work

  23. Myerson’s optimal auction design • A mechanism is truthful if and only if for every buyer i and bids of other agents b-i fixed: • C1)allocation xi(b-i, bi) is non-decreasing with bi • C2)payment:

  24. Incentive compatibility • Buyers’ profit: B B A A

  25. But how do I determine an optimal allocation?

  26. Under independent buyer’s valuations, every optimal allocation is a solution ofthe virtual surplus maximization Virtual valuation:

  27. Virtual valuation • Ex. 2 Fi(v) = 1 - exp(-li) • Ex. 1 Fi(v) uniform on [0, hi]

  28. Optimality of Vickery auction with reserve price • Single-item auction • Independent and identical buyers • Strictly increasing virtual valuations The optimal is Vickery auction with the reserve price r:

  29. Optimality of Vickery auction with reserve price (cont’d) • Ex.F uniform [0, h],

  30. Competitive framework for auctions • Competitiveness to a profit benchmark B(v) Ex. 1 sum valuation Ex. 3 uniform pricing with at least two winners Ex. 2 max valuation Competitive ratio for an auction A =

  31. Random reserve price auction (Lu at al 2006) Run the second-price auction 1- d d Sample reserve price r from Ifb1 ≥ r thenallocate the item to a buyer with highest bid

  32. Random reserve price (cont’d) E[profit] = E[social welfare] = h = max valuation • A tighter expected revenue can be obtained using a successive composition of log(x+1) • Can’t do a better expected revenue !

  33. Why incentive compatibility as a requirement? • Pros • Simplifies buyer’s strategy – just report the type • Simplifies the problem for the designer • Cons • Computational complexity

  34. This tutorial agenda • Design objectives • Vickery & Myerson auctions • Prior-free auctions • Auctions for resource allocation

  35. Every time you send an email you participate in an auction

  36. Kelly’s resource allocation b1 bi bn C allocation to buyer i: payment by buyer i = bi

  37. Kelly’s resource allocation (cont’d) • Extensions to networks of links: the mechanism applied by each link • Two user models scalar bids (TCP like) vector bids

  38. Kelly’s resource allocation (cont’d) • Price-taking users: • Underprice-taking users with concave, utilityfunctions, efficiency is 100%.

  39. Johari & Tsitsiklis’ price-anticipating users User: • Underprice-anticipating users with concave, non-negative utility functions, and vector bids, the worst-case efficiency is 75%.

  40. Full efficiency loss under scalar bids • (Hajek & Yang 2004) Underprice-anticipating users with concave, non-negative utility functions, and scalar bids, theworst-case efficiency is 0. • A worst-case: serial network of unit capacity links

  41. But how do I maximize my profit?

  42. Profit maximization by price discrimination

  43. I’d like to run an auction!

  44. The weighted proportional allocation mechanism • Guarantees on social welfare and seller’s profit - Thanh-V. 2009 • Allocation to buyer i: • Payment by buyer i = bi

  45. Some important aspects not discussed in this tutorial • When truthfulness requires side-payments • Frugality, envy-freeness • Competitive guarantees of some auctions, ex. digital-goods auctions • Computational complexity under incentive compatibility

  46. Thank you for your attention!

  47. Some references • Aggarwal G., Fiat A., Goldberg A. V., Hartline J. D., Immorlica N., Sudan Madhu, Derandomization of auctions, STOC 2005. • Archer A. and Tardos E, Truthful Mechanisms for one-parameter agents, FOCS 2001. • Balcan M.-F., Blum A., Harline J. D., Mansour Y., Mechanism Design via Machine Learning, FOCS 2005. • Bulow J. and Klemperer P., Auctions versus negotiations, The American Economic Review, Vol 86, No 1, 1996. • DiPalantino D. and Vojnovic M., Crowdsourcing and all-pay auctions, ACM EC ‘09. • Edelman B., Ostrovsky M., Schwartz M., Internet Advertising and the Generalized Second Price Auction: Selling Billion of Dollars Worth of Keywords, Working Paper, 2005. • Fiat A., Goldberg A. V., Hartline J. D., and Karlin A. R., Competitive Generalized Auctions, STOC 2002. • Goldberg A. V., Hartline J. D., Karlin A. R., Saks M., A lower bound on the competitive ratio of truthful auctions, FOCS 2004. • Goldberg A. V, Hartline J. D., Wright A., Competitive Auctions and Digital Goods, SODA 2001. • Hajek B. and Yang S., Strategic buyers in a sum bid game for flat networks, IMA Workshop, 2004. • Hartline J. D., The Lectures on Optimal Mechanism Design, 2006. • Hartline J. D., Roughgarden T., Simple versus Optimal Mechanisms, ACM EC ’09.

  48. Some references (cont’d) • Johari R. And Tsitsiklis J. N., Efficiency Loss in a Network Resource Allocation Game, Mathematics of Operations Research, Vol 29, No 3, 2004. • Kelly F., Charing and rate control for elastic traffic, European Trans. on Telecommunications, Vol 8, 1997. • Levin D., LaCurts K., Spring N., Bhattacharjee B., Bittorrent is an auction: analyzing and improving Bittorrent’s incentives, ACM Sigcomm 2008. • Lu P., Teng S.-H., Yu C., Truthful Auctions with Optimal Profit, WINE 2006 • Lucier B. And Borodin A., Price of Anarchy for Greedy Auctions, SODA 2009. • Migrom P. R. And Weber R. J., A Theory of Auctions and Competitive Bidding, Econometrica, Vol 50, No 5, 1982. • Myerson R. B., Optimal Auction Design, Mathematics of Operations Research, Vol 6, No 1, 1981. • The Prize Committee of the Royal Swedish Academy of Sciences, Mechanism Design Theory, 2007. • Papadimitriou C., Schapira M., Singer Y., On the hardness of being truthful, FOCS 2008. • Ronen A., On approximating optimal auctions, ACM EC ‘01. • Ronen A. And Saberi A., Optimal auctions are hard. • Thanh N. and Vojnovic M., The Weighted Proportional Allocation Mechanism, MSR Technical Report, MSR-TR-2009-123, 2009. • Varian H. R., Position auctions, Int’l Journal of Industrial Organization, Vol 25, 2007. • Vickery W., Counterspeculation, auctions, and competitive sealed tenders, The Journal of Finance, 1961.