The Pseudo-Dimension of Near Optimal Auctions Jamie Morgenstern - PowerPoint PPT Presentation

The Pseudo-Dimension of Near Optimal Auctions Jamie Morgenstern Tim Roughgarden Presented by Shubhang Kulkarni

Previously v Auctions that maximize revenue v Performance measure w.r.t. prior distribution Myerson’s auction – prior known Myerson’s empirical auction – prior unknown v Sample complexity amount of data necessary for near optimal expected revenue

Motivation – what makes a mechanism “simple” v Virtual welfare maximizers don’t resemble everyday auction formats v We have notions of auction optimality – need notions of simplicity v Realistic auctions with theoretical approximation guarantees [Hartline, Roughgarden ‘09] 𝟐 𝐹𝑦𝑞𝑓𝑑𝑢𝑓𝑒 𝑠𝑓𝑤𝑓𝑜𝑣𝑓 𝑝𝑔 ≥ 𝟑 ⋅ 𝑃𝑞𝑢𝑗𝑛𝑣𝑛 𝑓𝑦𝑞𝑓𝑑𝑢𝑓𝑒 𝑠𝑓𝑤𝑓𝑜𝑣𝑓 𝑊𝐷𝐻 𝑥𝑗𝑢ℎ 𝑁𝑝𝑜𝑝𝑞𝑝𝑚𝑧 𝑆𝑓𝑡𝑓𝑠𝑤𝑓𝑡 (~ 𝑜 𝑒𝑓𝑕𝑠𝑓𝑓𝑡 𝑝𝑔 𝑔𝑠𝑓𝑓𝑒𝑝𝑛) (→ 𝑗𝑜𝑔𝑗𝑜𝑗𝑢𝑓 𝑒𝑓𝑕𝑠𝑓𝑓𝑡 𝑝𝑔 𝑔𝑠𝑓𝑓𝑒𝑝𝑛)

Goal v Quantitative definition of mechanism simplicity v Introduce t-level auctions to interpolate between auction complexity and optimality v Give tools to solve simple vs optimal bicriteria optimization problems v Show that poly-t-level auctions are simple (more on this later)

VC Dimension Measure of capacity that can be learned by a statistical classification algorithm v Let H be Hypothesis set (set of boolean functions) v Let S be a sample set v Define 𝑇 ∩ 𝐼 = ℎ ∩ 𝑇 ℎ ∈ 𝐼} v 𝑌 ⊆ 𝑇 is shattered by H if 𝑌 ∩ 𝐼 contains all subsets of 𝑌 i.e. 𝑌 ∩ 𝐼 = 2 𝑌 v V.C. dimension of H is the largest integer D s.t. ∃ 𝑌 ⊆ 𝑇, 𝑌 = 𝐸, 𝐼 shatters X

VC Dimension – Examples Cardinality of the largest set of data the class can overfit to in boolean classification

Psuedo Dimension v Let H be Hypothesis set (set of real functions) v Let S be a sample set v 𝑌 ⊆ 𝑇 is shattered by H if • there is a witness 𝑈 = (𝑢 1 , … , 𝑢 𝑌 ) s.t. • For every subset K ⊆ 𝑌 , there is a function ℎ ∈ 𝐼 s.t. • ℎ 𝑦 𝑗 ≥ 𝑢 𝑗 ↔ 𝑦 𝑗 ∈ 𝐿 v Auction 𝐵: 𝐼 𝑜 → 𝑆 𝑀𝑝𝑥 𝑄𝑡𝑣𝑓𝑒𝑝 𝐸𝑗𝑛𝑓𝑜𝑡𝑗𝑝𝑜 ≡ 𝑇𝑗𝑛𝑞𝑚𝑓 𝐵𝑣𝑑𝑢𝑗𝑝𝑜 𝐷𝑚𝑏𝑡𝑡

Psuedo Dimension – Performance Auction Pseudo Dimension Intuitive degrees of freedom Vickery Auction, Anonymous 𝑃(1) 1 Reserve Vickery Auction, Bidder Specific 𝑃(𝑜 log 𝑜) 𝑜 Reserve Virtual Welfare Maximizer ∞ ∞

t-level Auctions – Intuition v Each bidder faces one of t possible prices v Prices depend on other bidders v Define each price to be a threshold v t-level auction is thus defined by 𝑜 ⋅ 𝑢 thresholds

t-level Auctions – Allocation Rule v For each bidder 𝑗, v 𝑢 𝑗 𝑤 𝑗 = 𝜐 index of largest threshold 𝑚 𝑗,𝜐 that lower bounds the value 𝑤 𝑗 , v -1 if 𝑤 𝑗 < 𝑚 𝑗,0 v 𝑢 𝑗 (𝑤 𝑗 ) is the level of bidder 𝑗 v Sort bidders in descending values of levels, lexicographic tie breaking v Award highest bidder the item. v No sale if ∀𝑗 𝑢 𝑗 𝑤 𝑗 = −1

t-level Auctions – Payment Rule v Unique one that renders auction DSIC v Winner pays lowest bid at which he/she continues to win v Losers pay 0 v Authors note three interesting cases

t-level Auctions – Payment Rule: Case Monop t = 4 t = 3 t = 1 t = 2 𝒎 𝒃 2 𝒘 𝒃 4 6 8 𝒎 𝒄 𝒘 𝒄 1.5 5 9 10 𝒎 𝒅 𝒘 𝒅 1.7 3.9 6 7 2 𝒒 𝒃 𝒒 𝒄 0 𝒒 𝒅 0

t-level Auctions – Payment Rule: Case Mult t = 4 t = 3 t = 1 t = 2 𝒎 𝒃 2 4 6 8 𝒘 𝒃 𝒎 𝒄 1.5 5 9 10 𝒘 𝒄 𝒎 𝒅 1.7 3.9 𝒘 𝒅 6 7 8 𝒒 𝒃 𝒒 𝒄 0 𝒒 𝒅 0

t-level Auctions – Payment Rule: Case Unique t = 4 t = 3 t = 1 t = 2 𝒎 𝒃 2 4 6 8 𝒘 𝒃 𝒎 𝒄 1.5 5 𝒘 𝒄 9 10 𝒎 𝒅 1.7 3.9 𝒘 𝒅 6 7 4 𝒒 𝒃 𝒒 𝒄 0 𝒒 𝒅 0

t-level Auctions – Payment Rule: Case Unique t = 4 t = 3 t = 1 t = 2 𝒎 𝒃 2 4 𝒘 𝒃 6 8 𝒎 𝒄 1.5 5 𝒘 𝒄 9 10 𝒎 𝒅 1.7 3.9 6 𝒘 𝒅 7 0 𝒒 𝒃 𝒒 𝒄 0 𝒒 𝒅 6

Connections to virtual functions v Discrete approximations to virtual welfare maximizers v Each level interpreted as a constraint of form: “if any bidder has level at least 𝜐 , don’t sell to bidder less than 𝜐 ” v Levels map to common virtual values v 1-level auctions treat all values below single thresholds as –ve virtual value. Above threshold uses values as proxies for virtual values. v 2-level auctions refine virtual value estimates with 2 nd threshold v t → ∞ , possible to estimate virtual valuations to arbitrary accuracy.

Theorem: For a fixed tie-breaking ordering ≻ the pseudo-dimension of the set of n-bidder single item t-level auctions is 𝑃(𝑜𝑢 ⋅ log 𝑜𝑢) Proof Sketch: v Need to bound size of every set shatterable by t-level auctions v Fix sample 𝑇 = (𝑡 1 , 𝑡 2 … 𝑡 𝑛 ) , witness 𝑆 = 𝑠 1 , 𝑠 2 , … 𝑠 𝑛 v Each auction C of the class induces binary labelling on S w.r.t. R S is shattered w.r.t. R iff distinct labelings of 𝑇 = 2 𝑛 1. 2. Define equivalence classes of t-level auctions. Number of equivalence classes ≤ 𝑜𝑛 + 𝑜𝑢 2𝑜𝑢 3. Upper bound number of distinct labelings of S that can be generated by any auction in a single equivalence class ≤ 𝑛 𝑜𝑢 4. Combining the above, we get 2 𝑛 < 𝑜𝑛 + 𝑜𝑢 3𝑜𝑢 → 𝑛 = 𝑃(𝑜𝑢 ⋅ log 𝑜𝑢)

Lemma: Consider Bidders with valuations in 0, 𝐼 and with 𝑄 max 𝑤 y > 𝛽 ≥ 𝛿 , then 𝐷 } contains a single item auction with expected revenue at least 1 − 𝜗 times the optimal € „ revenue for 𝑢 = Θ • + log €‚ƒ … Theorem: For a fixed tie-breaking ordering ≻ the pseudo-dimension of the set of n-bidder single item t-level auctions is 𝑃(𝑜𝑢 ⋅ log 𝑜𝑢) Tunable Sweet Spot 𝑢 = Θ 1 𝐼 𝑢 → ∞ 𝛿 + log €‚ƒ 𝑢 = 1 𝛽 Complex Simple Simple for polynomial t Optimal Non-optimal 1 − 𝜗 times optimal

Unformatted References J 1. The Pseudo-Dimension of Near-Optimal Auctions 2. The Sample Complexity of Revenue Maximization: https://www.youtube.com/watch?v=-qjzrAxkoew 3. VC dimension: • http://www.cs.cmu.edu/~guestrin/Class/15781/slides/learningtheory-bns- annotated.pdf • https://en.wikipedia.org/wiki/VC_dimension 4. http://theory.stanford.edu/~tim/f16/l/l13.pdf