1. Optimal auction design Roger Myerson Mathematics of Operations research 1981

2. Auctions What is an auction? – Agreement between seller and bidders Who gets the item? How much does everyone pay?

3. Optimal auction design problem The seller has a single item to sale She doesn’t know how bidders value the item She wants to make as much money as possible

4. The setting A seller has 1 item for sale, which she values at 0 A set of bidders: bidder i’s valuation (type) t i towards the item is private info Others view t i as a random variable in [a i, b i ] drawn from F i (t i ) An outcome: a probability p i of allocation and a payment x i, for each i – Who gets the item at what price Bidder’s utility: u i = p i t i -x i Seller’s goal: maximizes her expected utility/revenue thru a mechanism Bidders maximize their expected utility 4

5. Auction mechanisms A mechanism – Specifies a set A i of actions for each bidder i – Outcome function: a 1 ×…a n  outcome A bidder i’s strategy s i (): [a i, b i ]  A i Bidders’ strategies forms a (Bayes) Nash equilibrium Infinite space: action can be anything! 5

6. Revelation principle Direct revelation mechanisms – Everyone’s action is to report a valuation (A i =[a i,b i ]) – Being truthful is an equilibrium (incentive compatible) Revelation principle – It is WLOG to focus on direct revelation mechanism – In other words, anything outcome implemented by a mechanism can also be implemented by a direct revelation mechanism

7. Proof

8. The seller’s problem Design direct revelation mechanism (p(t),x(t)), so as to maximize E t (∑ i x i (t)) where (t=(t 1,…t n )) Subject to – Incentive compatibility (IC) truthful is NE – Individual rationality (IR) participation – Resource feasibility (RF) Seller should never

9. Analysis: constraints simplification Interim allocation probability Lemma: Constraints simplification – IC, IR and RF iff

10. Proof

11. Analysis: objective simplification lemma Subject to RF and Q being increasing and

12. Proof

14. Optimal auction: the regular case Virtual value: t i -(1-F i (t i ))/f i (t i ) Regularity: t i -(1-F i (t i ))/f i (t i ) is increasing in t i – So that Q is increasing (last constraint satisfied) Allocation rule: give the item to the highest non- negative virtual value Payment rule: max {0, Inverse of 2 nd highest VV }

15. Summary Upon receiving bids t i from each bidder i The seller calculates VV: t i -(1-F i (t i ))/f i (t i ) The seller gives the item to j who has the highest non-negative VV The seller charges j the amount that would tie him to the 2 nd highest VV If all VV are negative, the seller keeps the item

16. Discussion: bidders may have different virtual valuation functions 16 Actual valuation ranking Virtual valuation ranking winner is 2 t1t1 t2t2 t3t3 ~t 1 ~t 3 ~t 2

17. Discussion: symmetric bidders Assume F i =F j (symmetric bidders) Every bidder has the same virtual valuation function Myerson auction is 2 nd -price auction with a reserve price 17

18. Discussion: F i =F j 18 Actual valuation ranking Virtual valuation ranking winner is 3 0 reserve t1t1 t2t2 t3t3 ~t 1 ~t 2 ~t 3

19. Exercise: envelope theorem [Milgron, ]

20. Recent progresses Optimal auction – Single item setting (Myerson) – Multiple identical item (Maskin&Riley) – Combinatorial items with single parameter (Levin) – Two items with discrete distribution (Armstrong) Approximate optimal auction – 2 nd -price auction 2-approximates Myerson (Hartline and Roughgarden, EC-09) – VCG 2-approximates Levin (Tang and Sandholm, IJCAI-11) – One bidder, two item: Separate Myerson 2-approximates optimal (Hart-Nisan, EC-12) Unfortunately: even for 1 bidder, 2 item case, the optimal auction is unknown! Two sellers?