The Sample Complexity of Revenue Maximization in the Hierarchy of Deterministic Combinatorial Auctions Ellen Vitercik Joint work with Nina Balcan and Tuomas Sandholm Theory Lunch 27 April 2016
Combinatorial (multi-item) auctions : $5 : $5 : $6 Combinatorial auctions allow bidders to express preferences for bundles of goods
Real-world examples • US Government wireless spectrum auctions [FCC] • Sourcing auctions [Sandholm 2013] • Airport time slot allocation [Rassenti 1982] • Building development, e.g. office space in GHC (no money) • Property sales
Mechanism design • Mechanism designer must determine: – Allocation function: Who gets what? – Payment function: What does the auctioneer charge? • Goal: design strategy-proof mechanisms – Easy for the bidders to compute the optimal strategy – Easy for designer to analyze possible outcomes
Warm-up: single-item auctions Second-Price Auction : $5 , -$3 Allocation (N:$5, T:$3) N INA = give carrot to Nina : $3 Payment (N:$5, T:$3) = charge Nina $3 ø, -$0 T UOMAS Second-price auction: the classic strategy-proof, single-item auction.
Revenue-maximizing combinatorial auctions • Standard assumptions: bidders’ valuations drawn from distribution 𝑬 , mechanism designer knows 𝑬 – Allocation and payment rules often depend on 𝑬
Revenue-maximizing combinatorial auctions Design Challenges Feasible Solutions Support of 𝐄 might be doubly- Draw samples from 𝐄 instead exponential NP-hard to determine the Fix a rich class of auctions. Can revenue-maximizing we learn the revenue- deterministic auction with maximizing combinatorial respect to 𝐄 auction in that class with respect to 𝐄 given samples [Conitzer and Sandholm 2002] drawn from 𝐄 ? • Central problem in Automated Mechanism Design [Conitzer and Sandholm 2002, 2003, 2004, Likhodedov and Sandholm 2004, 2005, 2015, Sandholm 2003] No theory that relates the performance of the designed mechanism on the samples to that mechanism’s expected performance on 𝑬 , until now.
Outline • Introduction • Hierarchy of deterministic combinatorial auction classes • Our contribution: how many samples are needed to learn over the hierarchy of auctions? • Affine maximizer auctions and Rademacher complexity • Mixed-bundling auctions and pseudo-dimension • Summary and future directions
Combinatorial auctions N INA T UOMAS : $1 : 50¢ : $0 : 50¢ : $1 : 50¢ • 𝟒 𝟑 possible outcomes 𝒑 = (𝒑 𝟐 , 𝒑 𝟑 ) • For example, 𝒑 = ( , )
A natural generalization of second price N INA T UOMAS : $1 : 50¢ : $0 : 50¢ : $1 : 50¢ • Social Welfare 𝒑 𝒑 ∗ maximizes SW 𝒑 = SW 𝒑 = 𝒘 𝒋 𝒑 𝒋∈𝑪𝒋𝒆𝒆𝒇𝒔𝒕 𝒑 −𝒋 maximizes SW -i 𝒑 • SW -i 𝒑 = 𝒘 𝒌 𝒑 𝒌∈𝑪𝒋𝒆𝒆𝒇𝒔𝒕− 𝒋 • Allocation: 𝒑 ∗ • Payment: Nina pays SW − 𝑶𝒋𝒐𝒃 𝒑 −𝑶𝒋𝒐𝒃 − SW − 𝑶𝒋𝒐𝒃 𝒑 ∗ The “ Vickrey-Clarke- Groves mechanism” (VCG).
VCG in action N INA T UOMAS : $1 : 50¢ : $0 : 50¢ : $1 : 50¢ • 𝒑 ∗ = , • 𝒑 −𝑶𝒋𝒐𝒃 = (∅, , ) • Nina pays 𝒘 𝑼𝒗𝒑𝒏𝒃𝒕 ( , ) −𝒘 𝑼𝒗𝒑𝒏𝒃𝒕 ( ) = 0 How do we get the bidders to pay more?
Outcome boosting N INA T UOMAS : $1 : 50¢ : $0 : 50¢ : $1 : 50¢ • value ∅, , = 𝒘 𝑶𝒋𝒐𝒃 ∅ + 𝒘 𝑼𝒗𝒑𝒏𝒃𝒕 ( , ) = 50¢
Outcome boosting N INA T UOMAS : $1 : 50¢ : $0 : 50¢ : $1 : 50¢ • value ∅, , = 𝒘 𝑶𝒋𝒐𝒃 ∅ + 𝒘 𝑼𝒗𝒑𝒏𝒃𝒕 ( , ) = 50¢ + 99¢ • 𝒑 ∗ = , • 𝒑 −𝑶𝒋𝒐𝒃 = (∅, , )
Outcome boosting N INA T UOMAS : $1 : 50¢ : $0 : 50¢ : $1 : 50¢ • value ∅, , = 𝒘 𝑶𝒋𝒐𝒃 ∅ + 𝒘 𝑼𝒗𝒑𝒏𝒃𝒕 ( , ) = 50¢ + 99¢ • 𝒑 ∗ = , • 𝒑 −𝑶𝒋𝒐𝒃 = (∅, , ) • Nina pays 𝒘 𝑼𝒗𝒑𝒏𝒃𝒕 ( , ) + 99¢ −𝒘 𝑼𝒗𝒑𝒏𝒃𝒕 ( ) = 99¢
Affine maximizer auctions (AMAs) • Boost outcomes: 𝝁(𝒑) Take bids 𝒘 • • Compute outcome: 𝒐 𝝁 𝒑 ∗ = 𝒃𝒔𝒉𝒏𝒃𝒚 𝒑 𝑻𝑿 𝒑 + 𝝁 𝒑 𝒌∈𝑪𝒋𝒆𝒆𝒇𝒔𝒕 Compute Bidder 𝒋 ’s payment: • 𝑻𝑿 −𝒋 𝒑 −𝒋 + 𝝁 𝒑 −𝒋 − 𝑻𝑿 −𝒋 𝒑 ∗ + 𝝁 𝒑 ∗
Affine maximizer auctions (AMAs) • Boost outcomes: 𝝁(𝒑) Take bids 𝒘 • • Compute outcome: 𝒐 𝒑 ∗ = 𝒃𝒔𝒉𝒏𝒃𝒚 𝒑 𝒘 𝒌 𝒑 + 𝝁 𝒑 𝒌∈𝑪𝒋𝒆𝒆𝒇𝒔𝒕 Compute Bidder 𝒋 ’s payment: • 𝒘 𝒌 𝒑 −𝒋 + 𝝁 𝒑 −𝒋 𝒘 𝒌 𝒑 ∗ + 𝝁 𝒑 ∗ − 𝒌∈𝑪𝒋𝒆𝒆𝒇𝒔𝒕−{𝒋} 𝒌∈𝑪𝒋𝒆𝒆𝒇𝒔𝒕−{𝒋}
Affine maximizer auctions (AMAs) • Boost outcomes: 𝝁(𝒑) ; Weight bidders: 𝒙 𝒋 Take bids 𝒘 • • Compute outcome: 𝒐 𝒑 ∗ = 𝒃𝒔𝒉𝒏𝒃𝒚 𝒑 𝒘 𝒌 𝒑 + 𝝁 𝒑 𝒌∈𝑪𝒋𝒆𝒆𝒇𝒔𝒕 Compute Bidder 𝒋 ’s payment: • 𝒘 𝒌 𝒑 −𝒋 + 𝝁 𝒑 −𝒋 𝒘 𝒌 𝒑 ∗ + 𝝁 𝒑 ∗ − 𝒌∈𝑪𝒋𝒆𝒆𝒇𝒔𝒕−{𝒋} 𝒌∈𝑪𝒋𝒆𝒆𝒇𝒔𝒕−{𝒋}
Affine maximizer auctions (AMAs) • Boost outcomes: 𝝁(𝒑) ; Weight bidders: 𝒙 𝒋 Take bids 𝒘 • • Compute outcome: 𝒐 𝒑 ∗ = 𝒃𝒔𝒉𝒏𝒃𝒚 𝒑 𝒙 𝒌 𝒘 𝒌 𝒑 + 𝝁 𝒑 𝒌∈𝑪𝒋𝒆𝒆𝒇𝒔𝒕 Compute Bidder 𝒋 ’s payment: • 𝒘 𝒌 𝒑 −𝒋 + 𝝁 𝒑 −𝒋 𝒘 𝒌 𝒑 ∗ + 𝝁 𝒑 ∗ − 𝒌∈𝑪𝒋𝒆𝒆𝒇𝒔𝒕−{𝒋} 𝒌∈𝑪𝒋𝒆𝒆𝒇𝒔𝒕−{𝒋}
Affine maximizer auctions (AMAs) • Boost outcomes: 𝝁(𝒑) ; Weight bidders: 𝒙 𝒋 Take bids 𝒘 • • Compute outcome: 𝒐 𝒑 ∗ = 𝒃𝒔𝒉𝒏𝒃𝒚 𝒑 𝒙 𝒌 𝒘 𝒌 𝒑 + 𝝁 𝒑 𝒌∈𝑪𝒋𝒆𝒆𝒇𝒔𝒕 Compute Bidder 𝒋 ’s payment: • 𝟐 𝒙 𝒌 𝒘 𝒌 𝒑 −𝒋 + 𝝁 𝒑 −𝒋 𝒙 𝒌 𝒘 𝒌 𝒑 ∗ + 𝝁 𝒑 ∗ − 𝒙 𝒋 𝒌∈𝑪𝒋𝒆𝒆𝒇𝒔𝒕−{𝒋} 𝒌∈𝑪𝒋𝒆𝒆𝒇𝒔𝒕−{𝒋}
Hierarchy of parameterized auction classes 𝒙 𝒋 , 𝝁 𝒑 ∈ ℝ Affine maximizer auctions [R79] ∪ ∪ Virtual valuation • 𝒙 𝒋 = 𝟐 𝝁 𝒑 = 𝝁 𝒋 𝒑 𝝁 -auctions [J07] combinatorial auctions 𝝁 𝒑 ∈ ℝ • 𝒋∈𝑪𝒋𝒆𝒆𝒇𝒔𝒕 [SL03] ∪ ∪ • 𝒙 𝒋 = 𝟐 𝝁 𝒑 = 𝟏 except any • Mixed bundling auctions outcome where a with reserve prices [TS12] bidder gets all items • item reserve prices ∪ 𝒙 𝒋 = 𝟐 • Mixed bundling auctions 𝝁 𝒑 = 𝟏 except • [J07] outcome where a bidder gets all items
Outline • Introduction • Hierarchy of deterministic combinatorial auction classes • Our contribution: how many samples are needed to learn over the hierarchy of auctions? • Affine maximizer auctions and Rademacher complexity • Mixed-bundling auctions and pseudo-dimension • Summary and future directions
Our contribution Optimize 𝝁 𝒑 and 𝒙 given a sample 𝑻~𝑬 𝑶 • – (Automated Mechanism Design) • We want: – The auction with best revenue over the sample has almost optimal expected revenue – Any approximately revenue-maximizing auction over the sample will have approximately optimal expected revenue For any auction we output, we want |𝑻| large enough such that: • |empirical revenue – expected revenue| < 𝝑 • In other words, how many samples 𝐓 = 𝑶 do we need to ensure that |empirical revenue – expected revenue| = 𝟐 𝑶 𝒔𝒇𝒘 𝑩 𝒘 − 𝔽 𝒘~𝑬 𝒔𝒇𝒘 𝑩 𝒘 < 𝝑 𝒘∈𝑻 for all auctions 𝑩 in the class? • (We can only do this with high probability.)
How many samples do we need? Affine maximizer auctions [R79] 𝟑 𝑽𝒐 𝒏 𝒏 𝑽 + 𝒐 𝒏/𝟑 /𝝑 𝑶 = 𝑷 ∪ ∪ Virtual valuation combinatorial auctions 𝝁 -auctions [J07] [SL03] 𝟑 𝑽𝒐 𝒏 𝒏 𝑽 + 𝒐 𝒏/𝟑 /𝝑 𝑶 = 𝑷 𝟑 𝑽𝒐 𝒏 𝒏 𝑽 + 𝒐 𝒏/𝟑 /𝝑 𝑶 = 𝑷 ∪ ∪ Mixed bundling auctions with reserve prices [TS12] Variables 𝑽/𝝑 𝟑 𝒏 𝟒 𝑶 = 𝑷 𝑶 : sample size 𝒐 : # bidders ∪ 𝒏 : # items 𝑽 : maximum revenue achievable over the Mixed bundling auctions [J07] support of the bidders’ valuation 𝑽/𝝑 𝟑 𝑶 = 𝑷 distributions
How many samples do we need? Affine maximizer auctions [R79] 𝟑 𝑽𝒐 𝒏 𝒏 𝑽 + 𝒐 𝒏/𝟑 /𝝑 𝑶 = 𝑷 ∪ ∪ Virtual valuation combinatorial auctions 𝝁 -auctions [J07] [SL03] 𝟑 𝑽𝒐 𝒏 𝒏 𝑽 + 𝒐 𝒏/𝟑 /𝝑 𝑶 = 𝑷 𝟑 𝑽𝒐 𝒏 𝒏 𝑽 + 𝒐 𝒏/𝟑 /𝝑 𝑶 = 𝑷 ∪ ∪ Mixed bundling auctions with reserve prices [TS12] Variables 𝑽/𝝑 𝟑 𝒏 𝟒 𝑶 = 𝑷 𝑶 : sample size 𝒐 : # bidders ∪ 𝒏 : # items 𝑽 : maximum revenue achievable over the Mixed bundling auctions [J07] support of the bidders’ valuation 𝑽/𝝑 𝟑 𝑶 = 𝑷 distributions Nearly-matching exponential lower bounds.
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