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## Mechanism Design

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**Mechanism Design**Milan Vojnović Lab tutorial, March 2010**Mechanism design is about designing a game so as to achieve**a desired goal**b1**b2 bn ... Input: Output: (x, p) other input allocation: payment:**Ex 1: sponsored search (cont’d)**a Position 1 Position 2 Position 3 Position 4 advertisers Position 5**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**... 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...**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)**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**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)**This tutorial agenda**• Design objectives • Vickery & Myerson auctions • Prior-free auctions • Auctions for resource allocation**Standard goals**Max seller’s profit “optimal auction design” Max social welfare “efficient”**Examples of other goals**min makespan, max flow, max weighted flow machines ... v2 vn v1 processing speed jobs**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**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**Vickery auctionfor allocation of a single item**• Allocation to the buyer with highest bid • Payment equal to the second highest bid**Incentive compatibility**equal profit equal profit win only by overbidding dominated by truthful win only if truthful equal profit = 0 lose in either case**Vickery auction is a truthful efficient auction**But how do I maximize my profit?**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:**Incentive compatibility**• Buyers’ profit: B B A A**Under independent buyer’s valuations, every optimal**allocation is a solution ofthe virtual surplus maximization Virtual valuation:**Virtual valuation**• Ex. 2 Fi(v) = 1 - exp(-li) • Ex. 1 Fi(v) uniform on [0, hi]**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:**Optimality of Vickery auction with reserve price (cont’d)**• Ex.F uniform [0, h],**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 =**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**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 !**Why incentive compatibility as a requirement?**• Pros • Simplifies buyer’s strategy – just report the type • Simplifies the problem for the designer • Cons • Computational complexity**This tutorial agenda**• Design objectives • Vickery & Myerson auctions • Prior-free auctions • Auctions for resource allocation**Kelly’s resource allocation**b1 bi bn C allocation to buyer i: payment by buyer i = bi**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**Kelly’s resource allocation (cont’d)**• Price-taking users: • Underprice-taking users with concave, utilityfunctions, efficiency is 100%.**Johari & Tsitsiklis’ price-anticipating users**User: • Underprice-anticipating users with concave, non-negative utility functions, and vector bids, the worst-case efficiency is 75%.**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**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**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**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.**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.