Introduction to Auction Design via Machine Learning

Transcription

1 Introduction to Auction Design via Machine Learning Ellen Vitercik December 4, 2017 CMU Advanced Introduction to Machine Learning

2 Ad auctions contribute a huge portion of large internet companies revenue.

3 Ad revenue in 2016 Total revenue in 2016 Google $79 billion $89.46 billion Facebook $27 billion $27.64 billion

4 Leonardo da Vinci painting sells for $450.3m, shattering auction highs [NY Times 17]

5 Diamond fetches $33.7m at Christie s auction in Geneva [BBC 17]

6 The FCC spectrum auction is sending $10 billion to broadcasters [NiemanLab 17]

7 Very-large-scale generalized combinatorial multi-attribute auctions: Lessons from conducting $60 billion of sourcing [Sandholm 13]

8 How can we design revenuemaximizing auctions using machine learning?

9 1. Introduction 2. Auction design background 3. Revenue-maximizing auctions 4. Learning-theoretic auction design 5. Tools 6. Case study: Second price auctions with reserve prices 7. Related work

10 Common misconception: There s only one way to hold an auction.

11 There are infinitely many ways to design auctions.

12 Ascending price auction: Auctioneer calls out the opening bid. Bidders call out bids until there is no new bid within a period of time. Last bidder to bid wins the item and pays their bid.

13 Ascending price auction: Auctioneer calls out the opening bid. Bidders call out bids until there is no new bid within a period of time. Last bidder to bid wins the item and pays their bid. Online auctions can only take a fraction of a second, so we need faster auction formats.

14 2 1 Second price auction for a single good : Each bidder submits a bid. Highest bidder wins. They pay the second highest bid. pays 1

15 The second price auction is incentive compatible (every bidder will maximize their utility by bidding truthfully). value payment 1 wins item Why not bid above value? If winner, will stay winner and price won t change. If loser, might become winner, but will have to pay a price larger than value, which will result in negative utility.

16 The second price auction is incentive compatible (every bidder will maximize their utility by bidding truthfully). value payment 1 wins item Why not bid below value? If winner, might become loser, and will shift from non-negative to zero utility. If loser, will still be loser, so utility will still be zero.

17 Second price auction with a reserve price r R for a single good : Each bidder submits a bid. Highest bidder wins if their bid is above r. They pay maximum of r and the second highest bid.

18 2 1 Second price auction with a reserve price of 1.5 for a single good : Each bidder submits a bid. Highest bidder wins if their bid is above 1.5. They pay maximum of 1.5 and the second highest bid. pays 1.5 Optimizing reserve prices can lead to higher revenue. But how should we choose the reserve price?

19 1. Introduction 2. Auction design background 3. Revenue-maximizing auctions 4. Learning-theoretic auction design 5. Tools 6. Case study: Second price auctions with reserve prices 7. Related work

20 In 1981, Roger Myerson discovered the revenue-maximizing singleitem, multi-bidder auction. He won the 2007 Nobel prize in economics for his work on mechanism design. R. Myerson. Optimal auction design. Mathematics of Operations Research, 6(1):58 73, 1981.

21 Standard assumption A buyer's value for a bundle of goods is defined by a probability distribution over all the possible valuations she might have for that bundle. value ~ Uniform(1, 3)

22 Standard assumption A buyer's value for a bundle of goods is defined by a probability distribution over all the possible valuations she might have for that bundle. For simplicity, suppose all bidders have the same valuation distribution, with PDF f, CDF F, and support in [0, 1]. Myerson s optimal auction Let φ x = x 1 F(x) f(x). Run a second price auction with a reserve price of φ 1 0.

23 Myerson also analyzed the case where the bidders values are not identically distributed.

24 The revenue-maximizing multi-item auction is unknown!

25 Myerson s optimal auction Let φ x = x 1 F(x) f(x). Run a second price auction with a reserve price of φ 1 0. This requires that we know the distribution over bidders values. Where does this knowledge come from? In multi-item cases, the support of the distribution might be doubly exponential!

26 1. Introduction 2. Auction design background 3. Revenue-maximizing auctions 4. Learning-theoretic auction design 5. Tools 6. Case study: Second price auctions with reserve prices 7. Related work

27 value ~ There s an unknown distribution over valuations. Use a sample to learn a mechanism with high expected revenue.

28 value ~ Research papers: [Likhodedov and Sandholm 04, 05, Elkind 07, Dhangwatnotai et al. 10, Medina and Mohri 14, Cole and Roughgarden 14, Huang et al. 15, Morgenstern and Roughgarden 15, 16, Roughgarden and Schrijvers 16, Devanur et al. 16, Balcan, Sandholm, and V 16, 17, Hartline and Taggart 17, Syrgkanis 17, Alon et al. 17, Cai and Daskalakis 17, Gonczarowski and Nisan 17, Medina and Vassilvitskii 17, Dütting et al. 17]

29 value ~ Classes: Nina Balcan Connections between Learning, Game Theory, and Optimization Georgia Tech Fall 2010 Constantinos Daskalakis and Vasilis Syrgkanis Algorithmic Game Theory and Data Science MIT Spring 2017

30 value ~ Workshops and Tutorials: 2016 EC Tutorial on Algorithmic Game Theory and Data Science 2015, 2016, 2017 EC Workshop on Algorithmic Game Theory and Data Science 2017 NIPS Workshop on Learning in Presence of Strategic Behavior 2017 Dagstuhl Workshop on Game Theory Meets Computational Learning Theory

31 Fix an auction class M Look at the samples Optimize over M Field auction Central question: Is the resulting auction nearly optimal?

32 Fix an auction class M Look at the samples Optimize over M Field auction Central question: Does the resulting auction approximately maximize revenue in expectation over the unknown distribution?

33 Central question: Does the resulting auction approximately maximize revenue in expectation over the unknown distribution? Depends on the class M. Let M be the optimal auction and let M M be the auction returned by the learning algorithm. E Rev M E Rev M = Rev M max Total Revrevenue M + max loss Rev(M) Rev M M M M M

34 Central question: Does the resulting auction approximately maximize revenue in expectation over the unknown distribution? Depends on the class M. Let M be the optimal auction and let M M be the auction returned by the learning algorithm. = E Rev M max M M E Rev M E Rev M E Rev M + E Rev M E Rev M max M M Approximation loss Estimation loss

35 Central question: Does the resulting auction approximately maximize revenue in expectation over the unknown distribution? Depends on the class M. Simple class Complex class Bad approximation loss Good estimation loss Good approximation loss Bad estimation loss

36 These are the same problems we face in machine learning. Simple class Complex class Bad approximation loss Good estimation loss Good approximation loss Bad estimation loss

37 Say M is the class of second price auctions with reserve prices. Myerson s optimal auction for i.i.d. bidders Let φ x = x 1 F(x) f(x). Run a second price auction with a reserve price of φ 1 0. M has zero approximation loss when the bidders values are i.i.d.! This talk: What is the estimation loss of M?

38 1. Introduction 2. Auction design background 3. Revenue-maximizing auctions 4. Learning-theoretic auction design 5. Tools 6. Case study: Second price auctions with reserve prices 7. Related work

39 VC dimension: A complexity measure for binary-valued functions h: X 0,1

40 Example. For a, b R, a < b, let h a,b x = Let H int = h a,b : a, b R, a < b. 1, x (a, b) &0, x (a, b) No set S R of size 3 can be labeled in all 2 3 ways by H int.

41 The VC dimension of a class H is the size of the largest set S that can be labeled in all 2 S ways by functions in H. VC dimension A set S = x 1,, x m is shattered by H if for all b {0,1} S, there exists a function h H such that for all i [m], h x i = b i. The VC dimension of H is the size of the largest set S that can be shattered by H.

42 The VC dimension of a class H is the size of the largest set S that can be labeled in all 2 S ways by functions in H The VC dimension of H int is at least The VC dimension of H int is at most 2.

43 The VC dimension of a class H is the size of the largest set S that can be labeled in all 2 S ways by functions in H The VC dimension of H int is

44 Pseudo-dimension: A complexity measure for real-valued functions h: X R

45 x 1 h x 1 z (1) 0 x 2 h x 2 z (2) 0 x 3 h x 3 > z (3) 1 x 4 h x 4 z (4) 0 x 5 h x 5 > z (5) 1 x 6 h x 6 z (6) 0 x 7 h x 7 > z (7) 1 x 8 h x 8 > z (8) 1 x 9 h x 9 z (9) 0 Set S = x 1,, x m Class of functions H = h: X R z = z (1),, z (m) R m witnesses the shattering of S&by H if for all b {0,1} m, there exists h H such that h x i z (i) if b i = 0 and h x i > z (i) if b i = 1

46 x 1 h x 1 z (1) 0 x 2 h x 2 z (2) 0 x 3 h x 3 > z (3) 1 x 4 h x 4 z (4) 0 x 5 h x 5 > z (5) 1 x 6 h x 6 z (6) 0 x 7 h x 7 > z (7) 1 x 8 h x 8 > z (8) 1 h x 9 z (9) 0 Set S = x 1,, x m Class of functions H = h: X R z = z (1),, z (m) R m witnesses the shattering of S&by H if for all b {0,1} m, there exists h H such that h x i z (i) if b i = 0 and h x i > z (i) if b i = 1

47 x 1 h x 1 z (1) 0 x 2 h x 2 z (2) 0 x 3 h x 3 > z (3) 1 x 4 h x 4 z (4) 0 x 5 h x 5 > z (5) 1 x 6 h x 6 z (6) 0 x 7 h x 7 > z (7) 1 x 8 h x 8 > z (8) 1 Set S = x 1,, x m Class of functions H = h: X R z = z (1),, z (m) R m witnesses the shattering of S&by H if for all b {0,1} m, there exists h H such that h x i z (i) if b i = 0 and h x i > z (i) if b i = 1

48 x 1 h x 1 z (1) 0 x 2 h x 2 z (2) 0 x 3 h x 3 > z (3) 1 x 4 h x 4 z (4) 0 x 5 h x 5 > z (5) 1 x 6 h x 6 z (6) 0 x 7 h x 7 > z (7) 1 x 8 h x 8 > z (8) 1 Set S = x 1,, x m Class of functions H = h: X R z = z (1),, z (m) R m witnesses the shattering of S&by H if for all b {0,1} m, there exists h H such that h x i z (i) if b i = 0 and h x i > z (i) if b i = 1

49 x 1 h x 1 z (1) 0 x 2 h x 2 z (2) 0 x 3 h x 3 > z (3) 1 x 4 h x 4 z (4) 0 x 5 h x 5 > z (5) 1 x 6 h x 6 z (6) 0 x 7 h x 7 > z (7) 1 x 8 h x 8 > z (8) 1 Set S = x 1,, x m Class of functions H = h: X R z = z (1),, z (m) R m witnesses the shattering of S&by H if for all b {0,1} m, there exists h H such that h x i z (i) if b i = 0 and h x i > z (i) if b i = 1 The pseudo-dimension of H is the cardinality of the largest set that can be shattered by H.

50 x 1 h x 1 z (1) 0 x 2 h x 2 z (2) 0 x 3 h x 3 > z (3) 1 x 4 h x 4 z (4) 0 x 5 h x 5 > z (5) 1 x 6 h x 6 z (6) 0 x 7 h x 7 > z (7) 1 x 8 h x 8 > z (8) 1 Set S = x 1,, x m Class of functions H = h: X R z = z (1),, z (m) R m witnesses the shattering of S&by H if for all b {0,1} m, there exists h H such that h x i z (i) if b i = 0 and h x i > z (i) if b i = 1 The pseudo-dimension of H is the VC dimension of the set of below the graph functions (x, z) 1 h x z>0 &h H.

51 Theorem [Haussler 1992] Suppose H = h: X [ U, U] For every ε, δ > 0, and distribution D, if m = O U ε 2 Pdim(H), then with probability 1 δ over the draw of S = x 1,, x m ~D m, for all h H, 1 m m i=1 h x i E x~d [h x ] < ε

52 Corollary [Haussler 1992] Suppose H = h: X [ U, U] Given a sample S = x 1,, x m, let h S = max h H m i=1 h x i. For all ε, δ > 0, m = O U ε 2 Pdim(H) samples are sufficient to ensure that w.p. 1 δ over the draw of S = x 1,, x m ~D m, max h H E x~d h x E x~d h S x ε

53 Central question: Does the resulting auction approximately maximize revenue in expectation over the unknown distribution? Depends on the class M. Let M be the optimal auction and let M M be the auction returned by the learning algorithm. = E Rev M max M M E Rev M E Rev M E Rev M + E Rev M E Rev M max M M Approximation loss Estimation loss

54 Corollary [Haussler 1992] Suppose H = h: X [ U, U] Given a sample S = x 1,, x m, let h S = max h H m i=1 h x i. For all ε, δ > 0, m = O U ε 2 Pdim(H) samples are sufficient to ensure that w.p. 1 δ over the draw of S = x 1,, x m ~D m, max h H E x~d h x E x~d h S x ε Pseudo-dimension allows us to bound estimation loss.

55 Example. Let H be the set of all constant functions H = h x X, h x = c&for&some&c R

56 Example. For S = x 1, x 2 X, let z (1) and z (2) be two potential witnesses. WLOG, suppose z (1) z (2). There is no constant function h x = c where z (2) < h x 2 = c = h x 1 z (1). S is not shatterable. z (2) z (1) x 1 x 2

57 Example. For S = x 1, x 2 X, let z (1) and z (2) be two candidate witnesses. WLOG, suppose z (1) z (2). There is no constant function h x = c where z (2) < h x 2 = c = h x 1 z (1). S is not shatterable. Clearly, a single point is shatterable. Therefore, the pseudo-dimension of H is 1.

58 1. Introduction 2. Auction design background 3. Revenue-maximizing auctions 4. Learning-theoretic auction design 5. Tools 6. Case study: Second price auctions with reserve prices a) Recap b) Result 7. Related work

59 Fix an auction class M Look at the samples Optimize over M Field auction Central question: Does the resulting auction approximately maximize revenue in expectation over the unknown distribution?

60 Central question: Does the resulting auction approximately maximize revenue in expectation over the unknown distribution? Depends on the class M. Let M be the optimal auction and let M M be the auction returned by the learning algorithm. = E Rev M max M M E Rev M E Rev M E Rev M + E Rev M E Rev M max M M Approximation loss Estimation loss

61 Suppose M is the class of second price auctions with reserve prices. Myerson s optimal auction for i.i.d. bidders Let φ x = x 1 F(x) f(x). Run a second price auction with a reserve price of φ 1 0. M has zero approximation loss when the bidders valuations are i.i.d.! What is the estimation loss of M?

62 1. Introduction 2. Auction design background 3. Revenue-maximizing auctions 4. Learning-theoretic auction design 5. Tools 6. Case study: Second price auctions with reserve prices a) Recap b) Result 7. Related work

63 Notation: v = v 1,, v n ~D, where v i is the value of bidder i Let v (1) = max v 1,, v n and v (2) = max v 1,, v n v 1 For a reserve price r, rev r v = max r, v (2) 1 v 1 r. Define the class H = rev r v r& 0. We ve seen that the approximation loss for learning over H is zero when the bidders have i.i.d. valuations.

64 Notation: v = v 1,, v n ~D, where v i is the value of bidder i Let v (1) = max v 1,, v n and v (2) = max v 1,, v n v 1 For a reserve price r, rev r v = max r, v (2) 1 v 1 r. Define the class H = rev r v r& 0. What is the estimation loss for learning over H? To answer this, we ll bound the pseudo-dimension of H.

65 Theorem* The pseudo-dimension of the class H = rev r v r& 0 is 2. *Special case of a theorem by Morgenstern and Roughgarden [2016]

66 Proof. Suppose S = v (1),, v (m) is shatterable by H = rev r v r 0. There is a set of witnesses z (1),, z (m) such that for every b 0,1 m, there exists r b 0 such that rev rb v (i) z (i) if b i = 0 and rev rb v (i) > z (i) if b i = 1.

67 Proof. Let v S and let z be its witness. Write rev v r = rev r v. rev v r v (2) v (2) v (1) r

68 Proof. Suppose z < v (2). When r v (1), rev v r > z and when r > v (1), rev v r z. rev v r v (2) z v (1) r

69 Proof. Suppose z v (2). When r 0, z or r v 1,, rev v r z. When r z, v 1, rev v r > z. rev v r z z v (1) r

70 Proof. Notation: Let v (i) S and let z (i) be its witness. rev v (i) r Second highest value of v (i) (i) v (2) Highest value of v (i) (i) v (1) r

71 Proof. Sort z 1, v 1 1,, z m, v m 1. For all v (i) S, between these sorted values, either rev v (i) r > z (i) or rev v (i) r z (i). rev v (i) r (i) v (2) z i (i) v (1) r

72 Proof. Sort z 1, v 1 1,, z m, v m 1. For all v (i) S, between these sorted values, either rev v (i) r > z (i) or rev v (i) r z (i). rev v (i) r z i z i (i) v (1)

73 Proof. Recall: Suppose S = v (1),, v (m) is shatterable by H = rev r v r 0. There is a set of witnesses z (1),, z (m) such that for every b 0,1 m, there exists r b 0 such that rev rb v (i) z (i) if b i = 0 and rev rb v (i) > z (i) if b i = 1. Let R = r b b 0,1 m.

74 Proof. Sort z 1, v 1 1,, z m, v m 1. For all v (i) S, between these sorted values, either rev v (i) r > z (i) or rev v (i) r z (i). rev v (i) r (i) v (2) z i (i) v (1) r

75 Proof. Sort z 1, v 1 1,, z m, v m 1. For all v (i) S, between these sorted values, either rev v (i) r > z (i) or rev v (i) r z (i). rev v (i) r z i z i (i) v (1)

76 Proof. Sort z 1, v 1 1,, z m, v m 1. For all v (i) S, between these sorted values, either rev v (i) r > z (i) or rev v (i) r z (i). At most one r R is from each interval between sorted values. Therefore, 2 m = R 2m + 1, so m 2. Theorem The pseudo-dimension of the class H = rev r v r& 0 is 2.

77 Corollary Suppose the bidders valuations are i.i.d with support in [0,1]. Let H be the class of second price auctions with reserves. Approximation loss: 0 Estimation loss: O 1 S 2

78 Corollary Suppose the bidders valuations are i.i.d with support in [0,1]. Let H be the class of second price auctions with reserves. For a sample S = v (1),, v (m) ~D, let r S be the reserve price that m maximizes rev r v i i=1. O 1 ε2 samples are sufficient to ensure that with probability at least 1 δ, E v~d rev rs v max E v~d rev r v ε. r 0 This is tight [Cesa-Bianchi et al. 2015]!

79 1. Introduction 2. Auction design background 3. Revenue-maximizing auctions 4. Learning-theoretic auction design 5. Tools 6. Case study: Second price auctions with reserve prices 7. Related work

80 Other single-item auction batch learning papers: Elkind 07, Dhangwatnotai et al. 10, Medina and Mohri 14, Cole and Roughgarden 14, Huang et al. 15, Morgenstern and Roughgarden 15, Roughgarden and Schrijvers 16, Devanur et al. 16, Hartline and Taggart 17, Alon et al. 17, Gonczarowski and Nisan 17 Multi-item auction batch learning: Likhodedov and Sandholm 04, 05, Morgenstern and Roughgarden 16, Balcan, Sandholm, and V 16, 17, Syrgkanis 17, Cai and Daskalakis 17, Medina and Vassilvitskii 17 Online auction design: Kleinberg and Leighton 03, Blum and Hartline 05, Cesa-Bianchi et al. 15, Dudík et al. 17, Balcan, Dick and V 17

81 Auction design via deep learning: Dütting et al. 17 Solid regions represent allocation probability learned by neural network when there s a single bidder with v 1, v 2 ~U[0,1]. The optimal mechanism of Manelli and Vincent [ 06] is described by the regions separated by the dashed orange lines.