Announcements Ø Collect HW1 grading (see Collab for sample solution) Ø HW 2 is due next Tuesday • No class on next Tuesday, but TAs will be here to collect HW Ø HW 3 will be out by the end of this week • Likely will have a very light HW 4 or no HW 4 Ø Instructions for course project will be out by the end of this week 1
CS6501: T opics in Learning and Game Theory (Fall 2019) Simple Auctions Instructor: Haifeng Xu
Outline Ø Prior-Independent Auctions for I.I.D. Buyers Ø Intricacy of Optimal Auction for Independent Buyers Ø Simple Auction for Independent Buyers 3
IID Buyers: What Have We Learned So Far? Ø Optimal auction is a second-price auction with reserve 𝜚 "# (0) • Notation: buyer value 𝑤 ( ∼ 𝑔 (regular) and 𝜚 𝑤 = 𝑤 − #"-(.) /(.) Ø Optimal auction (unrealistically) requires completely knowing 𝑔 4
IID Buyers: What Have We Learned So Far? Ø Optimal auction is a second-price auction with reserve 𝜚 "# (0) • Notation: buyer value 𝑤 ( ∼ 𝑔 (regular) and 𝜚 𝑤 = 𝑤 − #"-(.) /(.) Ø Optimal auction (unrealistically) requires completely knowing 𝑔 Ø Last lecture – prior-independent auction • Still assume 𝑤 ( ∼ 𝑔 , but do not know 𝑔 • Guarantee roughly 1/2 of the optimal revenue for any 𝑜 ≥ 2 • Like ML: data drawn from unknown distributions Second-Price auction with Random Reserve (SP-RR) 1. Solicit buyer values 𝑤 # , ⋯ , 𝑤 7 2. Pick 𝑘 ∈ [𝑜] uniformly at random as the reserve buyer 3. Run second-price auction with reserve 𝑤 < but only among bidders in 𝑜 ∖ 𝑘 . 5
IID Buyers: What Have We Learned So Far? Key insights from the proof of ½ approximation: Ø Discarding a buyer does not hurt revenue much Lemma 1. The expected optimal revenue for an environment with 7"# (𝑜 − 1) buyers is at least fraction of the optimal expected 7 revenue for 𝑜 buyers. 6
IID Buyers: What Have We Learned So Far? Key insights from the proof of ½ approximation: Ø Discarding a buyer does not hurt revenue much Lemma 1. The expected optimal revenue for an environment with 7"# (𝑜 − 1) buyers is at least fraction of the optimal expected 7 revenue for 𝑜 buyers. Ø Using random reserve is not bad • SP-OR: second price auction with optimal reserve 𝑠 ∗ = 𝜚 "# (0) • SP-RR: second price auction with random reserve 𝑠 ∼ 𝐺 Lemma 2. Rev(SP-RR) ≥ # > Rev(SP-OR) for any 𝑜 ≥ 1 and regular 𝐺 . 7
IID Buyers: What Have We Learned So Far? Next, we show that even directly running second-price auction without reserve is not bad for i.i.d. buyers Ø Built upon a fundamental result by [Bulow-Klemperer, ’96] Ø Can be used to strengthen previous approximation guarantee • Drawback: this technique does not easily generalize to independent buyers 8
IID Buyers: What Have We Learned So Far? Next, we show that even directly running second-price auction without reserve is not bad for i.i.d. buyers Ø Built upon a fundamental result by [Bulow-Klemperer, ’96] Ø Can be used to strengthen previous approximation guarantee • Drawback: this technique does not easily generalize to independent buyers Ø Inspired the whole research agenda on simple yet approximately optimal auction design Note: “Simple” is a subjective judge, no formal definition 9
The Bulow-Klemperer Theorem Theorem. For any 𝑜(≥ 1) i.i.d. buyers with regular 𝐺 , we have 𝑆𝑓𝑤 7D# 𝑇𝑄 ≥ 𝑆𝑓𝑤 7 (𝑇𝑄 - 𝑃𝑆) Notations Ø SP – second-price auction; Ø 𝑆𝑓𝑤 7 (𝑁) – revenue of any mechanism 𝑁 for 𝑜 i.i.d buyers 10
The Bulow-Klemperer Theorem Theorem. For any 𝑜(≥ 1) i.i.d. buyers with regular 𝐺 , we have 𝑆𝑓𝑤 7D# 𝑇𝑄 ≥ 𝑆𝑓𝑤 7 (𝑇𝑄 - 𝑃𝑆) Ø That is, second-price auction with an additional buyer achieves higher revenue than the optimal auction Ø Insight: more competition is better than finding the right auction format 11
The Bulow-Klemperer Theorem Theorem. For any 𝑜(≥ 1) i.i.d. buyers with regular 𝐺 , we have 𝑆𝑓𝑤 7D# 𝑇𝑄 ≥ 𝑆𝑓𝑤 7 (𝑇𝑄 - 𝑃𝑆) Proof: an application of Myerson’s Lemma Lemma. Consider any BIC mechanism 𝑁 with interim allocation 𝑦 and interim payment 𝑞 , normalized to 𝑞 ( 0 = 0 . The expected revenue of 𝑁 is equal to the expected virtual welfare served 7 ∑ (L# 𝔽 . N ∼/ N 𝜚 ( 𝑤 ( 𝑦 ( (𝑤 ( ) 12
The Bulow-Klemperer Theorem Theorem. For any 𝑜(≥ 1) i.i.d. buyers with regular 𝐺 , we have 𝑆𝑓𝑤 7D# 𝑇𝑄 ≥ 𝑆𝑓𝑤 7 (𝑇𝑄 - 𝑃𝑆) Proof: an application of Myerson’s Lemma Ø Consider the following auction for 𝑜 + 1 buyers: 1. Run SP-OR for first 𝑜 buyers; 2. If not sold, give the item to bidder 𝑜 + 1 for free Ø Two observations a. This auction always allocates the item, and is BIC b. Achieves the same revenue as 𝑆𝑓𝑤 7 (𝑇𝑄 - 𝑃𝑆) Ø We argue that SP for 𝑜 + 1 buyers achieves higher revenue 13
The Bulow-Klemperer Theorem Theorem. For any 𝑜(≥ 1) i.i.d. buyers with regular 𝐺 , we have 𝑆𝑓𝑤 7D# 𝑇𝑄 ≥ 𝑆𝑓𝑤 7 (𝑇𝑄 - 𝑃𝑆) Proof: an application of Myerson’s Lemma Ø Consider the following auction for 𝑜 + 1 buyers: 1. Run SP-OR for first 𝑜 buyers; 2. If not sold, give the item to bidder 𝑜 + 1 for free Claim. SP has highest revenue among auctions that always allocate item ü Myerson’s lemma: revenue = virtual welfare served ü SP always gives the item to the one with highest virtual welfare 14
The Bulow-Klemperer Theorem Theorem. For any 𝑜(≥ 1) i.i.d. buyers with regular 𝐺 , we have 𝑆𝑓𝑤 7D# 𝑇𝑄 ≥ 𝑆𝑓𝑤 7 (𝑇𝑄 - 𝑃𝑆) Corollary. For any 𝑜 ≥ 2 , 𝑆𝑓𝑤 7 𝑇𝑄 ≥ (1 − # 7 )𝑆𝑓𝑤 7 (𝑇𝑄 - 𝑃𝑆) Remarks: Ø SP is prior-independent, simple and approximately optimal Ø Recovers previous result when 𝑜 = 2 • With even better guarantee when 𝑜 ≥ 3 15
The Bulow-Klemperer Theorem Theorem. For any 𝑜(≥ 1) i.i.d. buyers with regular 𝐺 , we have 𝑆𝑓𝑤 7D# 𝑇𝑄 ≥ 𝑆𝑓𝑤 7 (𝑇𝑄 - 𝑃𝑆) Corollary. For any 𝑜 ≥ 2 , 𝑆𝑓𝑤 7 𝑇𝑄 ≥ (1 − # 7 )𝑆𝑓𝑤 7 (𝑇𝑄 - 𝑃𝑆) Proof: 𝑆𝑓𝑤 7 𝑇𝑄 ≥ 𝑆𝑓𝑤 7"# (𝑇𝑄 - 𝑃𝑆) # ≥ (1 − 7 )𝑆𝑓𝑤 7 (𝑇𝑄 - 𝑃𝑆) Since discarding a bidder does not hurt revenue much 16
Outline Ø Prior-Independent Auctions for I.I.D. Buyers Ø Intricacy of Optimal Auction for Independent Buyers Ø Simple Auction for Independent Buyers 17
Optimal Auction for Independent Buyers Theorem. For single-item allocation with regular value distribution 𝑤 ( ∼ 𝑔 ( independently, the following auction is BIC and optimal: Solicit buyer values 𝑤 # , ⋯ , 𝑤 7 1. #"- N (. N ) Transform 𝑤 ( to “virtual value” 𝜚 ( (𝑤 ( ) where 𝜚 ( 𝑤 ( = 𝑤 ( − 2. / N (. N ) If 𝜚 ( 𝑤 ( < 0 for all 𝑗 , keep the item and no payments 3. Otherwise, allocate item to 𝑗 ∗ = arg max (∈[7] 𝜚 ( (𝑤 ( ) and charge him 4. "# max max the minimum bid needed to win, i.e., 𝜚 ( <X( ∗ 𝜚 < (𝑤 < ) , 0 . 18
An Example Ø Two bidders, 𝑤 # ∼ 𝑉[0,1] , 𝑤 > ∼ 𝑉[0,100] Ø 𝜚 # (𝑤 # ) = 𝑤 # − #"- Z . Z = 2𝑤 # − 1 , 𝜚 > (𝑤 > ) = 2𝑤 > − 100 / Z . Z Optimal auction has the following rules: ü When 𝑤 # > ½, 𝑤 > < 50 , allocate to bidder 1 and charge ½ ü When 𝑤 # < ½, 𝑤 > > 50 , allocate to bidder 2 and charge 50 ü When 0 < 2𝑤 # − 1 < 2𝑤 > − 100 , allocate to bidder 2 and charge (99 + 2𝑤 # )/2 (a tiny bit above 50 ) ü When 0 < 2𝑤 > − 100 < 2𝑤 # − 1 , allocate to bidder 1 and charge (2𝑤 > − 99)/2 (a tiny bit above 1/2 ) Ø Roughly, want to give it to bidder 2 for 50, and otherwise give it to bidder 1 for 0.5 Ø Optimal auction is less natural, especially with many buyers 19
An Example Ø Two bidders, 𝑤 # ∼ 𝑉[0,1] , 𝑤 > ∼ 𝑉[0,100] Ø 𝜚 # (𝑤 # ) = 𝑤 # − #"- Z . Z = 2𝑤 # − 1 , 𝜚 > (𝑤 > ) = 2𝑤 > − 100 / Z . Z Optimal auction has the following rules: ü When 𝑤 # > ½, 𝑤 > < 50 , allocate to bidder 1 and charge ½ ü When 𝑤 # < ½, 𝑤 > > 50 , allocate to bidder 2 and charge 50 ü When 0 < 2𝑤 # − 1 < 2𝑤 > − 100 , allocate to bidder 2 and charge (99 + 2𝑤 # )/2 (a tiny bit above 50 ) ü When 0 < 2𝑤 > − 100 < 2𝑤 # − 1 , allocate to bidder 1 and charge (2𝑤 > − 99)/2 (a tiny bit above 1/2 ) Q: Is there a simple auction that’s approximately optimal? Note: second-price auction alone does not work à The above example 20
Outline Ø Prior-Independent Auctions for I.I.D. Buyers Ø Intricacy of Optimal Auction for Independent Buyers Ø Simple Auction for Independent Buyers Notations: v ` ∼ f ` for i ∈ [n] • 21
Simple Auctions are Approximately Optimal Ø Second-price auction with a single reserve also achieves ≈ 1/4 fraction of OPT • The best reserve will depend on 𝑔 ( ’s Ø Second-price auction with personalized reserve (depending on the priors) achieves ≈ 1/2 fraction of OPT • Again, reserves will depend on 𝑔 ( ’s 22
Recommend
More recommend
