Recall Myerson Optimal Auction

1. Recall Myerson Optimal Auction • Myerson: Bayesian Single-item Auction problem: • Single item for sale (can be extended to ``service”) • bidders • Distribution from which bidder values are drawn • VCG on virtual values TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAAAA

2. Quick Question (to make sure everyone is awake): • What about Myerson when bidders are correlated? (Not a product distribution) • Answer: Can beat Myerson revenue (by cheating somewhat) • Exp. Utility to agent if value = 10 is zero • Exp. Utility to agent if value = 100 is 30

3. Joint distribution • Run VCG AND charge extra payments: • P(b) – extra payment if OTHER GUY bids b • Both bid 100: both get utility -60 • Both bid 10: both get utility +30 Better than Reserve price of 100 Revenue = TOTAL “surplus”

4. Papers today (very partially) • On Profit-Maximizing Envy-Free Pricing • Guruswami, Hartline, Karlin, Kempe, Kenyon, McSherry • SODA 05 • Algorithmic Pricing via Virtual Valuations • Chawla, Hartline, Kleinberg • arXiv 2008 • Pricing Randomized Allocations • Briest, Chawla, Kleinberg, Weinberg • arXiv, 2009 • Approximate Revenue Maximization with Multiple Items • Hart, Nisan • arXiv 2012

5. Bayesian Unit Demand Pricing Problem • Myerson: Bayesian Single-item Auction problem: • Single item for sale (can be extended to ``service”) • bidders • Distribution from which bidder values are drawn • VCG on virtual values • Bayesian Unit-demand Pricing Problem: • Single unit-demand bidder • items for sale • Distribution from which the bidder valuation for items is drawn

6. What’s the connection • How do n bidders single item relate to single bidder n items?

7. For the Bayesian Single item Auction and the Bayesian Unit-demand Pricing problem are the same problem. • Offer the item at a price of where is the virtual valuation function for distribution (For regular distributions, Use ironed virtual values if irregular)

8. Theorem: For any price vector p, the revenue of Myerson (when bidder value of the single item comes from is the revenue when the single bidder gets value for item from • Given price vector p, new mechanism M: • Allocate the item to the bidder with that maximizes • Have bidder pay the critical price (the lowest bid at which bidder would win. • The allocation function is monotone and so this is a truthful mechanism. • Myerson is optimal amongst all Bayes Nash IC mechanisms, • Myerson gets more revenue than M

9. Myerson Revenue is Opt revenue for Single bidder Bayesian values • Allocate the item to the bidder with that maximizes • Have bidder pay the critical price (the lowest bid at which bidder would win. • The minimum bid for bidder to win is This is the revenue to M when wins • The revenue from a single bidder who buys item with value at a price of is • QED

10. Reg distributions: almost tight (??) • Let - the probability that no item is sold at pricing vector • Use prices , where is chosen as follows: • Choose such that and • Theorem (Chawla, Hartline, B. Kleinberg): Gives no less than Myerson revenue / 3 • Algorithmic Pricing via Virtual Valuations

11. Discrete Distributions

12. Distribution vs. set of Bidders The following two problems are equivalent: • Given n items and m “value vectors” for m bidders compute the revenue-optimal item pricing • Given a single bidder whose valuations are chosen uniformly at random from compute the revenue-optimal item pricing

13. APX hardness of optimal (deterministic) item pricing • Reduction from vertex cover on graphs of maximum degree at most B (APX hard for B3). • Given (connected) Graph G with n vertices construct n items and m+n bidders (or bidder types) • For every edge add bidder with value 1 for items and zero otherwise • Additionally, for every item there is a bidder with value 2 for the item. • The optimal pricing gives profit, where is the smallest vertex cover of

14. APX hardness of (deterministic) item pricing • Given (connected) Graph G with n vertices construct n items and m+n bidders (or bidder types) • For every edge add bidder with value 1 for items and zero otherwise • Additionally, for every item there is a bidder with value 2 for the item. • The optimal pricing gives profit, where is the smallest vertex cover of • Let S be vertex cover, charge 1 for items in S, 2 otherwise • If and price(i)=price(j)=2 then no profit from bidder - reducing price of (say) to one keeps profit unchange.

15. Pricing Randomized Allocations • Patrick Briest,ShuchiChawla, Robert Kleinberg, S. Matthew Weinberg • Remark: When speaking, say “Bobby Kleinberg”, when writing write “Robert Kleinberg”.

16. Selling Lotteries • 2 items uniformly value uniformly distributed in [a,b] • Optimal item pricing sets • Offer lottery equal prob item 1 or item 2, at cost • Lose here • Gain here

17. Different models • Buy-one model: Consumer can only buy one option • Buy-many model: Consumer can buy any number of lotteries and get independent sample from each (will discard multiple copies)

18. Buy-one model • Polytime algorithm to compute optimal lottery pricing • For item pricing (also known as envy-free unit demand pricing) the optimal item pricing is APX-hard (earlier today) • There is no finite ratio between optimal lottery revenue and optimal item-pricing revenue (for

19. Lotteries • where , and price of lottery • – collection of lotteries • Bidder type is • Utility of picking lottery is • Utility maximizing lotteries • Max payment for utility maximizing lottery: • Profit of is

20. Poly time optimal algorithm in case of finite support values of type j bidders - prob of type j bidders – lottery designed for type j - price for lottery Feasible Affordable Type j prefer lottery j

21. Compare lottery revenue with item pricing revenue • - maximal expected revenue for single bidder when offered optimal revenue maximizing lotteries, bidder values from (joint) distribution • - maximal expected revenue for single bidder when offered optimal revenue maximizing item prices, bidder values from (joint) distribution

22. Theorem: Assume optimal WLOG Valuation chooses to purchase

23. Valuation chooses to purchase

24. Making sense of it (again) • Correlated distributions, unit demand, finite distribution over types: • Deterministic, item pricing • We saw: APX hardness • Lower bound of (m- number of items) or - number of different agent types – not standard complexity assumptions Briest • We saw: Lotteries: optimal solvable Briest et al • Product distributions: • 2 approximation Chawla HatlineMalec Sivan • PTAS – Cai and Daskalakis

25. 2 Approximation product distribution • bidders, 1 item • 1 bidder, n items

26. Prophet Inequality • Gambler • games • Each game has a payoff drawn from an independent distribution • After seeing payoff of ith game, can continue to next game or take payoff and leave • Each payoff comes from a distribution • Optimal gambler stategy: backwards induction • Theorem: There is a threshold t such that if the gambler takes the first prize that exceeds t then the gambler gets ½ of the maximal payoff

27. Proof Choose such that (exists?) Benchmark (highest possible profit for gambler): For

28. Profit of gambler = “extras” • - event that for all :

29. Proof of Prophet Inequality Benchmark Gambler profit

30. More generally Find t such that increases and is continuous decreases and is continuous

31. Relationship to Auctions • Many bidders, one item, valuation for bidder from • Vickrey 2nd price auction maximzes social welfare ( • Myerson maximize virtual social welfare maximizes revenue • Many items, one bidder, value for item from

32. Simpler social welfare maximization • Many bidders, 1 item: • There is an anonymous (identical) posted price that gives a approximation to welfare. Directly from Prophet inequality. • Who gets the item? Could be anyone that wants it. There is no restriction on tie breaking. • Many items, one bidder: • There is a single posted price (for all items) that gives a approximation to social welfare.

33. Simpler revenue maximization • Many bidders, 1 item: • There is a single virtual price that gives a approximation to welfare. Directly from Prophet inequality and that the revenue is the expected virtual welfare. • This means non-anonymous prices • Who gets the item? Could be anyone that wants it. There is no restriction on tie breaking. • Many items, one bidder: • Use same prices for approximation