Announcements Ø HW1 grading will be out next Tuesday, and sample solution is out on Collab Ø HW 2 is due next Tuesday 1
CS6501: T opics in Learning and Game Theory (Fall 2019) Mechanism Design from Samples Instructor: Haifeng Xu
Outline Ø Optimal Auction and its Limitations Ø The Sample Mechanism and its Revenue Guarantee 3
Recap: Optimal Auction for Single Item Theorem. For single-item allocation with regular value distribution 𝑤 " ∼ 𝑔 " independently, the following auction is BIC and optimal: Solicit buyer values 𝑤 % , ⋯ , 𝑤 ( 1. %./ 0 (1 0 ) Transform 𝑤 " to “virtual value” 𝜚 " (𝑤 " ) where 𝜚 " 𝑤 " = 𝑤 " − 2. 2 0 (1 0 ) If 𝜚 " 𝑤 " < 0 for all 𝑗 , keep the item and no payments 3. Otherwise, allocate item to 𝑗 ∗ = arg max 4. "∈[(] 𝜚 " (𝑤 " ) and charge him .% max max the minimum bid needed to win, i.e., 𝜚 " ?@" ∗ 𝜚 ? (𝑤 ? ) , 0 . Ø Recall, “regular” means 𝜚 " 𝑤 " is monotone non-decreasing Ø Will always assume distributions are regular and “nice” henceforth 4
Recap: Optimal Auction for Single Item An important special case: 𝑤 " ∼ 𝐺 i.i.d. Ø The second-price auction with reserve 𝜚 .% (0) is optimal 1. Solicit buyer values 𝑤 % , ⋯ , 𝑤 ( 2. If 𝑤 " < 𝜚 .% (0) for all 𝑗 , keep the item and no payments 3. Otherwise, allocate to 𝑗 ∗ = arg max "∈[(] 𝑤 " and charge him the minimum ?@" ∗ 𝑤 ? , 𝜚 .% 0 bid needed to win, i.e., max( max ) 5
Recap: Optimal Auction for Single Item An important special case: 𝑤 " ∼ 𝐺 i.i.d. Ø The second-price auction with reserve 𝜚 .% (0) is optimal 1. Solicit buyer values 𝑤 % , ⋯ , 𝑤 ( 2. If 𝑤 " < 𝜚 .% (0) for all 𝑗 , keep the item and no payments 3. Otherwise, allocate to 𝑗 ∗ = arg max "∈[(] 𝑤 " and charge him the minimum ?@" ∗ 𝑤 ? , 𝜚 .% 0 bid needed to win, i.e., max( max ) Intuitions about why second-price auction with reserve is good Ø Incentive compatibility requires payment to not depend on bidder’s own bid à second highest bid is pretty much the best choice Ø Use the reserve to balance between “charging a higher price” and “disposing the item” 6
Recap: Optimal Auction for Single Item Myerson’s Lemma is central to the proof 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 ( ∑ "F% 𝔽 1 0 ∼2 0 𝜚 " 𝑤 " 𝑦 " (𝑤 " ) 7
Drawbacks of the Optimal Auction Buyer’s value 𝑤 " is assumed to be drawn from a distribution 𝑔 1. " The precise distribution 𝑔 2. " is assumed to be known to seller Ø In this lecture, we will keep Assumption 1, but relax Assumption 2 8
Drawbacks of the Optimal Auction Buyer’s value 𝑤 " is assumed to be drawn from a distribution 𝑔 1. " The precise distribution 𝑔 2. " is assumed to be known to seller Ø In this lecture, we will keep Assumption 1, but relax Assumption 2 Ø This is precisely the machine learning perspective • ML assumes data drawn from distributions • The precise distribution is unknown; instead samples are given 9
Task and Goal of This Lecture Ø Will focus on setting with 𝑜 buyer, i.i.d. values Ø Buyer value 𝑤 " is drawn from regular distribution 𝑔 , which is unknown to the seller Goal: design an auction that has revenue close to the optimal revenue when knowing 𝑔 Ø Optimal auction is a second-price auction with reserve 𝜚 .% (0) Ø “Closeness” will be measured by guaranteed approximation ratio 10
Task and Goal of This Lecture Ø Will focus on setting with 𝑜 buyer, i.i.d. values Ø Buyer value 𝑤 " is drawn from regular distribution 𝑔 , which is unknown to the seller Goal: design an auction that has revenue close to the optimal revenue when knowing 𝑔 Ø Optimal auction is a second-price auction with reserve 𝜚 .% (0) Ø “Closeness” will be measured by guaranteed approximation ratio But wait . . . we cannot have any guarantee without assumptions on bidder values – is this a contradiction? No, we assumed 𝑤 " ∼ 𝑔 • 11
A Natural First Attempt Ø Since 𝑤 " ’s are all drawn from 𝑔 , these 𝑜 i.i.d. samples can be used to estimate 𝑔 Ø This results in the following “empirical Myerson” auction Empirical Myerson Auction 1. Solicit buyer values 𝑤 % , ⋯ , 𝑤 ( 2. Use 𝑤 % , ⋯ , 𝑤 ( to estimate an empirical distribution ̅ 𝑔 3. Run second-price auction with reserve J 𝜚 .% (0) where J 𝜚 𝑔 instead is calculated using ̅ Q: does this mechanism work? No, may fail in multiple ways 12
Issues of Empirical Myerson Empirical Myerson Auction 1. Solicit buyer values 𝑤 % , ⋯ , 𝑤 ( problematic 2. Use 𝑤 % , ⋯ , 𝑤 ( to estimate an empirical distribution ̅ 𝑔 3. Run second-price auction with reserve J 𝜚 .% (0) where J 𝜚 𝑔 instead is calculated using ̅ Ø Not incentive compatible – reserve depends on bidder’s report • This is a crucial difference from standard machine learning tasks where samples are assumed to be correctly given 13
Issues of Empirical Myerson Empirical Myerson Auction 1. Solicit buyer values 𝑤 % , ⋯ , 𝑤 ( problematic 2. Use 𝑤 % , ⋯ , 𝑤 ( to estimate an empirical distribution ̅ 𝑔 3. Run second-price auction with reserve J 𝜚 .% (0) where J 𝜚 𝑔 instead is calculated using ̅ Ø Not incentive compatible – reserve depends on bidder’s report • This is a crucial difference from standard machine learning tasks where samples are assumed to be correctly given 𝑔 may not be regular Ø Even bidders report true values, ̅ 𝑔 is regular, J 𝜚 .% (0) may not be close to 𝜚 .% (0) Ø Even ̅ • Depend on how large is 𝑜 , and shape of 𝑔 14
Outline Ø Optimal Auction and its Limitations Ø The Sample Mechanism and its Revenue Guarantee 15
The Basic Idea Ø Want to use second-price auction with an estimated reserve Ø Lesson from previous example – if a bidder’s bid is used to estimate the reserve, we cannot use this reserve for him Ø Main idea: pick a “reserve buyer” à use his bid to estimate the reserve but never sell to this buyer • I.e., we give up any revenue from the reserve buyer 16
The Basic Idea Ø Want to use second-price auction with an estimated reserve Ø Lesson from previous example – if a bidder’s bid is used to estimate the reserve, we cannot use this reserve for him Ø Main idea: pick a “reserve buyer” à use his bid to estimate the reserve but never sell to this buyer • I.e., we give up any revenue from the reserve buyer Q: why only pick one reserve buyer, not two or more? We have to give up revenue from reserve buyers, better not too many 17
The Basic Idea Ø Want to use second-price auction with an estimated reserve Ø Lesson from previous example – if a bidder’s bid is used to estimate the reserve, we cannot use this reserve for him Ø Main idea: pick a “reserve buyer” à use his bid to estimate the reserve but never sell to this buyer • I.e., we give up any revenue from the reserve buyer Q: why only pick one reserve buyer, not two or more? We have to give up revenue from reserve buyers, better not too many Q: which buyer to choose as the reserve buyer? A-priori, they are the same à pick one uniformly at random 18
The Basic Idea Ø Want to use second-price auction with an estimated reserve Ø Lesson from previous example – if a bidder’s bid is used to estimate the reserve, we cannot use this reserve for him Ø Main idea: pick a “reserve buyer” à use his bid to estimate the reserve but never sell to this buyer • I.e., we give up any revenue from the reserve buyer Q: how to use a single buyer’s value to estimate reserve? Not much we can do . . . just use his value as reserve 19
The Mechanism Second-Price auction with Random Reserve (SP-RR) 1. Solicit buyer values 𝑤 % , ⋯ , 𝑤 ( 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 . 20
The Mechanism Second-Price auction with Random Reserve (SP-RR) 1. Solicit buyer values 𝑤 % , ⋯ , 𝑤 ( 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 . Claim. SP-RR is dominant-strategy incentive compatible. For any bidder 𝑗 Ø If 𝑗 is picked as reserve, his bid does not matter to him, so truthful bidding is an optimal strategy Ø If 𝑗 is not picked, he faces a second-price auction with reserve. Again, truthful bidding is optimal 21
The Mechanism Theorem. Suppose 𝐺 is regular. In expectation, SP-RR achieves % (.% M ⋅ at least ( fraction of the optimal expected revenue. Remarks % (.% M ⋅ ( is a worst-case guarantee Ø Ø The first time we use approximation as a lens to analyze algorithms in this class Ø It is possible to have a good auction even without knowing 𝐺 • But we still assumed 𝑤 " ∼ 𝐺 i.i.d. 22
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