1. Advanced Subjects in GT - 096226 Prepared by Rina Talisman Introduction Revenue Equivalence The Optimal Auction (Myerson 1981) Auctions

2. Advanced Subjects in GT - 096226 What is an Auction? A public sale in which property or merchandise are sold to the highest bidder. resource allocationprices A market institution with explicit rules determining resource allocation and prices on the basis of bids from participants. The seller and the buyers want to maximize their utility – opposite objectives

3. Advanced Subjects in GT - 096226 Going Once, Going Twice, … Seller Seller sells an object – claims the starting bid at $1 buyers, There are N potential buyers, each one with his own valuation for the object. The bidding process starts: Who will make the bid?  valuation > current bid Winner Winner – the bidder with the highest bid payment What will be the payment? Seller’s goal: reveal the buyer with the maximal valuation  to reveal the real valuations (Mechanism Design) Buyers goal: to pay as less as possible

4. Advanced Subjects in GT - 096226 Overview of Auctions private value Different private value auction mechanisms: Single UnitMulti Unit Sealed Bid First Price Second price ….. Pay your bid Uniform Price …. Open Auction Descending (Dutch) Ascending (English) …. Descending (Dutch) Ascending (English) ….

5. Advanced Subjects in GT - 096226 Common Value Auction The object is worth the same - common value to every bidder No bidder knows exactly what this common value is Each bidder has some private knowledge (guess) about the real realization of the value Example: oil tract leases The government auctions oil tracts in the Golf of Mexico. No one knows how much oil is in given oil tract, though each firm bidding has some guess

6. Advanced Subjects in GT - 096226 Sources of Uncertainty Private Value Auction Private Value Auction Each bidder knows his or her value for the object Bidders differ in their values for the object e.g., arts auctions, e-commerce Common Value Auction Common Value Auction The item has a single though unknown value Bidders differ in their estimates of the true value of the object e.g., FCC spectrum, drilling, disciplinary corporate takeovers

7. Advanced Subjects in GT - 096226 Overview of Auctions (cont.) Ascending - English Seller Seller claims the starting bid (low) – buyer/s, object at th If there is a buyer/s, which agree to buy the object at the specified price, the seller raises the price to The bidding process continues till the moment that nobody will accept the price  the object will be sold at price Open auction – easy to analyze: dominant strategy for each buyer i: if  stay otherwise  quit

8. Advanced Subjects in GT - 096226 Descending - Dutch Seller Seller claims the starting bid (high) – buyer/s, object at th If there is no buyer/s, which agree to buy the object at the specified price, the seller lowers the price to The bidding process continues till the moment that someone will accept the price  the object will be sold at price Overview of Auctions (cont.)

9. Advanced Subjects in GT - 096226 Sealed-Bid First Price Auctions All buyers submit bids (simultaneously) Buyer submitting the highest bid wins and pays the price his bid $700 $400 $500 $300 WINNER! Pays $700

10. Advanced Subjects in GT - 096226 First-price sealed-bid auction

11. Advanced Subjects in GT - 096226 First-price sealed-bid auction

12. Advanced Subjects in GT - 096226 First-price sealed-bid auction

13. Advanced Subjects in GT - 096226 First-price sealed-bid auction

14. Advanced Subjects in GT - 096226 Sealed-Bid Second Price Auctions All buyers submit bids Buyer submitting the highest bid wins and pays the second highest bid $700 $400 $500 $300 WINNER! Pays $500

15. Advanced Subjects in GT - 096226 Why Second Price? It is strategically equivalent to an English Auction $300 $400 $500

16. Advanced Subjects in GT - 096226 Why Second Price? Bidding strategy is easy Bidding one’s true valuation is a dominant strategy Intuition: The amount a bidder pays is not dependent on her bid

17. Advanced Subjects in GT - 096226 Optimal Bidding Strategy in Second Price Auctions You Lose You Win higher Your bid Others’ bids Your value

18. Advanced Subjects in GT - 096226 Bidding Higher Than My Valuation Case 1Case 2Case 3 No difference Lose money

19. Advanced Subjects in GT - 096226 Bidding Lower Than My Valuation Case 1Case 2Case 3 No difference Lose money

20. Advanced Subjects in GT - 096226 Second Price Auction always bid your true valuation In a second price auction, the dominant strategy is - always bid your true valuation Winning bidder’s surplus Difference between the winner’s valuation and the second highest valuation

21. Advanced Subjects in GT - 096226 Which is Better? In a second price auction bidders bid their true value auctioneer receives the second highest bid In a first price auction bidders bid below their true value auctioneer receives the highest bid

22. Advanced Subjects in GT - 096226 Buyers properties Risk averse Risk neutral Risk seeking vU(v)v U(v) U(v)v

23. Advanced Subjects in GT - 096226 Buyers properties Risk Aversion Risk Aversion Does not influence 2 nd price auctions (dominant strategies) Risk averse bidders are more aggressive in first price auctions ( afraid to lose ) Risk aversion  1 st price or Dutch are better to the seller ( bits are higher )

24. Advanced Subjects in GT - 096226 AIPV Setting: asymmetric independent private values Seller has a single object N risk-neutral buyers with type and declaration Buyers have quasi linear utility functions: Each buyer type Buyers’ types are independent and with density on

25. Advanced Subjects in GT - 096226 Revenue Equivalence Suppose that a given pair of Bayesian NE of two different AIPV auction procedures are such that: then these equilibriums of the two auctions generate the same expected revenue for the seller

26. Advanced Subjects in GT - 096226 Optimal Auction (Myerson 1981) Consider the following AIPV auction procedure: - each bidder is asked to report his value - the auction procedure consists of a pair

27. Advanced Subjects in GT - 096226 Optimal Auction – Optimal Mechanism Common knowledge auction function: auction function: - is a Bayesian incentive compatible social choice function  - maximizes seller’s expected revenue b (= v)

28. Advanced Subjects in GT - 096226 Optimal Auction (cont.) definition: definition: a virtual valuation of buyer i is a non decreasing function and is defined as: Auction maximize seller’s expected revenue:

29. Advanced Subjects in GT - 096226 Optimal Auction (cont.) Auction maximize seller’s expected revenue: - the object is allocated to the buyer with the highest virtual valuation and exceeds 0 - only the winner pays, and the price is equal to the lowest virtual value the winner can have and still win