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Mechanism Design via Machine Learning

Mechanism Design via Machine Learning. Maria-Florina Balcan. Joint work with Avrim Blum, Jason Hartline, and Yishay Mansour. An Overview of the Results. Reduce problems of incentive-compatible mechanism design to standard algorithmic questions.

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Mechanism Design via Machine Learning

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  1. Mechanism Design via Machine Learning Maria-Florina Balcan Joint work with Avrim Blum, Jason Hartline, and Yishay Mansour Maria-Florina Balcan

  2. An Overview of the Results • 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 • Often, improve previous known bounds. 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]. 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. • A mapping : Digital Good • (p,privi)= p if p · privi • (p,privi)= 0 if p>privi (offer, privi) ! profiti Goal: Profit Maximization • G - pricing functions, g 2 G maps the pubi to an offer. • Goal: IC mech to do nearly as well as the best g 2 G. • Profit of g: i(g(pubi),privi) 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 info of S and private info of S n{i}. Try to compete with best g 2 G. Maria-Florina Balcan

  10. 20$ 5$ 30$ 25$ 2$ 5$ Our Contributions • 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. 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) 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 also OPTG¸ n, get that our profit g1(S2) +g2(S1) is at least (1-)OPTG/. Maria-Florina Balcan

  14. 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+). 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+). Corollary (roughly) Maria-Florina Balcan

  16. 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

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

  18. 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). Theorem (roughly) If G’ is -cover of G, then the previous theorems hold with |G| replaced by |G’|. Maria-Florina Balcan

  19. Conclusions • 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. Maria-Florina Balcan

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