Mechanism Design

1. Mechanism Design Milan Vojnović Microsoft Research Summer Research Institute, EPFL, June 2011

2. Mechanism design - designing a game to achieve a desired outcome

3. 2 Input: Output: other input payment: allocation:

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 • Resource = communication network, computing resources in a data centre, …

9. Computer science Economics

10. 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)

11. Active research area • Algorithmic design problems • Efficient and user-friendly mechanisms • Prior-free and online learning • Computational / communication complexity • Used to better understand and design systems and services

12. Outline • Overview of design objectives • Standard auctions • Bayesian mechanism design • Prior-free framework

13. Standard goals Seller’s profit “optimal auction design” Social welfare “efficient” Buyer’s profit

14. Standard constraints • Resource • Single item (indivisible or infinitely divisible) • Multiple items • Polyhedral constraints • Solution concepts • Dominant strategy: truthful • Nash equilibrium: no truthfulness constraints

15. Available information • Complete information • Incomplete information • Valuations are private information • Valuations are common knowledgeamong buyers • Valuations drawn from prior distribution • is public information • Common priors assumption

16. Outline • Overview of design objectives • Standard auctions • Bayesian mechanism design • Prior-free framework

17. Standard auctions • Standard auction: item allocated to highest bidder • First price auction - winner pays own bid • Second price auction - winner pays the second highest bid • All-pay - pay own bid

18. Bayes Nash equilibrium • Strategy: bid value given the valuation value • Common prior assumption • Every agent has the same belief about the prior distribution of a buyer’s valuation • A strategy profile is in Bayes-Nash equilibrium if is a best response when other players play

19. Second price auction • Def. • wins and pays • Bidding truthfully is a dominant strategy: max social welfare (Vickery’61) Profit Profit Case Case

20. First price auction • Has no dominant strategy equilibrium • Ex. i.i.d. valuations, symmetric Bayes-Nash equilibrium: = largest value among i.i.d. valuation samples

21. Social welfare • Second price auction • Max social welfare in dominant strategy equilibrium for any distribution • First price auction • Max social welfare in Bayes-Nash equilibrium for i.i.d. value distributions

23. Revenue optimal auction (Myerson ’81) • A strategy profile is in Bayes-Nash equilibrium iff: • 1)monotonic allocation: is non-decreasing with • 2)paymentidentity: payment

24. Revenue equivalence • Corollary: two auctions with the same outcome in Bayes-Nash equilibrium yield the same expected revenue • True for i.i.d. valuations • In this case standard auctions such as first-price, second-price and all-pay auctions are revenue equivalent

25. But how do I determine a revenue-maximizing allocation?

26. Revenue optimal allocation (Myerson ’81) • Goal: find a monotone that maximizes • Myerson’s lemma: virtual valuation: • is said to be regular if is monotone • If is regular, the optimal auction is to allocate the item to the bidder with the highest positive virtual value.

27. Examples of virtual valuations • Ex. 2 = • Ex. 1 uniform on

28. Second price auction with reserve price • Assume i.i.d. valuations, regular • Bidder wins   • Second price auction with reserve price maximizes expected revenue

29. Example reserve prices • Ex. 2 = • Ex. 1 uniform on

30. Setting the reserve price

31. An example multi-item auction: parallel auctions (DiPalantino and V. ’09) • Participation constraint: each bidder participates in at most one auction • Standard auctions • Used as a model of crowdsourcing services value of unit reward reward

32. Parallel auctions (cont’d) Existence and full characterization of Bayes-Nash equilibrium for the following two cases: • where is i.i.d. • for each i, i.i.d. over with having a density on

33. Parallel auctions (cont’d) In a many auctions limit, the expected participation per class of auction is: • = expected number per auction • = fraction of auctions of class • Diminishing returns participation with the offered reward

34. Taskcn service • Conditional on the rate at which workers submit solutions • Conditioning on more experienced users, the better the prediction by the model any rate once a month every fourth day every second day

35. Summary • Pros • Dominant strategy equilibrium simplifies user’s decision: just tell the truth • Mechanism is simple in some cases • Ex. second price auction • Cons • Requires prior knowledge • Common priors assumption • Many auctions in practice are not truthful • Ex. GSP, first price, proportional allocation

36. Outline • Overview of design objectives • Standard auctions • Bayesian mechanism design • Prior-free framework

37. Prior-free framework • Seller assumed to have no prior information about buyers’ valuations • Examples of prior-free mechanisms • First-price auction • Second-price auction • Quasi-proportional allocation: for concave increasing Proportional allocation:

38. Revenue • Second price auction revenue = • Quasi-proportional revenue for two-buyer case: (Nadav et al ’10) • What best revenue can be guaranteed in the prior-free framework? • Could be guaranteed? (Nadav et al, ’10)

40. Weakly monotonic auctions • Best response: continuous and differentiable with • Allocation monotonicity: For every such that there exists such that and • Payment monotonicity: For every and there exists such that if then for every • Ex. Truthful auctions, quasi-proportional auctions Profit = = best response

41. Revenue upper bound (Nguyen and V., ’11) For every weakly monotonic auction and constant , there exists a valuation vector such that the revenue in Nash equilibrium is smaller than where is the highest buyer’s valuation. • cannot be guaranteed

42. Achievability For the case of two buyers, there exists a prior-free auction such that for given there is such that the revenue is at least (Lu et al, ’06) • For single-item auctions, no gain by allowing for non-truthful auctions • Not the case for more general auctions (Nguyen and V. ’11)

43. Two buyers auction (Lu et al, 2006) • Input: buyers’ bids and • Random price sampled from: • Allocation: allocate to highest bidder if the highest bid is at least , otherwise no sale • Payment: if item is allocated

44. Proof sketchrevenue upper bound • Valuations: • Bids: • Payments: Assume revenue Then,

45. Revenue of quasi-proportional auctions (Nguyen and V., ’11) • Revenue of quasi-proportional auctions is • Proportional allocation revenue • Worst cases for many buyers

46. Proportional allocation b1 bi bn C allocation: payment:

47. Proportional allocation (cont’d) • Buyers’ bid: • Vector (individual bid for each desired resource) • Scalar (same bid applied to each desired resource) • User models: • Price taking given • Price anticipating (fully rational)

48. Social welfare • Price taking users: social efficiency (Kelly ’97) • Price anticipating users with vector bids: social efficiency (Johari and Tsitsiklis ’04) • Price anticipating users with scalar bids: full efficiency loss in worst case (Hajek and Yang ’04)

49. Full efficiency loss under scalar bids (Hajek and Yang ’04) • A worst-case: serial network of unit capacity links Efficiency = , for as grows large

50. But how do I maximize my revenue?