When are Welfare Guarantees Robust? I-CORE DAY 2016 INBAL - PowerPoint PPT Presentation

When are Welfare Guarantees Robust? I-CORE DAY 2016 INBAL TALGAM-COHEN (I (I-CORE POST-DOC) BASED ON JOINT WORK WITH TIM ROUGHGARDEN & & JAN VONDRAK

The Welfare Maximization Problem Central problem in AGT; interesting algorithmically ◦ [e.g., Murota’96, Lehmann -Lehmann- Nisan’06, Vondrak’08, Khot et al.’08, Feige’09…] Setting: 𝑛 items, 𝑜 buyers, valuation functions 𝑤 1 , … , 𝑤 𝑜 𝑤 𝑗 (bundle 𝑇 𝑗 of items) = value of buyer 𝑗 for 𝑇 𝑗 Monotonicity: 𝑤 𝑗 𝑇 𝑗 ≤ 𝑤 𝑗 𝑈 𝑗 whenever 𝑇 𝑗 ⊂ 𝑈 𝑗 The algorithmic problem: Given oracle access to the valuations Output an allocation of items 𝑇 1 , … , 𝑇 𝑜 Goal: maximize welfare σ 𝑗∈𝑂 𝑤 𝑗 (𝑇 𝑗 ) 2 TALGAM-COHEN WHEN ARE WELFARE GUARANTEES ROBUST?

Hardness Determined by Valuation Class Monotone 𝑤 𝑗 ′ is submodular Submodular Gross Substitutes Linear 𝑤 𝑗 is linear 3 TALGAM-COHEN WHEN ARE WELFARE GUARANTEES ROBUST?

A Trichotomy Valuations Welfare Maximization Gross substitutes Easy Submodular (more generally, Easy to approximate within constant complement-free) General Hard 4 TALGAM-COHEN WHEN ARE WELFARE GUARANTEES ROBUST?

In Economics Substitutes (“coffee and tea” rather than “coffee and cream") are a very central but stringent assumption ◦ [e.g., Kelso- Crawford’82] For mechanism and market design: “Easy” “Hard” Valuations (Gross) Complements with a dominant Substitutes substitutes component??? Example supporting folklore belief : Spectrum auctions seem to work… 5 TALGAM-COHEN WHEN ARE WELFARE GUARANTEES ROBUST?

Research Question Agenda: Evaluate rigorously whether the main takeaways from the study of gross substitutes remain valid if the substitutes condition holds approximately This work (ongoing): Evaluate whether the nice algorithmic properties hold Meta-question: Is algorithmic approximation tied to economic approximation? 6 TALGAM-COHEN WHEN ARE WELFARE GUARANTEES ROBUST?

Overview of Results For a natural notion of 𝜗 -close to gross substitutes: Negative results in the value query model ◦ Optimal welfare cannot be approximated, demand queries cannot be simulated Positive results with ◦ Additional structure on the valuations ◦ Demand queries ◦ Subclasses of gross substitutes Bottom line: The main take-aways for GS don’t hold AS IS. Standard methods may fail. A more delicate story waiting to be uncovered? 7 TALGAM-COHEN WHEN ARE WELFARE GUARANTEES ROBUST?

Related Work [Karande- Devanur’07] : For divisible items, “markets do not suddenly become intractable if they slightly violate the weak GS property” [Singer- Hassidim’16] : Robust welfare maximization for submodular valuations; related techniques [Lehmann-Lehmann- Nisan’06, Feige et al ’14] : Notions of closeness to submodular (and more general) valuations 8 TALGAM-COHEN WHEN ARE WELFARE GUARANTEES ROBUST?

GS & Approximate GS 9 TALGAM-COHEN WHEN ARE WELFARE GUARANTEES ROBUST?

Gross Substitutes Let Ԧ 𝑞 ≤ Ԧ 𝑟 be vectors of item prices A bundle 𝑇 is in demand given Ԧ 𝑞 if it maximizes the utility 𝑤 𝑗 𝑇 − 𝑞(𝑇) 𝑤 𝑗 is GS if: ◦ for every 𝑇 in demand given Ԧ 𝑞 , ◦ there exists 𝑈 in demand given Ԧ 𝑟 which contains every 𝑘 ∈ 𝑇 whose price didn’t increase 𝑻 Seems brittle… $ $ $ $ 𝑼 10 TALGAM-COHEN WHEN ARE WELFARE GUARANTEES ROBUST?

Subclass and Superclass of GS Subclass: 𝑏 of item values such that 𝑤 𝑗 𝑇 = 𝑑 + σ 𝑘∈𝑇 𝑏 𝑘 𝑤 𝑗 is linear if there is a vector Ԧ ◦ Additive if 𝑑 = 0 ◦ Welfare maximization easy 𝑻 Superclass: 𝑤 𝑗 is submodular if its marginal values 𝑼 𝒌 𝑤 𝑗 𝑘 ∣ 𝑇 = 𝑤 𝑗 𝑘 ∪ 𝑇 − 𝑤 𝑗 (𝑇) are decreasing 𝑤 𝑗 𝑘 ∣ 𝑈 ≥ 𝑤 𝑗 𝑘 ∣ 𝑇 whenever 𝑈 ⊂ 𝑇 ◦ Greedy gives 2- approximation [LLN’06,FNW’78] 11 TALGAM-COHEN WHEN ARE WELFARE GUARANTEES ROBUST?

Approximate GS Idea: 𝑤 is a 2 𝑛 -vector of values; add small pointwise perturbations ◦ Either true valuation is perturbed, or oracle access is erroneous Definition: 𝑤 is 𝜗 -close to GS if there is a GS valuation 𝑤′ such that 𝑤′ 𝑇 ≤ 𝑤 𝑇 ≤ 1 + 𝜗 𝑤′(𝑇) for every 𝑇 Main question, restated: Do the laudable properties of gross substitutes degrade gracefully with 𝜗 ? (e.g. can welfare be approximated up to 1 − 𝜗 ?) Focus in this talk: 𝜗 -close to linear 12 TALGAM-COHEN WHEN ARE WELFARE GUARANTEES ROBUST?

Summary of Answer for Value Queries Monotone Submodular GS Linear 𝑤 is 𝜗 -close to linear but also (close to) submodular - easy 𝑤 is 𝜗 -close to linear - hard 13 TALGAM-COHEN WHEN ARE WELFARE GUARANTEES ROBUST?

Value and Demand Queries 2 types of oracle access to valuations: 𝑇 𝑞 Ԧ GS Welfare-Max Value Demand Value Algo Oracle Oracle Oracle 𝑇 in optimal poly- 𝑤(𝑇) allocation demand many In general: A value query reduces to poly-many demand queries ◦ [Blumrosen-Nisan] For GS: ◦ Welfare can be maximized with poly-many value queries [Murota’96] ◦ A demand query reduces to poly-many value queries [Bertelsen’04] 14 TALGAM-COHEN WHEN ARE WELFARE GUARANTEES ROBUST?

Negative Results for Value Queries For 𝜗 -close to GS: we show 2 impossibilities 𝜗 -close to GS Welfare-Max Value Algo Oracle subpoly- subexp- many approximation 𝑞 Ԧ Demand Value Oracle Oracle 𝑇 in subexp- many demand Conclusion: No sweeping generalization of GS properties 15 TALGAM-COHEN WHEN ARE WELFARE GUARANTEES ROBUST?

What Goes Wrong: Linear vs. Additive If 𝑤 is 𝜗 -close to additive ( 𝑤′ 𝑇 = σ 𝑘∈𝑇 𝑏 𝑘 ) ◦ 𝑏 𝑘 ≤ 𝑤 𝑘 ≤ 1 + 𝜗 𝑏 𝑘 ◦ Can recover 𝑏 𝑘 up to 1 + 𝜗 If 𝑤 is 𝜗 -close to linear ( 𝑤′ 𝑇 = 𝑑 + σ 𝑘∈𝑇 𝑏 𝑘 ) ◦ 𝑏 𝑘 − 𝜗𝑑 ≤ 𝑤 𝑘 ∣ ∅ ≤ (1 + 𝜗)𝑏 𝑘 + 𝜗𝑑 ◦ Cannot recover an approximate version of 𝑤′ ◦ Information theoretic impossibility Possible solution: Strengthen notion of approximate GS so marginals are approximated ⟹ From pointwise approximation to approximation at marginal level 16 TALGAM-COHEN WHEN ARE WELFARE GUARANTEES ROBUST?

Marginal Closeness and Submodularity 𝑻 Notation: Marginal 𝑤(𝑘| ⋅) maps 𝑇 to 𝑤 𝑘 𝑇 Definition: 𝑤 is marginal- 𝜗 -close to decreasing if 𝑼 𝒌 Equivalent for ∀𝑘 ∶ 𝑤(𝑘| ⋅) is 𝜗 -close to decreasing 𝜷 = 𝟐 + 𝝑 Definition [LLN’06] : 𝑤 is 𝛽 -submodular 𝛽𝑤 𝑘 ∣ 𝑈 ≥ 𝑤 𝑘 ∣ 𝑇 whenever 𝑈 ⊂ 𝑇 Proposition: For 𝜗 -close to linear and 𝛽 -submodular valuations, greedy achieves 1−3𝜗 a 𝛽 -approximation to welfare with value queries in poly-time ◦ [compare to Sviridenko et al.’15] 17 TALGAM-COHEN WHEN ARE WELFARE GUARANTEES ROBUST?

Demand Queries: Positive Results Optimal welfare can be estimated with demand queries: Proposition: For every class 𝐷 of valuations for which the integrality gap of the configuration LP is 𝛿 , can estimate welfare within (1 + 𝜗)𝛿 for 𝜗 -close to 𝐷 GS 𝜗 -close to GS Configuration Configuration Demand Demand LP LP Oracle Oracle Fractional allocation Fractional allocation Integrality gap = 1 Integrality gap = 1 + 𝜗 Oblivious rounding techniques (linear, XOS) also enable approximation 18 TALGAM-COHEN WHEN ARE WELFARE GUARANTEES ROBUST?

Performance of Standard Algorithms Proposition: Approximation guarantee of Kelso-Crawford ascending auction degrades gracefully for 𝜗 -close to subclasses of GS In general, can fail miserably… 19 TALGAM-COHEN WHEN ARE WELFARE GUARANTEES ROBUST?

Conclusion: A Cautionary Tale Ideally, approximation guarantees degrade gracefully under small deviations Negative results for 𝜗 -close to linear with value queries ⟹ No “generic reason” for belief that “close to substitutes is easy” Positive results for 𝜗 -close to linear with additional structure or demand queries ⟹ I t matters how closeness is defined ⟹ Robust results can require new ideas 20 TALGAM-COHEN WHEN ARE WELFARE GUARANTEES ROBUST?

The Bigger Picture [Lehmann-Lehmann- Nisan’06]: “This paper’s main message is that the case of submodular valuation functions should be the focus of interest.” Led to a lot of groundbreaking research over past decade It’s a good time to be working on GS (and its approximations )… [Hsu et al. ‘16; Paes Leme and Wong ’16; Cohen -Addad et al ’16; …] 21 TALGAM-COHEN WHEN ARE WELFARE GUARANTEES ROBUST?

Open Problems 1) Is it possible to get a more sweeping positive result? E.g. for ◦ 𝜗 -close to GS and submodular valuations with value queries ◦ 𝜗 -close to GS with a normalization guarantee ◦ Other natural notions of closeness to GS 2) Do nice economic aspects of GS hold for 𝜗 -close to GS? ◦ Possibly need notions of approximate market equilibrium, robust prices, robust auction formats ◦ [ cf . Bikhchandani- Ostroy’02, Ben -Zwi et al. ’13, Feldman et al.’14, Duetting -Roughgarden- T.C.’14 Roughgarden - T.C.’15] 22 TALGAM-COHEN WHEN ARE WELFARE GUARANTEES ROBUST?