and Pricing Problems

Transcription

1 Mechanism Design, Machine Learning, and Pricing Problems Maria-Florina Balcan Carnegie Mellon University

2 Overview Pricing and Revenue Maimization Software Pricing Digital Music

3 Pricing Problems One Seller, Multiple Buyers with Comple Preferences. Seller s Goal: maimize profit. Algorithm Design Problem (AD) Incentive Compatible Auction (IC) Version 1: Seller knows the true values. Version 2: values given by selfish agents. BBHM 05: Generic Reduction based on ML techniques

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

5 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: Approimation Algorithms for Item Pricing. [Balcan-Blum, EC 2006, TCS 2007] Revenue maimization in combinatorial auctions with single-minded consumers.

6 MP3 Selling Problem Seller of some digital good (or any item of fied marginal cost), e.g. MP3 files. Goal: Profit Maimization

7 MP3 Selling Problem Seller/producer of some digital good, e.g. MP3 files. Goal: Profit Maimization Digital Good Auction (e.g., [GHW01]) Compete with fied price. or Use bidders attributes: country, language, ZIP code, etc. Compete with best simple function. Attribute Auctions [BH05]

8 Eample 2, Boutique Selling Problem $20 $5 $100 30$ $25 $30 $1 $20

9 Eample 2, Boutique Selling Problem $20 $5 $100 30$ $25 $30 $1 $20 Combinatorial Auctions Goal: Profit Maimization Compete with best item pricing [GH01]. (unit demand consumers)

10 S set of n bidders. Bidder i: Generic Setting (I) priv i (e.g., how much i is willing to pay for the MP3 file) pub i (e.g., ZIP code) bid i ( reported priv i ) O outcome space. Incentive Compatible: bid i =priv i Space of legal offers/pricing functions. g maps the pub i 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 bid i g(i)= 0 if p>bid i

11 Generic Setting (I) S set of n bidders. Bidder i: priv i, pub i, bid i Space of legal offers/pricing functions. g maps the pub i to pricing over the outcome space. g(i) profit obtained from making offer g to bidder i Goal: Profit Maimization G - pricing functions. Goal: Incentive Compatible mechanism to do nearly as well as the best g 2 G. Unlimited supply Profit of g: i g(i)

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

13 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: bid i =priv i 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.

14 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.

15 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 g 1 =OPT(S 1 ) S 1 S 2 g 2 =OPT(S 2 )

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

17 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 also OPT G n, get that our profit g 1 (S 2 ) +g 2 (S 1 ) is at least (1- )OPT G /.

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

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

20 Structural Risk Minimization Reduction What if different functions at different levels of compleity? Don t know best compleity level in advance. SRM Reduction Let Randomly split the bidders into S 1 and S 2. Compute g i to maimize Apply g 1 over S 2 and g 2 over S 1. Theorem

21 Attribute Auctions, Linear Pricing Functions Assume X=R d. N= (n+1)(1/ ) ln h. G N d+1 attributes valuations

22 Covering Arguments What if G is infinite w.r.t S? valuations 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). attributes Theorem (roughly) Analysis Technique If G is -cover of G, then the previous theorems hold with G replaced by G.

23 Summary [BBHM05] Eplicit 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. See also upcoming paper of [Morgenstern, Roughgarden, NIPS 15] for other settings (e.g., limited supply)!

24 Outline Part I: Generic Framework for reducing problems of incentive-compatible mechanism design to standard algorithmic questions. Part II: Approimation Algorithms for Item Pricing. [Balcan-Blum, EC 2006, TCS 2007] Revenue maimization in combinatorial auctions with single-minded consumers

25 Algorithmic Problem, Single-minded Bidders [BB 06] m item types with unlimited supply of each. n single-minded customers. Customer i: shopping list L i, will only shop if the total cost of items in L i is at most w i All marginal costs are 0, and we know all the (L i, w i ). 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?

26 Algorithmic Problem, Single-minded Bidders [BB 06] A multigraph G with values w e on edges e. 5 Goal: assign prices on vertices to maimize total profit, where: Unlimited supply APX hard [GHKKKM 05].

27 A Simple 2-Appro. in the Bipartite Case Given a multigraph G with values w e on edges e. Goal: assign prices on vertices to maimize total profit, where: Algorithm Set prices in R to 0 and separately fi prices for each node on L. Set prices in L to 0 and separately fi prices for each node on R. Take the best of both options. L 5 40 R Proof simple! OPT=OPT L +OPT R

28 A 4-Appro. for Graph Verte Pricing Given a multigraph G with values w e on edges e. Goal: assign prices on vertices to maimize 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 epectation half of OPT s profit is from edges with one endpoint in L and one endpoint in R.

29 Algorithmic Pricing, Single-minded Bidders, k-hypergraph Problem List of size k. 15 Algorithm Put each node in L with prob. 1/k, in R with prob. 1 1/k. Let GOOD = set of edges with eactly 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 epected profit at least:

30 4 appro for graph case. Summary [BB06]: O(k) appro for k-hypergraph case. Improves the O(k 2 ) approimation of Briest and Krysta, SODA 06. Also simpler and can be naturally adapted to the online setting. Other known results: O(log mn) appro. by picking the best single price [GHKKKM05]. (log n) hardness for general case [DFHS06].

31 Overall Summary Revenue Maimization in a wide range of settings. Both Algorithmic and Incentive Compatible Aspects. Natural Connections to Machine Learning.

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