1. Maria-Florina Balcan Mechanism Design, Machine Learning, and Pricing Problems Maria-Florina Balcan 11/13/2007

2. Maria-Florina Balcan Outline Reduce problems of incentive-compatible mechanism design to standard algorithmic questions. Focus on revenue-maximization, unlimited supply. - Digital Good Auction - Attribute Auctions - Combinatorial Auctions Use ideas from Machine Learning. –Sample Complexity techniques in MLT for analysis. Approximation Algorithms for Item Pricing. Revenue maximization, unlimited supply combinatorial auctions with single-minded consumers [BBHM05] [BB06]

3. Maria-Florina Balcan MP3 Selling Problem We are seller/producer of some digital good (or any item of fixed marginal cost), e.g. MP3 files. Goal: Profit Maximization

4. Maria-Florina Balcan MP3 Selling Problem We are seller/producer of some digital good, e.g. MP3 files. Compete with fixed price. or… Use bidders’ attributes: country, language, ZIP code, etc. Goal: Profit Maximization Digital Good Auction (e.g., [GHW01]) Attribute Auctions [BH05] Compete with best “simple” function.

5. Maria-Florina Balcan Example 2, Boutique Selling Problem 20$ 30$ 5$ 25$ 20$ 100$ 1$

6. Maria-Florina Balcan Example 2, Boutique Selling Problem Goal: Profit Maximization Combinatorial Auctions Compete with best item pricing [GH01]. 20$ 30$ 5$ 25$ 20$ 100$ 1$ (unit demand consumers)

7. Maria-Florina Balcan Generic Setting (I) S set of n bidders. Bidder i: –priv i (e.g., how much is willing to pay for the MP3 file) –pub i (e.g., ZIP code) Unlimited supply Goal: Profit Maximization Profit of g:  i g(i) Space of legal offers/pricing functions. G - pricing functions. Goal: IC mech to do nearly as well as the best g 2 G. g(i) – profit obtained from making offer g to bidder i g(i)= p if p · priv i g(i)= 0 if p>priv i Digital Good g maps the pub i to pricing over the outcome space. g=“ take the good for p, or leave it”

8. Maria-Florina Balcan Attribute Auctions one item for sale in unlimited supply (e.g. MP3 files). bidder i has public attribute a i 2 X Example: X=R 2, G - linear functions over X G - a class of ‘’natural’’ pricing functions. Attr. space attributes valuations

9. Maria-Florina Balcan Generic Setting (II) Our results: reduce IC to AD. Algorithm Design: given (priv i, pub i ), for all i 2 S, find pricing function g 2 G of highest total profit. Incentive Compatible mechanism: offer for bidder i based on the public information of S and private info of S n {i}. Focus on one-shot mechanisms, off-line setting

10. Maria-Florina Balcan Main Results [BBHM05] Generic Reductions, unified analysis. General Analysis of Attribute Auctions: –not just 1-dimensional Combinatorial Auctions: –First results for competing against opt item-pricing in general case (prev results only for “unit-demand”[GH01]) –Unit demand case: improve prev bound by a factor of m. could offer great improvement over single price

11. Maria-Florina Balcan Basic Reduction: Random Sampling Auction RSOPF (G,A) Reduction Bidders submit bids. Randomly split the bidders into S 1 and S 2. Run A on S i to get (nearly optimal) g i 2 G w.r.t. S i. Apply g 1 over S 2 and g 2 over S 1. S S1S1 S2S2 g 1 =OPT(S 1 ) g 2 =OPT(S 2 )

12. Maria-Florina Balcan Basic Analysis, RSOPF (G, A) Theorem 1 1) Consider a fixed g and profit level p. Use McDiarmid ineq. to show: Proof sketch Lemma 1 h - maximum valuation, G - finite

13. Maria-Florina Balcan Basic Analysis, RSOPF (G,A), cont 2) Let g i be the best over S i. Know g i (S i ) ¸ g OPT (S i )/ . In particular, Using get that our profit g 1 (S 2 ) +g 2 (S 1 ) is at least (1-  )OPT G / .

14. Maria-Florina Balcan Basic Analysis, RSOPF (G, A) Theorem 2 Theorem 1 h - maximum valuation, G - finite

15. Maria-Florina Balcan Attribute Auctions, RSOPF (G k, A) G k : k markets defined by Voronoi cells around k bidders & fixed price within each market. Assume we discretize prices to powers of (1+  ). attributes

16. Maria-Florina Balcan Attribute Auctions, RSOPF (G k, A) G k : k markets defined by Voronoi cells around k bidders & fixed price within each market. Assume we discretize prices to powers of (1+  ). Corollary (roughly)

17. Maria-Florina Balcan Structural Risk Minimization Reduction SRM Reduction Let Randomly split the bidders into S 1 and S 2. Compute g i to maximize Apply g 1 over S 2 and g 2 over S 1. What if we have different functions at different levels of complexity? Don’t know best complexity level in advance. Theorem

18. Maria-Florina Balcan Attribute Auctions, Linear Pricing Functions Assume X=R d. N= (n+1)(1/  ) ln h. |G’| · N d+1 attributes valuations x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x

19. Maria-Florina Balcan Covering Arguments Definition: G’  -covers G wrt to S if for 8 g 9 g’ 2 G’ s.t. 8 i |g(i)-g’(i)| ·  g(i). What if G is infinite w.r.t S? Use covering arguments: find G’ that covers G, show that all functions in G’ behave well Theorem (roughly) If G’ is  -cover of G, then the previous theorems hold with |G| replaced by |G’|. attributes valuations Analysis Techniqu e

20. Maria-Florina Balcan Conclusions and Open Problems [BBHM05] Explicit connection between machine learning and mechanism design. Use of ideas in MLT for both design and analysis in auction/pricing problems. Unique challenges & particularities: Loss function discontinuous and asymmetric. Range of valuations large. Apply similar techniques to limited supply. Study Online Setting. Open Problems

21. Maria-Florina Balcan Outline Reduce problems of incentive-compatible mechanism design to standard algorithmic questions. Focus on revenue-maximization, unlimited supply. Use ideas from Machine Learning. –Sample Complexity techniques in MLT for analysis. Approximation Algorithms for Item Pricing. Revenue maximization, unlimited supply combinatorial auctions with single-minded bidders [BBHM05] [BB06]

22. Maria-Florina Balcan Algorithmic Problem, Single-minded Customers m item types (coffee, cups, sugar, apples, …), with unlimited supply of each. n customers. Say all marginal costs to you are 0 [revisit this in a bit], and you know all the (L i, w i ) pairs. Each customer i has a shopping list L i and will only shop if the total cost of items in L i is at most some amount w i (otherwise he will go elsewhere). What prices on the items will make you the most money? Easy if all L i are of size 1. What happens if all L i are of size 2?

23. Maria-Florina Balcan Algorithmic Pricing, Single-minded Customers A multigraph G with values w e on edges e. Goal: assign prices on vertices p v ¸ 0 to maximize total profit, where: APX hard [GHKKKM’05]. 10 40 15 20 30 5 10 5

24. Maria-Florina Balcan A Simple 2-Approx. in the Bipartite Case Goal: assign prices on vertices p v ¸ 0 as to maximize total profit, where: Set prices in R to 0 and separately fix prices for each node on L. Set prices in L to 0 and separately fix prices for each node on R Take the best of both options. Algorithm Given a multigraph G with values w e on edges e. Proof simple ! OPT=OPT L +OPT R 40 15 25 35 15 25 5 LR

25. Maria-Florina Balcan A 4-Approx. for Graph Vertex Pricing Goal: assign prices on vertices p v ¸ 0 to maximize total profit, where: Randomly partition the vertices into two sets L and R. Ignore the edges whose endpoints are on the same side and run the alg. for the bipartite case. Algorithm Proof In expectation half of OPT’s profit is from edges with one endpoint in L and one endpoint in R. Given a multigraph G with values w e on edges e. simple ! 10 40 15 20 30 5 10 5

26. Maria-Florina Balcan Algorithmic Pricing, Single-minded Customers, k-hypergraph Problem What about lists of size · k? –Put each node in L with probability 1/k, in R with probability 1 – 1/k. –Let GOOD = set of edges with exactly one endpoint in L. Set prices in R to 0 and optimize L wrt GOOD. Let OPT j,e be revenue OPT makes selling item j to customer e. Let X j,e be indicator RV for j 2 L & e 2 GOOD. Our expected profit at least: Algorithm 10 15 20

27. Maria-Florina Balcan Algorithmic Pricing, Single-minded Customers What if items have constant marginal cost to us? 10 40 15 20 3 5 7 5 We can subtract these from each edge (view edge as amount willing to pay above our cost). But, one difference: Can now imagine selling some items below cost in order to make more profit overall. 2 3 32 8 Reduce to previous problem.

28. Maria-Florina Balcan Algorithmic Pricing, Single-minded Customers What if items have constant marginal cost to us? We can subtract these from each edge (view edge as amount willing to pay above our cost). Reduce to previous problem. But, one difference: Can now imagine selling some items below cost in order to make more profit overall. Previous results only give good approximation wrt best “non-money-losing” prices. Can actually give a log(m) gap between the two benchmarks. 4 48 1 12 2 4 48 1 12 2 0 44 0

29. Maria-Florina Balcan Conclusions and Open Problems [BB06] Summary: 4 approx for graph case. O(k) approx for k-hypergraph case. Improves the O(k 2 ) approximation of Briest and Krysta, SODA’06. –Also simpler and can be naturally adapted to the online setting. 4 - , o(k). How well can you do if pricing below cost is allowed? Open Problems

30. Maria-Florina Balcan More On Revenue Maximization in Combinatorial Auctions Item Pricing in Unlimited Supply Combinatorial Auctions General bidders. Item Pricing in Limited Supply Combinatorial Auctions Bidders with subadditive valuations. [Balcan-Blum-Mansour’07]

31. Maria-Florina Balcan General Bidders Can extend [GHKKKM05] and get a log-factor approx for general bidders by an item pricing. There exists a price a p which gives a log(m) +log (n) approximation to the total social welfare. Theorem Can we say anything at all??

32. Maria-Florina Balcan General Bidders Can we do this via Item Pricing? Can extend [GHKKKM05] and get a log-factor approx for general bidders by an item pricing. Note: if bundle pricing is allowed, can do it easily. –Pick a random admission fee from {1,2,4,8,…,h} to charge everyone. –Once you get in, can get all items for free. For any bidder, have 1/log chance of getting within factor of 2 of its max valuation.

33. Maria-Florina Balcan Unlimited Supply, General Bidders Focus on a single customer. Analyze demand curve. Claim 1: # is monotone non-increasing with p. # items price n0n0 p 0 =0 p1p1 p2p2 p L-1 pLpL n1n1 nLnL - -

34. Maria-Florina Balcan Unlimited Supply, General Bidders Focus on a single customer. Analyze demand curve. price # items n0n0 p 0 =0 p1p1 p2p2 p L-1 pLpL n1n1 nLnL - - Claim 2: customer’s max valuation · integral of this curve.

35. Maria-Florina Balcan Unlimited Supply, General Bidders Focus on a single customer. Analyze demand curve. price n0n0 p 0 =0 p1p1 p2p2 p L-1 pLpL n1n1 nLnL - - Claim 2: customer’s max valuation · integral of this curve. # items

36. Maria-Florina Balcan Unlimited Supply, General Bidders Focus on a single customer. Analyze demand curve. price n0n0 p 0 =0 p1p1 p2p2 p L-1 pLpL n1n1 nLnL - - Claim 2: customer’s max valuation · integral of this curve. # items

37. Maria-Florina Balcan Unlimited Supply, General Bidders Focus on a single customer. Analyze demand curve. price n0n0 p 0 =0 p1p1 p2p2 p L-1 pLpL n1n1 nLnL - - Claim 2: customer’s max valuation · integral of this curve. # items

38. Maria-Florina Balcan Unlimited Supply, General Bidders Focus on a single customer. Analyze demand curve. price n0n0 p 0 =0 p1p1 p2p2 p L-1 pLpL n1n1 nLnL - - Claim 2: customer’s max valuation · integral of this curve. # items

39. Maria-Florina Balcan Unlimited Supply, General Bidders Focus on a single customer. Analyze demand curve. price n0n0 0 h/4 h/2 h n1n1 nLnL - - Claim 3: random price in {h, h/2, h/4,…, h/(2n)} gets a log(n)-factor approx. # items

40. Maria-Florina Balcan Unlimited Supply, General Bidders Focus on a single customer. Analyze demand curve. price n0n0 0 h/4 h/2 h n1n1 nLnL - - Claim 3: random price in {h, h/2, h/4,…, h/(2n)} gets a log(n)-factor approx. # items

41. Maria-Florina Balcan What about the limited supply setting?

42. Maria-Florina Balcan What about Limited Supply? Assume one copy of each item. Fixed Price (p): Set R=J. For each bidder i, in some arbitrary order: Let S i be the demanded set of bidder i given the following prices: p for each item in R and for each item in J\R. Allocate S i to bidder i and set R=R \ S i. Goal: Profit Maximization Assume bidders with subadditive valuations.

43. Maria-Florina Balcan Limited Supply, Subadditive Valuations There exists a single price mechanism whose profit is a approximation to the social welfare. Can show a lower bound, for  =1/4. [DNS’06], [D’07] show that a single price mechanism provides a logarithmic approx. for social welfare in the submodular, subadditive case. [DNS’06] show a approximation to the total welfare for bidders with general valuations. welfare & revenue Other known results: welfare

44. Maria-Florina Balcan A Property of Subadditive Valuations Lemma 1 Let (T 1, …, T m ) be feasible allocation. There exists (L 1, …, L m ) and a price p such that : (2) (L 1, …, L m ) is supported at price p. (1) Assume v i subadditive. L i the subset that bidder i buys in a store where he sees only T i and every item is priced at p.

45. Maria-Florina Balcan Subadditive Valuations, Limited Supply Lemma 1 price p such that : and (L 1, …, L m ) is supported at price p. Lemma 2 be the allocation produced by FixedPrice (p/2). Then: Let (T 1, …, T m ) be feasible allocation. 9 (L 1, …, L m ) and Assume (L 1, …, L m ) is supported at p and let (S 1, …, S m ) There exists a single price mechanism whose profit is a Theorem approximation to the social welfare.

46. Maria-Florina Balcan Summary Item Pricing mechanism for limited supply setting. Matching upper and lower bounds.