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Introduction to Auction Design via Machine Learning
Ellen Vitercik
December 4, 2017 CMU 10-715
Advanced Introduction to Machine Learning
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Ad auctions contribute a huge portion of large internet companies’ revenue.
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Ad revenue in 2016 Total revenue in 2016
Google $79 billion $89.46 billion
Facebook $27 billion $27.64 billion
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Leonardo da Vinci painting sells for $450.3m, shattering auction highs [NY Times ‘17]
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Diamond fetches $33.7m at Christie’s auction in Geneva [BBC ‘17]
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The FCC spectrum auction is sending $10 billion to broadcasters [NiemanLab ‘17]
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Very-large-scale generalized combinatorial multi-attribute auctions: Lessons from conducting $60 billion of sourcing [Sandholm ‘13]
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How can we design revenue-maximizing auctions using
machine learning?
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1. Introduction
2. Auction design background
3. Revenue-maximizing auctions
4. Learning-theoretic auction design
5. Tools
6. Case study: Second price auctions with reserve prices
7. Related work
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Common misconception: There’s only one way to hold an auction.
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There are infinitely many ways to design auctions.
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Ascending price auction: Auctioneer calls out the opening bid. Bidders call out bids until there is no new bid within a period of time. Last bidder to bid wins the item and pays their bid.
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Ascending price auction: Auctioneer calls out the opening bid. Bidders call out bids until there is no new bid within a period of time. Last bidder to bid wins the item and pays their bid. Online auctions can only take a fraction of a second, so we need faster auction formats.
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Second price auction for a single good :
Each bidder submits a bid. Highest bidder wins. They
pay the second highest bid.
2
1 pays 1
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The second price auction is incentive compatible (every bidder will maximize their utility by bidding truthfully). Why not bid above value ? If winner, will stay winner and price won’t change. If loser, might become winner, but will have to pay a price larger than value , which will result in negative utility.
value ― payment • 1 wins item
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The second price auction is incentive compatible (every bidder will maximize their utility by bidding truthfully). Why not bid below value ? If winner, might become loser, and will shift from non-negative to zero utility. If loser, will still be loser, so utility will still be zero.
value ― payment • 1 wins item
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Second price auction with a reserve price 𝒓 ∈ ℝ for a
single good :
Each bidder submits a bid. Highest bidder wins if their
bid is above 𝑟. They pay maximum of 𝑟 and the
second highest bid.
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Optimizing reserve prices can lead to higher revenue. But how should we choose the reserve price?
Second price auction with a reserve price of 1.5 for a
single good :
Each bidder submits a bid. Highest bidder wins if their bid is above 1.5. They pay maximum of 1.5 and the
second highest bid.
pays 1.5
2
1
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1. Introduction
2. Auction design background
3. Revenue-maximizing auctions
4. Learning-theoretic auction design
5. Tools
6. Case study: Second price auctions with reserve prices
7. Related work
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In 1981, Roger Myerson discovered the revenue-maximizing single-item, multi-bidder auction. He won the 2007 Nobel prize in economics for his work on mechanism design. R. Myerson. Optimal auction design. Mathematics of Operations Research, 6(1):58–73, 1981.
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Standard assumption A buyer's value for a bundle of goods is defined by a probability distribution over all the possible valuations she might have for that bundle.
value ~ Uniform(1, 3)
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For simplicity, suppose all bidders have the same valuation distribution, with PDF 𝑓, CDF 𝐹, and support in [0, 1].
Standard assumption A buyer's value for a bundle of goods is defined by a probability distribution over all the possible valuations she might have for that bundle.
Myerson’s optimal auction
Let 𝜙 𝑥 = 𝑥 −1−𝐹(𝑥)
𝑓(𝑥). Run a second price auction with a reserve
price of 𝜙−1 0 .
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Myerson also analyzed the case where the bidders’ values are not identically distributed.
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The revenue-maximizing multi-item auction is unknown!
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This requires that we know the distribution over bidders’ values. Where does this knowledge come from? In multi-item cases, the support of the distribution might be doubly exponential!
Myerson’s optimal auction
Let 𝜙 𝑥 = 𝑥 −1−𝐹(𝑥)
𝑓(𝑥). Run a second price auction with a reserve
price of 𝜙−1 0 .
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1. Introduction
2. Auction design background
3. Revenue-maximizing auctions
4. Learning-theoretic auction design
5. Tools
6. Case study: Second price auctions with reserve prices
7. Related work
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There’s an unknown distribution over valuations. Use a sample to
learn a mechanism with high expected revenue.
value ~
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Research papers:
[Likhodedov and Sandholm ‘04, ‘05, Elkind ’07, Dhangwatnotai et al. ’10, Medina and
Mohri ‘14, Cole and Roughgarden ’14, Huang et al. ’15, Morgenstern and Roughgarden ’15, ‘16, Roughgarden and Schrijvers ’16, Devanur
et al. ’16, Balcan, Sandholm, and V ‘16, ’17, Hartline and Taggart ’17, Syrgkanis ‘17, Alon
et al. ’17, Cai and Daskalakis ’17, Gonczarowski and Nisan ’17, Medina and
Vassilvitskii ’17, Dütting et al. ‘17]
value ~
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Nina Balcan Connections between Learning, Game Theory, and Optimization
Georgia Tech
Fall 2010
Constantinos Daskalakis and Vasilis Syrgkanis
Algorithmic Game Theory and Data Science
MIT Spring 2017
Classes:
value ~
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Workshops and Tutorials:
value ~
2016 EC Tutorial on Algorithmic Game Theory and Data Science
2015, 2016, 2017
EC Workshop on Algorithmic Game Theory and Data Science
2017 NIPS Workshop on Learning in Presence of Strategic Behavior
2017 Dagstuhl Workshop on Game Theory Meets Computational Learning Theory
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Central question: Is the resulting auction nearly optimal?
Fix an auction class ℳ
Look at the samples
Optimize over ℳ
Field auction
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Central question: Does the resulting auction approximately maximize revenue in expectation over the unknown distribution?
Fix an auction class ℳ
Look at the samples
Optimize over ℳ
Field auction
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Depends on the class ℳ. Let 𝑀∗ be the optimal auction and let 𝑀 ∈ ℳ be the auction returned by the learning algorithm.
𝔼 Rev 𝑀∗ − 𝔼 Rev 𝑀
= Rev 𝑀∗ − max𝑀∈ℳ
Rev 𝑀 + max𝑀∈ℳ
Rev(𝑀) − Rev 𝑀
Central question: Does the resulting auction approximately maximize revenue in expectation over the unknown distribution?
Total revenue loss
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Approximation loss
Depends on the class ℳ. Let 𝑀∗ be the optimal auction and let 𝑀 ∈ ℳ be the auction returned by the learning algorithm.
𝔼 Rev 𝑀∗ − 𝔼 Rev 𝑀
= 𝔼 Rev 𝑀∗ − max𝑀∈ℳ
𝔼 Rev 𝑀 + max𝑀∈ℳ
𝔼 Rev 𝑀 − 𝔼 Rev 𝑀
Central question: Does the resulting auction approximately maximize revenue in expectation over the unknown distribution?
Estimation loss
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Depends on the class ℳ.
Central question: Does the resulting auction approximately maximize revenue in expectation over the unknown distribution?
Simple class
Bad approximation loss Good estimation loss
Good approximation loss Bad estimation loss
Complex class
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These are the same problems we face in machine learning.
Simple class
Bad approximation loss Good estimation loss
Good approximation loss Bad estimation loss
Complex class
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Say ℳ is the class of second price auctions with reserve prices. ℳ has zero approximation loss when the bidders’ values are i.i.d.! This talk: What is the estimation loss of ℳ?
Myerson’s optimal auction for i.i.d. bidders
Let 𝜙 𝑥 = 𝑥 −1−𝐹(𝑥)
𝑓(𝑥). Run a second price auction with a reserve
price of 𝜙−1 0 .
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1. Introduction
2. Auction design background
3. Revenue-maximizing auctions
4. Learning-theoretic auction design
5. Tools
6. Case study: Second price auctions with reserve prices
7. Related work
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VC dimension: A complexity measure for binary-valued functions ℎ:𝒳 → 0,1
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Example.
For 𝑎, 𝑏 ∈ ℝ, 𝑎 < 𝑏, let ℎ𝑎,𝑏 𝑥 = 1, 𝑥 ∈ (𝑎, 𝑏)&0, 𝑥 ∉ (𝑎, 𝑏)
Let ℋint = ℎ𝑎,𝑏: 𝑎, 𝑏 ∈ ℝ, 𝑎 < 𝑏 .
No set 𝑆 ⊂ ℝ of size 3 can be labeled in all 23 ways by ℋint.
1 1 0
1 0
1 0
0
1
0
1
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The VC dimension of a class ℋ is the size of the largest set 𝑆 that can be labeled in all 2|𝑆| ways by functions in ℋ.
VC dimension
A set 𝑆 = 𝑥1, … , 𝑥𝑚 is shattered by ℋ if for all 𝒃 ∈ {0,1}|𝑆|, there exists a function ℎ ∈ ℋ such that for all 𝑖 ∈ [𝑚], ℎ 𝑥𝑖 = 𝑏𝑖. The VC dimension of ℋ is the size of the largest set 𝑆 that can be shattered by ℋ.
1 0 0 0 1 0 1 1
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The VC dimension of a class ℋ is the size of the largest set 𝑆 that can be labeled in all 2|𝑆| ways by functions in ℋ.
1 0 0 0 1 0 1 1
The VC dimension of ℋint is at least 2.
1 0
1 0
0
1
0
1
The VC dimension of ℋint is at most 2. 1 1 0
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The VC dimension of a class ℋ is the size of the largest set 𝑆 that can be labeled in all 2|𝑆| ways by functions in ℋ.
1 0 0 0 1 0 1 1
1 0
1 0
0
1
0
1
1 1 0
The VC dimension of ℋint is 2.
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Pseudo-dimension: A complexity measure for real-valued functions ℎ:𝒳 → ℝ
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Set 𝑆 = 𝑥1, … , 𝑥𝑚 Class of functions ℋ = ℎ:𝒳 → ℝ
𝒛 = 𝑧(1), … , 𝑧(𝑚) ∈ ℝ𝑚 witnesses the shattering of 𝑆&by ℋ if for all 𝒃 ∈ {0,1}𝑚, there exists ℎ ∈ ℋ such that ℎ 𝑥𝑖 ≤ 𝑧(𝑖) if 𝑏𝑖 = 0 and ℎ 𝑥𝑖 > 𝑧(𝑖) if 𝑏𝑖 = 1
𝒙𝟏 𝒉 𝒙𝟏 ≤ 𝒛(𝟏) 𝟎
𝒙𝟐 𝒉 𝒙𝟐 ≤ 𝒛(𝟐) 𝟎
𝒙𝟑 𝒉 𝒙𝟑 > 𝒛(𝟑) 𝟏
𝒙𝟒 𝒉 𝒙𝟒 ≤ 𝒛(𝟒) 𝟎
𝒙𝟓 𝒉 𝒙𝟓 > 𝒛(𝟓) 𝟏
𝒙𝟔 𝒉 𝒙𝟔 ≤ 𝒛(𝟔) 𝟎
𝒙𝟕 𝒉 𝒙𝟕 > 𝒛(𝟕) 𝟏
𝒙𝟖 𝒉 𝒙𝟖 > 𝒛(𝟖) 𝟏
𝒙𝟗 𝒉 𝒙𝟗 ≤ 𝒛(𝟗) 𝟎
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Set 𝑆 = 𝑥1, … , 𝑥𝑚 Class of functions ℋ = ℎ:𝒳 → ℝ
𝒛 = 𝑧(1), … , 𝑧(𝑚) ∈ ℝ𝑚 witnesses the shattering of 𝑆&by ℋ if for all 𝒃 ∈ {0,1}𝑚, there exists ℎ ∈ ℋ such that ℎ 𝑥𝑖 ≤ 𝑧(𝑖) if 𝑏𝑖 = 0 and ℎ 𝑥𝑖 > 𝑧(𝑖) if 𝑏𝑖 = 1
𝑥1 ℎ 𝑥1 ≤ 𝑧(1) 0
𝑥2 ℎ 𝑥2 ≤ 𝑧(2) 0
𝑥3 ℎ 𝑥3 > 𝑧(3) 1
𝑥4 ℎ 𝑥4 ≤ 𝑧(4) 0
𝑥5 ℎ 𝑥5 > 𝑧(5) 1
𝑥6 ℎ 𝑥6 ≤ 𝑧(6) 0
𝑥7 ℎ 𝑥7 > 𝑧(7) 1
𝑥8 ℎ 𝑥8 > 𝑧(8) 1
ℎ 𝑥9 ≤ 𝑧(9) 0
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Set 𝑆 = 𝑥1, … , 𝑥𝑚 Class of functions ℋ = ℎ:𝒳 → ℝ
𝒛 = 𝑧(1), … , 𝑧(𝑚) ∈ ℝ𝑚 witnesses the shattering of 𝑆&by ℋ if for all 𝒃 ∈ {0,1}𝑚, there exists ℎ ∈ ℋ such that ℎ 𝑥𝑖 ≤ 𝑧(𝑖) if 𝑏𝑖 = 0 and ℎ 𝑥𝑖 > 𝑧(𝑖) if 𝑏𝑖 = 1
𝑥1 ℎ 𝑥1 ≤ 𝑧(1) 0
𝑥2 ℎ 𝑥2 ≤ 𝑧(2) 0
𝑥3 ℎ 𝑥3 > 𝑧(3) 1
𝑥4 ℎ 𝑥4 ≤ 𝑧(4) 0
𝑥5 ℎ 𝑥5 > 𝑧(5) 1
𝑥6 ℎ 𝑥6 ≤ 𝑧(6) 0
𝑥7 ℎ 𝑥7 > 𝑧(7) 1
𝑥8 ℎ 𝑥8 > 𝑧(8) 1
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Set 𝑆 = 𝑥1, … , 𝑥𝑚 Class of functions ℋ = ℎ:𝒳 → ℝ
𝒛 = 𝑧(1), … , 𝑧(𝑚) ∈ ℝ𝑚 witnesses the shattering of 𝑆&by ℋ if for all 𝒃 ∈ {0,1}𝑚, there exists ℎ ∈ ℋ such that ℎ 𝑥𝑖 ≤ 𝑧(𝑖) if 𝑏𝑖 = 0 and ℎ 𝑥𝑖 > 𝑧(𝑖) if 𝑏𝑖 = 1
𝑥1 ℎ 𝑥1 ≤ 𝑧(1) 0
𝑥2 ℎ 𝑥2 ≤ 𝑧(2) 0
𝑥3 ℎ 𝑥3 > 𝑧(3) 1
𝑥4 ℎ 𝑥4 ≤ 𝑧(4) 0
𝑥5 ℎ 𝑥5 > 𝑧(5) 1
𝑥6 ℎ 𝑥6 ≤ 𝑧(6) 0
𝑥7 ℎ 𝑥7 > 𝑧(7) 1
𝑥8 ℎ 𝑥8 > 𝑧(8) 1
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Set 𝑆 = 𝑥1, … , 𝑥𝑚 Class of functions ℋ = ℎ:𝒳 → ℝ
𝒛 = 𝑧(1), … , 𝑧(𝑚) ∈ ℝ𝑚 witnesses the shattering of 𝑆&by ℋ if for all 𝒃 ∈ {0,1}𝑚, there exists ℎ ∈ ℋ such that ℎ 𝑥𝑖 ≤ 𝑧(𝑖) if 𝑏𝑖 = 0 and ℎ 𝑥𝑖 > 𝑧(𝑖) if 𝑏𝑖 = 1
𝑥1 ℎ 𝑥1 ≤ 𝑧(1) 0
𝑥2 ℎ 𝑥2 ≤ 𝑧(2) 0
𝑥3 ℎ 𝑥3 > 𝑧(3) 1
𝑥4 ℎ 𝑥4 ≤ 𝑧(4) 0
𝑥5 ℎ 𝑥5 > 𝑧(5) 1
𝑥6 ℎ 𝑥6 ≤ 𝑧(6) 0
𝑥7 ℎ 𝑥7 > 𝑧(7) 1
𝑥8 ℎ 𝑥8 > 𝑧(8) 1
The pseudo-dimension of ℋ is the cardinality of the largest set that can be shattered by ℋ.
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Set 𝑆 = 𝑥1, … , 𝑥𝑚 Class of functions ℋ = ℎ:𝒳 → ℝ
𝒛 = 𝑧(1), … , 𝑧(𝑚) ∈ ℝ𝑚 witnesses the shattering of 𝑆&by ℋ if for all 𝒃 ∈ {0,1}𝑚, there exists ℎ ∈ ℋ such that ℎ 𝑥𝑖 ≤ 𝑧(𝑖) if 𝑏𝑖 = 0 and ℎ 𝑥𝑖 > 𝑧(𝑖) if 𝑏𝑖 = 1
𝑥1 ℎ 𝑥1 ≤ 𝑧(1) 0
𝑥2 ℎ 𝑥2 ≤ 𝑧(2) 0
𝑥3 ℎ 𝑥3 > 𝑧(3) 1
𝑥4 ℎ 𝑥4 ≤ 𝑧(4) 0
𝑥5 ℎ 𝑥5 > 𝑧(5) 1
𝑥6 ℎ 𝑥6 ≤ 𝑧(6) 0
𝑥7 ℎ 𝑥7 > 𝑧(7) 1
𝑥8 ℎ 𝑥8 > 𝑧(8) 1
The pseudo-dimension of ℋ is the VC dimension of the set of “below the graph” functions (𝑥, 𝑧) ⟼ 𝟏ℎ 𝑥 −𝑧>0|&ℎ ∈ ℋ .
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Theorem [Haussler 1992] Suppose ℋ = ℎ:𝒳 → [−𝑈,𝑈]
For every 𝜖, 𝛿 > 0, and distribution 𝐷, if 𝑚 = 𝑂 𝑈
𝜖
2∙ Pdim(ℋ) ,
then with probability 1 − 𝛿 over the draw of 𝑆 = 𝑥1, … , 𝑥𝑚 ~𝐷𝑚, for all ℎ ∈ ℋ,
1
𝑚 ℎ 𝑥𝑖 − 𝔼𝑥~𝐷[ℎ 𝑥 ]
𝑚
𝑖=1
< 𝜖
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Corollary [Haussler 1992] Suppose ℋ = ℎ:𝒳 → [−𝑈,𝑈] Given a sample 𝑆 = 𝑥1, … , 𝑥𝑚 , let ℎ𝑆 = max
ℎ∈ℋ ℎ 𝑥𝑖
𝑚𝑖=1 .
For all 𝜖, 𝛿 > 0, 𝑚 = 𝑂 𝑈
𝜖
2Pdim(ℋ) samples are sufficient to
ensure that w.p. 1 − 𝛿 over the draw of 𝑆 = 𝑥1, … , 𝑥𝑚 ~𝐷𝑚,
maxℎ∈ℋ
𝔼𝑥~𝐷 ℎ 𝑥 − 𝔼𝑥~𝐷 ℎ𝑆 𝑥 ≤ 𝜖
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Approximation loss
Depends on the class ℳ. Let 𝑀∗ be the optimal auction and let 𝑀 ∈ ℳ be the auction returned by the learning algorithm.
𝔼 Rev 𝑀∗ − 𝔼 Rev 𝑀
= 𝔼 Rev 𝑀∗ − max𝑀∈ℳ
𝔼 Rev 𝑀 + max𝑀∈ℳ
𝔼 Rev 𝑀 − 𝔼 Rev 𝑀
Central question: Does the resulting auction approximately maximize revenue in expectation over the unknown distribution?
Estimation loss
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Corollary [Haussler 1992] Suppose ℋ = ℎ:𝒳 → [−𝑈,𝑈] Given a sample 𝑆 = 𝑥1, … , 𝑥𝑚 , let ℎ𝑆 = max
ℎ∈ℋ ℎ 𝑥𝑖
𝑚𝑖=1 .
For all 𝜖, 𝛿 > 0, 𝑚 = 𝑂 𝑈
𝜖
2Pdim(ℋ) samples are sufficient to
ensure that w.p. 1 − 𝛿 over the draw of 𝑆 = 𝑥1, … , 𝑥𝑚 ~𝐷𝑚,
maxℎ∈ℋ
𝔼𝑥~𝐷 ℎ 𝑥 − 𝔼𝑥~𝐷 ℎ𝑆 𝑥 ≤ 𝜖
Pseudo-dimension allows us to bound estimation loss.
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Example. Let ℋ be the set of all constant functions
ℋ = ℎ ∶ ∀𝑥 ∈ 𝒳, ℎ 𝑥 = 𝑐&for&some&𝑐 ∈ ℝ
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Example. For 𝑆 = 𝑥1, 𝑥2 ⊆ 𝒳, let 𝑧(1) and 𝑧(2) be two potential witnesses. WLOG, suppose 𝑧(1) ≤ 𝑧(2). There is no constant function ℎ 𝑥 = 𝑐 where 𝑧(2) < ℎ 𝑥2 = 𝑐 = ℎ 𝑥1 ≤ 𝑧(1). ⇒ 𝑆 is not shatterable.
𝑥1 𝑥2
𝑧(1)
𝑧(2)
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Example. For 𝑆 = 𝑥1, 𝑥2 ⊆ 𝒳, let 𝑧(1) and 𝑧(2) be two candidate witnesses. WLOG, suppose 𝑧(1) ≤ 𝑧(2). There is no constant function ℎ 𝑥 = 𝑐 where 𝑧(2) < ℎ 𝑥2 = 𝑐 = ℎ 𝑥1 ≤ 𝑧(1). ⇒ 𝑆 is not shatterable. Clearly, a single point is shatterable. Therefore, the pseudo-dimension of ℋ is 1.
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1. Introduction
2. Auction design background
3. Revenue-maximizing auctions
4. Learning-theoretic auction design
5. Tools
6. Case study: Second price auctions with reserve prices a) Recap
b) Result
7. Related work
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Central question: Does the resulting auction approximately maximize revenue in expectation over the unknown distribution?
Fix an auction class ℳ
Look at the samples
Optimize over ℳ
Field auction
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Approximation loss
Depends on the class ℳ. Let 𝑀∗ be the optimal auction and let 𝑀 ∈ ℳ be the auction returned by the learning algorithm.
𝔼 Rev 𝑀∗ − 𝔼 Rev 𝑀
= 𝔼 Rev 𝑀∗ − max𝑀∈ℳ
𝔼 Rev 𝑀 + max𝑀∈ℳ
𝔼 Rev 𝑀 − 𝔼 Rev 𝑀
Central question: Does the resulting auction approximately maximize revenue in expectation over the unknown distribution?
Estimation loss
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Suppose ℳ is the class of second price auctions with reserve prices. ℳ has zero approximation loss when the bidders valuations are i.i.d.! What is the estimation loss of 𝓜?
Myerson’s optimal auction for i.i.d. bidders
Let 𝜙 𝑥 = 𝑥 −1−𝐹(𝑥)
𝑓(𝑥). Run a second price auction with a reserve
price of 𝜙−1 0 .
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1. Introduction
2. Auction design background
3. Revenue-maximizing auctions
4. Learning-theoretic auction design
5. Tools
6. Case study: Second price auctions with reserve prices a) Recap
b) Result
7. Related work
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Notation: 𝒗 = 𝑣1, … , 𝑣𝑛 ~𝐷, where 𝑣𝑖 is the value of bidder 𝑖
Let 𝑣(1) = max 𝑣1, … , 𝑣𝑛 and 𝑣(2) = max 𝑣1, … , 𝑣𝑛 ∖ 𝑣 1
For a reserve price 𝑟, rev𝑟 𝒗 = max 𝑟, 𝑣(2) ∙ 𝟏 𝑣 1 ≥ 𝑟 .
Define the class ℋ = rev𝑟 𝒗 ∶ 𝑟& ≥ 0 . We’ve seen that the approximation loss for learning over ℋ is zero when the bidders have i.i.d. valuations.
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Notation: 𝒗 = 𝑣1, … , 𝑣𝑛 ~𝐷, where 𝑣𝑖 is the value of bidder 𝑖
Let 𝑣(1) = max 𝑣1, … , 𝑣𝑛 and 𝑣(2) = max 𝑣1, … , 𝑣𝑛 ∖ 𝑣 1
For a reserve price 𝑟, rev𝑟 𝒗 = max 𝑟, 𝑣(2) ∙ 𝟏 𝑣 1 ≥ 𝑟 .
Define the class ℋ = rev𝑟 𝒗 ∶ 𝑟& ≥ 0 . What is the estimation loss for learning over ℋ? To answer this, we’ll bound the pseudo-dimension of ℋ.
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*Special case of a theorem by Morgenstern and Roughgarden [2016]
Theorem* The pseudo-dimension of the class ℋ = rev𝑟 𝒗 ∶ 𝑟& ≥ 0 is 2.
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Proof.
Suppose 𝑆 = 𝒗(1), … , 𝒗(𝑚) is shatterable by ℋ = rev𝑟 𝒗 ∶ 𝑟 ≥ 0 .
There is a set of witnesses 𝑧(1), … , 𝑧(𝑚) such that for every 𝒃 ∈ 0,1 𝑚, there exists 𝑟𝒃 ≥ 0 such that rev𝑟𝒃
𝒗(𝑖) ≤ 𝑧(𝑖) if 𝑏𝑖 = 0 and rev𝑟𝒃
𝒗(𝑖) > 𝑧(𝑖) if 𝑏𝑖 = 1.
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Proof. Let 𝒗 ∈ 𝑆 and let 𝑧 be its witness. Write rev𝒗 𝑟 = rev𝑟 𝒗 .
𝑣(2)
rev𝒗 𝑟
𝑟
𝑣(2)
𝑣(1)
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Proof. Suppose 𝑧 < 𝑣(2).
When 𝑟 ≤ 𝑣(1), rev𝒗 𝑟 > 𝑧 and when 𝑟 > 𝑣(1), rev𝒗 𝑟 ≤ 𝑧.
rev𝒗 𝑟
𝑟
𝑣(2)
𝑧
𝑣(1)
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Proof. Suppose 𝑧 ≥ 𝑣(2).
When 𝑟 ∈ 0, 𝑧 or 𝑟 ∈ 𝑣 1 , ∞ , rev𝒗 𝑟 ≤ 𝑧.
When 𝑟 ∈ 𝑧, 𝑣 1 , rev𝒗 𝑟 > 𝑧.
𝑣(1)
rev𝒗 𝑟
𝑟 𝑧
𝑧
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Proof. Notation: Let 𝒗(𝑖) ∈ 𝑆 and let 𝑧(𝑖) be its witness.
𝑟
rev𝒗(𝑖) 𝑟
Second highest value of
𝒗(𝑖)
𝑣(2)(𝑖)
𝑣(1)(𝑖)
Highest value of
𝒗(𝑖)
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Proof.
Sort 𝑧 1 , 𝑣 11, … , 𝑧 𝑚 , 𝑣 1
𝑚 . For all 𝒗(𝑖) ∈ 𝑆, between these sorted
values, either rev𝒗(𝑖) 𝑟 > 𝑧(𝑖) or rev𝒗(𝑖) 𝑟 ≤ 𝑧(𝑖).
rev𝒗(𝑖) 𝑟
𝑟
𝑣(2)(𝑖)
𝑧 𝑖
𝑣(1)(𝑖)
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Proof.
Sort 𝑧 1 , 𝑣 11, … , 𝑧 𝑚 , 𝑣 1
𝑚 . For all 𝒗(𝑖) ∈ 𝑆, between these sorted
values, either rev𝒗(𝑖) 𝑟 > 𝑧(𝑖) or rev𝒗(𝑖) 𝑟 ≤ 𝑧(𝑖).
rev𝒗(𝑖) 𝑟
𝑧 𝑖
𝑣(1)(𝑖)
𝑧 𝑖
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Proof. Recall:
Suppose 𝑆 = 𝒗(1), … , 𝒗(𝑚) is shatterable by ℋ = rev𝑟 𝒗 ∶ 𝑟 ≥ 0 .
There is a set of witnesses 𝑧(1), … , 𝑧(𝑚) such that for every 𝒃 ∈ 0,1 𝑚, there exists 𝑟𝒃 ≥ 0 such that rev𝑟𝒃
𝒗(𝑖) ≤ 𝑧(𝑖) if 𝑏𝑖 = 0 and rev𝑟𝒃
𝒗(𝑖) > 𝑧(𝑖) if 𝑏𝑖 = 1. Let 𝑅∗ = 𝑟𝒃 ∶ 𝒃 ∈ 0,1 𝑚 .
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Proof.
Sort 𝑧 1 , 𝑣 11, … , 𝑧 𝑚 , 𝑣 1
𝑚 . For all 𝒗(𝑖) ∈ 𝑆, between these sorted
values, either rev𝒗(𝑖) 𝑟 > 𝑧(𝑖) or rev𝒗(𝑖) 𝑟 ≤ 𝑧(𝑖).
rev𝒗(𝑖) 𝑟
𝑟
𝑣(2)(𝑖)
𝑧 𝑖
𝑣(1)(𝑖)
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Proof.
Sort 𝑧 1 , 𝑣 11, … , 𝑧 𝑚 , 𝑣 1
𝑚 . For all 𝒗(𝑖) ∈ 𝑆, between these sorted
values, either rev𝒗(𝑖) 𝑟 > 𝑧(𝑖) or rev𝒗(𝑖) 𝑟 ≤ 𝑧(𝑖).
rev𝒗(𝑖) 𝑟
𝑧 𝑖
𝑣(1)(𝑖)
𝑧 𝑖
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Proof.
Sort 𝑧 1 , 𝑣 11, … , 𝑧 𝑚 , 𝑣 1
𝑚 . For all 𝒗(𝑖) ∈ 𝑆, between these sorted
values, either rev𝒗(𝑖) 𝑟 > 𝑧(𝑖) or rev𝒗(𝑖) 𝑟 ≤ 𝑧(𝑖).
At most one 𝑟 ∈ 𝑅∗ is from each interval between sorted values. Therefore, 2𝑚 = 𝑅∗ ≤ 2𝑚 + 1, so 𝑚 ≤ 2.
Theorem The pseudo-dimension of the class ℋ = rev𝑟 𝒗 ∶ 𝑟& ≥ 0 is 2.
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Corollary Suppose the bidders’ valuations are i.i.d with support in [0,1]. Let ℋ be the class of second price auctions with reserves. Approximation loss: 0 Estimation loss: 𝑂 1
|𝑆|2
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Corollary Suppose the bidders’ valuations are i.i.d with support in [0,1]. Let ℋ be the class of second price auctions with reserves. For a sample 𝑆 = 𝒗(1), … , 𝒗(𝑚) ~𝐷, let 𝑟𝑆 be the reserve price that maximizes rev𝑟 𝒗 𝑖𝑚
𝑖=1 . 𝑂
1
𝜀2 samples are sufficient to ensure that with probability at
least 1 − 𝛿, 𝔼𝒗~𝐷 rev𝑟𝑆𝒗 ≥ max
𝑟≥0𝔼𝒗~𝐷 rev𝑟 𝒗 − 휀.
This is tight [Cesa-Bianchi et al. 2015]!
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1. Introduction
2. Auction design background
3. Revenue-maximizing auctions
4. Learning-theoretic auction design
5. Tools
6. Case study: Second price auctions with reserve prices
7. Related work
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Other single-item auction batch learning papers: Elkind ’07, Dhangwatnotai et al. ’10, Medina and Mohri ‘14, Cole and Roughgarden ’14, Huang et al. ’15, Morgenstern and Roughgarden ’15, Roughgarden and Schrijvers ’16, Devanur et al. ’16, Hartline and Taggart ’17, Alon et al. ’17, Gonczarowski and Nisan ’17 Multi-item auction batch learning: Likhodedov and Sandholm ‘04, ’05, Morgenstern and Roughgarden ‘16, Balcan, Sandholm, and V ‘16, ’17, Syrgkanis ’17, Cai and Daskalakis ’17, Medina and Vassilvitskii ’17 Online auction design: Kleinberg and Leighton ‘03, Blum and Hartline ‘05, Cesa-Bianchi et al. ‘15, Dudík et al. ‘17, Balcan, Dick and V ‘17
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Auction design via deep learning: Dütting et al. ’17
Solid regions represent allocation probability learned by neural network when there’s a single bidder with 𝑣1, 𝑣2~𝑈[0,1]. The optimal mechanism of Manelli and Vincent [‘06] is described by the regions separated by the dashed orange lines.