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Mechanism Design, Machine Learning, and Pricing Problems

Mechanism Design, Machine Learning, and Pricing Problems. Maria-Florina Balcan. Overview. Problems at the intersection of CS and Economics. Pricing and Revenue Maximization. Software Pricing. Digital Music. Overview. Problems at the intersection of CS and Economics. Advertising.

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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

  2. Overview Problems at the intersection of CS and Economics Pricing and Revenue Maximization Software Pricing Digital Music

  3. Overview Problems at the intersection of CS and Economics Advertising Pricing and Revenue Maximization

  4. Pricing Problems One Seller, Multiple Buyers with Complex Preferences. Seller’s Goal: maximize profit. Computer Science Economics Version 2: values given by selfish agents. Version 1: Seller knows the true values. Algorithm Design Problem (AD) Incentive Compatible Auction (IC) Previous Work on IC : very specific mechanisms for restricted settings.

  5. Pricing Problems One Seller, Multiple Buyers with Complex Preferences. Seller’s Goal: maximize profit. Computer Science Economics Version 2: values given by selfish agents. Version 1: Seller knows the true values. Algorithm Design Problem (AD) Incentive Compatible Auction (IC) BBHM (FOCS 2005): Generic Reduction Previous Work on IC : very specific mechanisms for restricted settings.

  6. Reduce IC to AD Generic Framework for reducing problems of incentive-compatible mechanism design to standard algorithmic questions. [Balcan-Blum-Hartline-Mansour, FOCS 2005, JCSS 2007] • Focus on revenue-maximization, unlimited supply. • - Digital Good Auction • - Attribute Auctions • - Combinatorial Auctions • Use ideas from Machine Learning. • Sample Complexity techniques in ML both for design and analysis .

  7. Outline Part I: Generic Framework for reducing problems of incentive-compatible mechanism design to standard algorithmic questions. [Balcan-Blum-Hartline-Mansour, FOCS 2005, JCSS 2007] Part II: Approximation Algorithms for Item Pricing. E.g, [Balcan-Blum, EC 2006, TCS 2007] Revenue maximizationin combinatorial auctions with single-minded consumers.

  8. MP3 Selling Problem • Seller of some digital good (or any item of fixed marginal cost), e.g. MP3 files. Goal: Profit Maximization

  9. MP3 Selling Problem • 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]

  10. $100 $20 $5 $30 $25 30$ $1 $20 Example 2, Boutique Selling Problem

  11. $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)

  12. Generic Setting (I) O outcome space. • Bidder i: • S set of n bidders. • privi(e.g., how much i is willing to pay for the MP3 file) • pubi(e.g., ZIP code) • bidi ( reported privi) • Incentive Compatible: bidi =privi • 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 · bidi • g(i)= 0 if p>bidi

  13. Generic Setting (I) • Bidder i: • privi • , pubi • , bidi • S set of n bidders. • 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 Goal:Profit Maximization • G - pricing functions. • Goal:Incentive Compatible mechanism to do nearly as well as the best g 2 G. Unlimited supply Profit of g: ig(i)

  14. 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

  15. Generic Setting (II) • Our results: reduce ICtoAD. • Algorithm Design:given (privi, pubi), for all i 2 S, find pricing function g 2 G of highest total profit. • Incentive Compatiblemechanism: bidi=privi • offer for bidder i based on the public information of S and reported private info of S n{i}. • Focus on one-shot mechanisms, off-line setting.

  16. 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.

  17. 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.

  18. Basic Analysis, RSOPF(G, A) h - maximum valuation,G - finite Theorem 1 Proof sketch 1) Fixed g and profit level p. Use a tail ineq. show: Lemma 1

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

  20. Attribute Auctions, RSOPF(Gk, A) Gk : k markets defined by Voronoi cells around k bidders & fixed price within each market. Discretize prices to powers of (1+). attributes

  21. Attribute Auctions, RSOPF(Gk, A) Gk : k markets defined by Voronoi cells around k bidders & fixed price within each market. Discretize prices to powers of (1+). Corollary (roughly)

  22. Structural Risk Minimization Reduction What if 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

  23. 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

  24. 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). Analysis Technique Theorem (roughly) If G’ is -cover of G, then the previous theorems hold with |G| replaced by |G’|.

  25. Summary [BBHM05] • Explicit connection between machine learning and mechanism design. • Use MLT both for design and analysis in auction/pricing problems. • Unique challenges & particularities: • Loss function discontinuous and asymmetric. • Range of valuations large.

  26. Thank you !

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