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Mechanism Design, Machine Learning, and Pricing Problems

Mechanism Design, Machine Learning, and Pricing Problems. Maria-Florina Balcan. 11/13/2007. Outline. Reduce problems of incentive-compatible mechanism design to standard algorithmic questions. [BBHM05]. Focus on revenue-maximization , unlimited supply . - Digital Good Auction

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Mechanism Design, Machine Learning, and Pricing Problems

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

  2. Outline • Reduce problems of incentive-compatible mechanism design to standard algorithmic questions. [BBHM05] • 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. [BB06] • Revenue maximization, unlimited supply combinatorial auctions with single-minded consumers Maria-Florina Balcan

  3. 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 Maria-Florina Balcan

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

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

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

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

  8. valuations Attr. space attributes Attribute Auctions • one item for sale in unlimited supply (e.g. MP3 files). • bidder i has public attribute ai 2 X • G - a class of ‘’natural’’ pricing functions. Example: X=R2, G - linear functions over X Maria-Florina Balcan

  9. Generic Setting (II) • Our results: reduce IC to AD. • Algorithm Design:given (privi, pubi), 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 Maria-Florina Balcan

  10. 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 Maria-Florina Balcan

  11. S1 S g1=OPT(S1) g2=OPT(S2) S2 Basic Reduction: Random Sampling Auction RSOPF(G,A) Reduction • Bidders submit bids. • Randomly split the bidders into S1 and S2. • Run A on Si to get (nearly optimal) gi2 G w.r.t. Si. • Apply g1 over S2 and g2 over S1. Maria-Florina Balcan

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

  13. Basic Analysis, RSOPF(G,A), cont 2) Let gi be the best over Si. Know gi(Si) ¸ gOPT(Si)/. In particular, Using get that our profit g1(S2) +g2(S1) is at least (1-)OPTG/. Maria-Florina Balcan

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

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

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

  17. Structural Risk Minimization Reduction What if we have different functions at different levels of complexity? Don’t know best complexity level in advance. SRM Reduction • Let • Randomly split the bidders into S1 and S2. • Compute gi to maximize • Apply g1 over S2 and g2 over S1. Theorem Maria-Florina Balcan

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

  19. valuations attributes Covering Arguments • 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 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). AnalysisTechnique Theorem (roughly) If G’ is -cover of G, then the previous theorems hold with |G| replaced by |G’|. Maria-Florina Balcan

  20. 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. Open Problems • Apply similar techniques to limited supply. • Study Online Setting. Maria-Florina Balcan

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

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

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

  24. L R 15 25 35 15 25 5 40 A Simple 2-Approx. in the Bipartite Case • Given a multigraph G with values we on edges e. • Goal: assign prices on vertices pv¸ 0 as to maximize total profit, where: Algorithm • 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. simple! Proof OPT=OPTL+OPTR Maria-Florina Balcan

  25. 5 15 10 10 40 20 30 5 A 4-Approx. for Graph Vertex Pricing • Given a multigraph G with values we on edges e. • Goal: assign prices on vertices pv¸ 0 to maximize total profit, where: Algorithm • 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. Proof simple! In expectation half of OPT’s profit is from edges with one endpoint in L and one endpoint in R. Maria-Florina Balcan

  26. 15 10 20 Algorithmic Pricing, Single-minded Customers,k-hypergraph Problem What about lists of size · k? Algorithm • 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 OPTj,e be revenue OPT makes selling item j to customer e. Let Xj,e be indicator RV for j 2 L & e 2 GOOD. • Our expected profit at least: Maria-Florina Balcan

  27. 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). 5 3 2 15 10 7 40 20 3 • Reduce to previous problem. 8 32 5 But, one difference: • Can now imagine selling some items below cost in order to make more profit overall. Maria-Florina Balcan

  28. 8 8 4 4 4 4 2 1 2 1 1 2 1 2 0 0 4 4 Algorithmic Pricing, Single-minded Customers • We can subtract these from each edge (view edge as amount willing to pay above our cost). What if items have constant marginal cost to us? • 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. Maria-Florina Balcan

  29. Conclusions and Open Problems [BB06] • 4 approx for graph case. • O(k) approx for k-hypergraph case. Summary: Improves the O(k2) approximation of Briest and Krysta, SODA’06. • Also simpler and can be naturally adapted to the online setting. Open Problems • 4 - , o(k). • How well can you do if pricing below cost is allowed? Maria-Florina Balcan

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

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

  32. General Bidders • Can extend[GHKKKM05]and get a log-factor approx for general bidders by anitem pricing. Can we do this via 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. Maria-Florina Balcan

  33. # items n0 n1 - nL - pL-1 pL p0=0 p1 p2 price Unlimited Supply, General Bidders Focus on a single customer. Analyze demand curve. • Claim 1: # is monotone non-increasing with p. Maria-Florina Balcan

  34. Unlimited Supply, General Bidders Focus on a single customer. Analyze demand curve. # items n0 n1 - nL - pL-1 pL p0=0 p1 p2 price • Claim 2: customer’s max valuation · integral of this curve. Maria-Florina Balcan

  35. Unlimited Supply, General Bidders Focus on a single customer. Analyze demand curve. # items n0 n1 - nL - pL-1 pL p0=0 p1 p2 price • Claim 2: customer’s max valuation · integral of this curve. Maria-Florina Balcan

  36. Unlimited Supply, General Bidders Focus on a single customer. Analyze demand curve. # items n0 n1 - nL - pL-1 pL p0=0 p1 p2 price • Claim 2: customer’s max valuation · integral of this curve. Maria-Florina Balcan

  37. Unlimited Supply, General Bidders Focus on a single customer. Analyze demand curve. # items n0 n1 - nL - pL-1 pL p0=0 p1 p2 price • Claim 2: customer’s max valuation · integral of this curve. Maria-Florina Balcan

  38. Unlimited Supply, General Bidders Focus on a single customer. Analyze demand curve. # items n0 n1 - nL - pL-1 pL p0=0 p1 p2 price • Claim 2: customer’s max valuation · integral of this curve. Maria-Florina Balcan

  39. Unlimited Supply, General Bidders Focus on a single customer. Analyze demand curve. # items n0 n1 - nL - 0 h/2 h/4 h price • Claim 3: random price in {h, h/2, h/4,…, h/(2n)} gets a log(n)-factor approx. Maria-Florina Balcan

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

  41. What about Limited Supply? Assume one copy of each item. Goal: Profit Maximization Fixed Price (p): Set R=J. For each bidder i, in some arbitrary order: • Let Si 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 Si to bidder i and set R=R \ Si. Assume bidders with subadditive valuations. Maria-Florina Balcan

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

  43. A Property of Subadditive Valuations Lemma 1 Assume visubadditive. Let (T1, …, Tm) be feasible allocation. There exists (L1, …, Lm) and a price p such that : (1) (2) (L1, …, Lm) is supported at price p. Lithe subset that bidder i buys in a store where he sees only Tiand every item is priced at p. Maria-Florina Balcan

  44. Subadditive Valuations, Limited Supply Lemma 1 Let (T1, …, Tm) be feasible allocation. 9(L1, …, Lm) and price p such that : and (L1, …, Lm) is supported at price p. Lemma 2 Assume (L1, …, Lm) is supported at p and let (S1, …, Sm) be the allocation produced by FixedPrice (p/2). Then: Theorem • There exists asingleprice mechanism whose profit is a approximation to the social welfare. Maria-Florina Balcan

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

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