1. 6.853: Topics in Algorithmic Game Theory Fall 2011 Matt Weinberg Lecture 24

2. Recap Myerson’s Lemma: The Expected Revenue of any BIC Mechanism is exactly its Expected Virtual Surplus Virtual Value φ(v): (defined on board) Virtual Surplus: The virtual valuation of the winner

3. A simple Corollary Upper Bound on Optimal Revenue: Always choose the allocation that maximizes virtual surplus – Proof: Any BIC allocation rule clearly obtains some virtual surplus, this virtual surplus cannot possibly exceed this bound – Is this bound attainable?

4. Virtual Second Price Auction Let φ i be any monotonically increasing functions. Let winner = argmax_i {φ i (v i )} (or no one if φ i (v i ) < 0 for all i) Charge the winner price φ i -1 (max{0,argmax_{j ≠ i} {φ j (v j )}}) Incentive Compatible! (Not just BIC) – Proof: Lowering your bid cannot affect the price you pay, may lose the item when willing to buy. Increasing bid cannot affect the price you pay, may win the item when unwilling.

5. Virtual Second Price Auction Candidate function for φ i ? – Virtual Valuation function? If Φ i is monotonically increasing, valid choice. – Always gives item to highest virtual value – Achieves maximum possible expected virtual surplus What if φ i isn’t monotonically increasing? – “Ironed Virtual Valuations” Picture on Board.

6. Virtual Second Price Auction Myerson’s Theorem: When bidders are single- dimensional, and each v i is sampled independently from a known distribution, the auction that maximizes expected revenue over all BIC auctions is to: – 1) Map each v i to φ i (v i ) using (ironed) virtual valuations – 2) Award the item to the bidder with the highest non- negative φ i (v i ), at price φ i -1 (max{0,argmax_{j ≠ i} φ j (v j )}) Furthermore, this auction is Incentive Compatible (not just BIC).

7. Regular Distributions What does it mean for φ i (v i ) to be monotonically increasing? If there were no other bidders and we were only selling the item to bidder i, then: – There exist unique prices p and q (might have p = q) such that: – 1) Selling the item at any price p ≤ x ≤ y ≤ q E[R(p)] = E[R(x)] = E[R(y)] = E[R(q)]. – 2) y x > q implies that E[R(y)] < E[R(x)] < E[R(p)]. – Proof: On Board. – Such distributions are called Regular

8. An Example 2 bidders, v 1 uniform in [0,1]. v 2 uniform in [0,100]. – Φ 1 (x) = 2x-1, Φ 2 (x) = 2x-100 – Intuition behind optimal auction: – If there was only bidder 2, would sell the item at price 50. Only bidder 1, sell at ½. – When v 1 > ½, v 2 < 50, sell to 1 at price ½. – When v 1 50, sell to 2 at price 50. – When 0 < 2v 1 -1 < 2v 2 – 100, sell to 2 at price: (99+2v 1 )/2, a tiny bit above 50 – When 0 < 2v 2 -100 < 2v 1 -1, sell to 1 at price: (2v 2 -99)/2, a tiny bit above ½. – Intuition: For each bidder, pretend they are the only bidder. Sometimes we want to give them both the item. In this case, we should increase the price charged to the winner. Myerson’s Lemma tells us exactly who should win.

9. I.I.D. Bidders Corollary of Myerson: When bidders are I.I.D., the revenue optimal auction is a second price auction with reserve. – Proof: Everyone can have the same (ironed) virtual valuation function, so the winner will always be the highest bidder.

10. When Does Myerson Apply? Single Item for sale. k copies of the same item (such as a CD) for sale, each bidder only wants one. – Myerson chooses the k bidders with the highest non-negative virtual values k copies of the same item (such as a CD) for sale, each bidder has value x i v i if they receive x i copies of the item. – Myerson chooses the bidder with the highest non-negative virtual value and gives her all k. Either of the above + combinatorial constraints on who can receive how many of each item. – Myerson chooses the feasible allocation that maximizes virtual surplus

11. What if we don’t know the distributions? Single Item, bidders I.I.D.? – Bulow-Klemperer Theorem (Next Slide) Infinitely many copies of the same item (IE: digital copy of a song), no prior information – Digital Goods Auctions (After)

12. Bulow-Klemperer Theorem Motivation: In the real world, bidders are usually i.i.d. – Have you ever seen an auction that sets a different price for different people? In the real world, learning distributions is a hard problem

13. Bulow-Klemperer Theorem Let VCG(F,n) denote the expected revenue of running the VCG (second price) auction with n bidders sampled independently from F. Let OPT(F,n) denote the expected revenue of running Myerson’s auction with n bidders sampled independently from F. Theorem (Bulow-Klemperer): If F is regular, VCG(F,n+1) ≥ OPT(F,n) Big Picture: With i.i.d. bidders, rather than learn the distributions, try to attract more bidders.

14. Bulow-Klemperer Theorem Proof of Theorem consequence of two simple claims. Let COPT(F,n) denote the maximum attainable expected revenue by an auction that always awards the item to someone. 1) COPT(F,n+1) ≥ OPT(F,n) 2) VCG(F,n+1) = COPT(F,n+1)

15. Bulow-Klemperer Theorem 1) COPT(F,n+1) ≥ OPT(F,n) – Proof: Run Myerson’s auction on the first n bidders. If Myerson doesn’t award the item, give it to bidder n+1 for free. Revenue is exactly OPT(F,n), item is always given away.

16. Bulow-Klemperer Theorem 2) VCG(F,n+1) = COPT(F,n+1) – Proof: We still know that expected revenue is exactly expected virtual surplus. If F is regular, then the highest bidder has the highest virtual value. So VCG chooses the highest virtual value always. – The only difference between VCG and Myerson is that Myerson sometimes withholds the good from sale if all virtual values are negative.

17. Worst-Case Guarantees Absolutely impossible to get any approximation with no prior, even with a single bidder and a single item. Value could be 1, could be 10, could be 1000000... Need a different benchmark

18. Digital Good Auctions Infinitely many copies of a single item (think digital copies of a song) Want a worst-case guarantee – Optimal Revenue is wrong benchmark: impossible to attain (highest bid could be 1, 10, 10000000…) F (2) benchmark: max p|(#bidders with v > p) > 1 {p*(#bidders with v > p)} – In English: Allowed to choose any price where at least two bidders are willing to purchase. – Why F (2) ? Anything stronger unattainable in worst- case. Simple approximation to F (2) possible (next slide)

19. Digital Good Auctions Random Sample Auction: For each bidder, flip a coin. Denote by T the set of bidders with tails, and H the set of bidders with heads. – Find the optimal price for bidders in T, p(T) and the optimal price for bidders in H, p(H). Charge price p(T) to bidders in H, and p(H) to bidders in T. – Truthful? Bidders are offered a price that they can’t control. Dominant strategy to tell the truth. – Optimal? Theorem: Random Sample Auction has expected revenue equal to at least F (2) /15. There are instances where the expected revenue is at most F (2) /4.

20. Summary Independent bidders, single item: – Myerson’s Lemma provides upper bound on possible expected revenue. – It is attainable using the virtual second price auction with (ironed) virtual valuations – Generalizes to single-dimensional settings with combinatorial constraints I.I.D. bidders, unknown distribution: Bulow-Klemperer: Recruiting an extra bidder is better than learning the distribution if distributions are regular No Prior, worst case guarantee: – Optimal revenue completely inapproximable in worst-case – F (2) /15 attainable by simple auction (Random Sample Auction)