2. Mechanism Design Milan Vojnović Lab tutorial, March 2010

3. Mechanism design is about designing a game so as to achieve a desired goal 3

4. ... b1b1 b2b2 bnbn Input: Output: (x, p) allocation: payment: other input

5. Ex 1: sponsored search 5

6. 6 a Ex 1: sponsored search (cont’d) Position 1 Position 2 Position 3 Position 4 Position 5 advertisers

7. Ex 2: online contests 7

8. Ex 2 (cont’d) 8

9. Ex 3: resource allocation... communication networks, data centres, distributed systems x1x1 x2x2 C/w C w P w 1 1 x1x1 x2x2 C C2C2 C2C2 C3C3 C1C1 x2x2 x1x1 P C2C2 C1C1 x1x1 C3C3 x2x2 P C C x2x2 x1x1 C x1x1 x2x2 9

10. Part of Computer Science 10

11. ... 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

12. Some developments 1999 Algorithmic mechanism design (Nisan & Ronen) 2001 Competitive auctions and digital goods (Goldberg et al) 1961 Vickery’s auction Algorithmic game theory (Nisan et al) 2007 1997 Overture’s auction; network resource allocation (Kelly) 2002 Generalized Second Price Auction... 1981 Myerson’s optimal auction design 12...

13. 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

14. The importance of irrelevant alternatives Economist.com 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. ÿ 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 Economist.com 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 & 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 Source: Ariely D. (2008) 14

15. This tutorial agenda Design objectives Vickery & Myerson auctions Prior-free auctions Auctions for resource allocation 15

16. Standard goals Max social welfare “efficient” 16 Max seller’s profit “optimal auction design”

17. Examples of other goals min makespan, max flow, max weighted flow jobs v1v1 v2v2 vnvn... processing speed 17 machines

18. 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 18 Individual rationality = ensure the agents’ profits are non-negative Also known as voluntary participation

19. Two kinds of games Incomplete information 19 Complete information – Types are private information – Types drawn from a distribution F – F is public information – Types are public information

20. Vickery auction for allocation of a single item 20 Allocation to the buyer with highest bid Payment equal to the second highest bid

21. Incentive compatibility 21 equal profit win only by overbidding dominated by truthful equal profit = 0 equal profit win only if truthful lose in either case

22. Vickery auction is a truthful efficient auction 22 But how do I maximize my profit?

23. 23... setting a reserve price should work

24. 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: 24 C1) allocation x i (b -i, b i ) is non-decreasing with b i C2) payment:

25. Incentive compatibility Buyers’ profit: A B A B 25

26. 26 But how do I determine an optimal allocation?

27. Under independent buyer’s valuations, every optimal allocation is a solution of the virtual surplus maximization Virtual valuation: 27

28. Virtual valuation Ex. 1 F i (v) uniform on [0, h i ] 28 Ex. 2 F i (v) = 1 - exp(- i )

29. 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

30. Optimality of Vickery auction with reserve price (cont’d) Ex. F uniform [0, h], 30

31. Competitive framework for auctions Competitiveness to a profit benchmark B(v) Competitive ratio for an auction A = 31 Ex. 1 sum valuation Ex. 2 max valuation Ex. 3 uniform pricing with at least two winners

32. Random reserve price auction (Lu at al 2006) 32 1- d d Run the second-price auction Sample reserve price r from If b 1 ≥ r then allocate the item to a buyer with highest bid

33. Random reserve price (cont’d) E[profit] = E[social welfare] = A tighter expected revenue can be obtained using a successive composition of log(x+1) Can’t do a better expected revenue ! 33 h = max valuation

34. 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

35. This tutorial agenda Design objectives Vickery & Myerson auctions Prior-free auctions Auctions for resource allocation 35

36. Every time you send an email you participate in an auction 36

37. Kelly’s resource allocation C b1b1 bnbn bibi payment by buyer i = b i allocation to buyer i: 37

38. Kelly’s resource allocation (cont’d) Extensions to networks of links: the mechanism applied by each link 38 scalar bids (TCP like) vector bids Two user models

39. Kelly’s resource allocation (cont’d) 39 Price-taking users: Under price-taking users with concave, utility functions, efficiency is 100%.

40. Johari & Tsitsiklis’ price-anticipating users 40 Under price-anticipating users with concave, non-negative utility functions, and vector bids, the worst-case efficiency is 75%. User:

41. Full efficiency loss under scalar bids (Hajek & Yang 2004) Under price-anticipating users with concave, non-negative utility functions, and scalar bids, the worst-case efficiency is 0. 41 A worst-case: serial network of unit capacity links

42. 42 But how do I maximize my profit?

43. Profit maximization by price discrimination 43

44. 44 I’d like to run an auction!

45. The weighted proportional allocation mechanism 45 Allocation to buyer i: Payment by buyer i = b i Guarantees on social welfare and seller’s profit - Thanh-V. 2009

46. 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

47. 47 Thank you for your attention!

48. 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

49. 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. 49