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 Myersons auction prior known
Jamie Morgenstern Tim Roughgarden
Presented by Shubhang Kulkarni
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
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
(~ 𝑜 𝑒𝑓𝑠𝑓𝑓𝑡 𝑝𝑔 𝑔𝑠𝑓𝑓𝑒𝑝𝑛) (→ 𝑗𝑜𝑔𝑗𝑜𝑗𝑢𝑓 𝑒𝑓𝑠𝑓𝑓𝑡 𝑝𝑔 𝑔𝑠𝑓𝑓𝑒𝑝𝑛) 𝐹𝑦𝑞𝑓𝑑𝑢𝑓𝑒 𝑠𝑓𝑤𝑓𝑜𝑣𝑓 𝑝𝑔 𝑊𝐷𝐻 𝑥𝑗𝑢ℎ 𝑁𝑝𝑜𝑝𝑞𝑝𝑚𝑧 𝑆𝑓𝑡𝑓𝑠𝑤𝑓𝑡 𝟐 𝟑 ⋅ 𝑃𝑞𝑢𝑗𝑛𝑣𝑛 𝑓𝑦𝑞𝑓𝑑𝑢𝑓𝑒 𝑠𝑓𝑤𝑓𝑜𝑣𝑓 ≥ [Hartline, Roughgarden ‘09]
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)
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
Measure of capacity that can be learned by a statistical classification algorithm
Cardinality of the largest set of data the class can overfit to in boolean classification
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
v Auction 𝐵: 𝐼 𝑜 → 𝑆 𝑀𝑝𝑥 𝑄𝑡𝑣𝑓𝑒𝑝 𝐸𝑗𝑛𝑓𝑜𝑡𝑗𝑝𝑜 ≡ 𝑇𝑗𝑛𝑞𝑚𝑓 𝐵𝑣𝑑𝑢𝑗𝑝𝑜 𝐷𝑚𝑏𝑡𝑡
Auction Pseudo Dimension Intuitive degrees of freedom Vickery Auction, Anonymous Reserve 𝑃(1) 1 Vickery Auction, Bidder Specific Reserve 𝑃(𝑜 log 𝑜) 𝑜 Virtual Welfare Maximizer ∞ ∞
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
v For each bidder 𝑗,
v 𝑢𝑗 𝑤𝑗 = 𝜐 index of largest threshold 𝑚 𝑗,𝜐 that lower bounds the value 𝑤𝑗, v -1 if 𝑤𝑗 < 𝑚 𝑗,0 v 𝑢𝑗(𝑤𝑗) 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 𝑢𝑗 𝑤𝑗 = −1
v Unique one that renders auction DSIC v Winner pays lowest bid at which he/she continues to win v Losers pay 0 vAuthors note three interesting cases
𝒎𝒃 2 𝒘𝒃 4 6 8 𝒎𝒄 𝒘𝒄 1.5 5 9 10 𝒎𝒅 𝒘𝒅 1.7 3.9 6 7 t = 1 t = 2 t = 3 t = 4 𝒒𝒃 2 𝒒𝒄 𝒒𝒅
𝒎𝒃 2 4 6 8 𝒘𝒃 𝒎𝒄 1.5 5 9 10 𝒘𝒄 𝒎𝒅 1.7 3.9 𝒘𝒅 6 7 t = 1 t = 2 t = 3 t = 4 𝒒𝒃 8 𝒒𝒄 𝒒𝒅
𝒎𝒃 2 4 6 8 𝒘𝒃 𝒎𝒄 1.5 5 𝒘𝒄 9 10 𝒎𝒅 1.7 3.9 𝒘𝒅 6 7 t = 1 t = 2 t = 3 t = 4 𝒒𝒃 4 𝒒𝒄 𝒒𝒅
𝒎𝒃 2 4 𝒘𝒃 6 8 𝒎𝒄 1.5 5 𝒘𝒄 9 10 𝒎𝒅 1.7 3.9 6 𝒘𝒅 7 t = 1 t = 2 t = 3 t = 4 𝒒𝒃 𝒒𝒄 𝒒𝒅 6
vDiscrete 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” vLevels map to common virtual values v 1-level auctions treat all values below single thresholds as –ve virtual
v 2-level auctions refine virtual value estimates with 2nd 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 = (𝑡1, 𝑡2 … 𝑡𝑛), witness 𝑆 = 𝑠
1, 𝑠 2, … 𝑠 𝑛
v Each auction C of the class induces binary labelling on S w.r.t. R 1. S is shattered w.r.t. R iff distinct labelings of 𝑇 = 2𝑛
≤ 𝑜𝑛 + 𝑜𝑢 2𝑜𝑢
auction in a single equivalence class ≤ 𝑛𝑜𝑢
2𝑛 < 𝑜𝑛 + 𝑜𝑢 3𝑜𝑢 → 𝑛 = 𝑃(𝑜𝑢 ⋅ log 𝑜𝑢)
Lemma: Consider Bidders with valuations in 0, 𝐼 and with 𝑄 max 𝑤y > 𝛽 ≥ 𝛿, then 𝐷} contains a single item auction with expected revenue at least 1 − 𝜗 times the optimal revenue for 𝑢 = Θ
€
„ …
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)
𝑢 = 1 Simple Non-optimal 𝑢 → ∞ Complex Optimal 𝑢 = Θ 1 𝛿 + log€‚ƒ 𝐼 𝛽 Simple for polynomial t 1 − 𝜗 times optimal Tunable Sweet Spot
https://www.youtube.com/watch?v=-qjzrAxkoew
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