ROBUST MARKET DESIGN: INFORMATION & COMPUTATION Inbal Talgam-Cohen Hebrew University, Tel-Aviv University EC/GAMES 2016
Talgam-Cohen Robust Market Design: Information & Computation 2 The Field of Market Design • Study of resource allocation with dispersed information by markets and auctions • Remarkably successful applications, 2012 Nobel Prize • Computer science involved in all aspects This talk focuses on: • Contributions of theoretical computer science to the theory of market design: • Relaxing assumptions • Tackling informational and computational challenges • In the context of indivisible resources, prices, welfare/revenue maximization
Talgam-Cohen Robust Market Design: Information & Computation 3 Major Challenges & Simplifying Assumptions Fundamental challenge for Mitigating assumption in Context market design classic market design theory Revenue- Extracting revenue when values Seller has information on the maximization are private information distributions of values Welfare- Achieving a welfare-maximizing Values have structure, maximization allocation of indivisible resources e.g., substitutes
Talgam-Cohen Robust Market Design: Information & Computation 4 Classic Results Rely on Simplifying Assumptions Context Assumption Classic result depending on assumption Revenue- Seller knows Characterization of optimal revenue [Mye’81] maximization distributions Welfare- Resources viewed Existence of welfare-maximizing equilibrium in markets with indivisible resources [KC’82] maximization as substitutes
Talgam-Cohen Robust Market Design: Information & Computation 5 Motivation for More Robust Market Design • The common knowledge assumption is too stringent: Contrary to much of current theory, the statistics of the data we observe shift very rapidly [Google Research white paper] • The substitutes assumption is too stringent: Complements across license valuations exist and complicate the design process [Byk’00 on spectrum licenses] Substitutes valuations form a zero-measure subset of (even submodular) valuations [Lehmann-Lehmann- Nisan’06]
Talgam-Cohen Robust Market Design: Information & Computation 6 Research Goal Use computer science theory to understand when central results in market design theory hold (at least approximately) in a robust way, without stringent assumptions What is really driving classic positive results?
Talgam-Cohen Robust Market Design: Information & Computation 7 My Research Domains Revenue-maximization and information: • Matching markets • Interdependent and correlated buyers • Ad auctions Welfare-maximization with complements: • Feasibility constraints in double auctions • Walrasian equilibrium • Bundling equilibrium • Purely computational and informational aspects
Talgam-Cohen Robust Market Design: Information & Computation 8 Plan for Rest of Talk 1. Model 2. Robust revenue-maximization in matching markets [RTY’12] • Main result : “Vickrey with increased competition” approximately extracts the (unknown) optimal revenue (a multi- parameter “Bulow - Klemperer” result) • Computational tools: approximation, probabilistic analysis 3. Non-existence of welfare-maximizing market equilibrium [RT’15] • Main result: Computational complexity explanation for non-existence of equilibrium • Computational tools: reduction, LP, complexity hierarchy
Talgam-Cohen Robust Market Design: Information & Computation 9 MODEL
Talgam-Cohen Robust Market Design: Information & Computation 10 A Resource Allocation Problem Maximize welfare 𝑛 indivisible items, priced 𝑜 buyers, where buyer 𝑗 has: Maximize Maximize • Private valuation 𝑤 𝑗 : 2 [𝑛] → ℝ ≥0 𝔽 [revenue] revenue • Demand 𝑇 ⊆ 2 𝑛 • 𝑇 maximizes 𝑗 ’s quasi-linear utility: 𝑤 𝑗 𝑇 − 𝑗′s payment Maximize own utility (demand) • Bayesian assumption for revenue: • Values for item 𝑘 are i.i.d. draws from a regular distribution 𝐺 𝑘 To solve, find allocation (𝑇 1 , … , 𝑇 𝑜 ) and set prices • Maximize welfare (sum of values ∑𝑤 𝑗 (𝑇 𝑗 ) ) or revenue (sum of payments)
Talgam-Cohen Robust Market Design: Information & Computation 11 Example: Unit-Demand Valuations (& Item Prices) Buyers Items 𝑞 = $3 1 𝑤 3 = $10 , 𝑞 = $5 2 𝑤 3 = $2 , … 3 4 𝑞 = $6 Unit-demand Allocation – Item prices valuations a matching 𝑤 𝑗 𝑇 = max 𝑘∈𝑇 𝑤 𝑗 (𝑘)
Talgam-Cohen Robust Market Design: Information & Computation 12 Example: More Complex Valuations (& Prices) Buyers Items 1 𝑤 3 = $15 , 𝑞 3 = $9 2 𝑤 3 = $4 , … 3 4 𝑞 3 = $6 General Allocation (not Personalized valuations a matching) bundle prices
Talgam-Cohen Robust Market Design: Information & Computation 13 Example: Bayesian Assumption Buyers Items 𝐺 𝐺 1 1 1 1 𝐺 𝑮 𝟑 2 2 𝑤 3 ∼ 𝑮 𝟑 2 3 3 𝐺 𝐺 3 3 4 4 𝐺 𝐺 𝑤 4 ∼ 𝑮 𝟑 4 4 “Regular” distributions
Talgam-Cohen Robust Market Design: Information & Computation 14 INFORMATIONAL CHALLENGES IN REVENUE MAXIMIZATION
Talgam-Cohen Robust Market Design: Information & Computation 15 Matching Settings and Revenue • Unit-demand valuations, Bayesian assumption 1 𝐺 𝑮 1 1 2 2 𝐺 𝑮 2 Known to market 3 3 𝐺 𝑮 3 maker 4 4 Matching 𝐺 𝑮 4 • 1 item : Myerson’s mechanism maximizes revenue (dominant -strategy) truthfully • Runs Vickrey (2 nd price) auction after using 𝑮 to fine-tune a reserve price • >1 items: No such mechanism known • (Revenue-maximizing, dominant-strategy truthful [ cf . Cai-Daskalakis- Weinberg’12] )
Talgam-Cohen Robust Market Design: Information & Computation 16 Simple Robust Approach: Increasing Competition • Design the market to determine pricing through competition Increase or: Limit demand supply … then maximize welfare by running simple special case of VCG Technical challenge: • VCG is inherently robust ; it’s left to quantify how much to increase competition such that VCG’s revenue is comparable to the (“unknown” ) optimal revenue
Talgam-Cohen Robust Market Design: Information & Computation 17 Main Results: Revenue • Define: • OPT = Optimal revenue with known distributions subject to (dominant-strategy) truthfulness Theorem: For 𝑛 items and 𝑜 buyers, assuming symmetry and regularity, 𝔽 revenue of VCG with 𝑜 + 𝑛 buyers ≥ OPT • “Multi -parameter Bulow- Klemperer theorem” [ cf. BK’96] • 𝑛 more buyers is tight in worst-case Theorem: For 𝑛 items and 𝑜 buyers, assuming symmetry and regularity, 𝔽 revenue of VCG with supply limit 𝑜/2 ≥ α ⋅ OPT • 𝛽 is at least ¼ when 𝑛 ≤ 𝑜
Talgam-Cohen Robust Market Design: Information & Computation 18 𝔽 revenue of VCG with 𝑜 + 𝑛 buyers ≥ OPT 𝔽 revenue of VCG with supply limit 𝑜/2 ≥ α ⋅ OPT Approximation Guarantee 𝑛 [revenue of VCG + ] 𝔽 𝐺 1 ,…,𝐺 Approximation ratio = min 𝔽 𝐺 𝑛 revenue of OPT 𝑮 𝟐 ,…,𝑮 𝒏 𝐺 1 ,…,𝐺 𝑛 1 ,…,𝐺 Where • 𝐺 1 , … , 𝐺 𝑛 = Arbitrary regular distributions • VCG + = Vickrey with increased supply, no knowledge of 𝐺 1 , … , 𝐺 𝑛 • OPT 𝑮 𝟐 ,…,𝑮 𝒏 = “Unknown” optimal mechanism for 𝐺 1 , … , 𝐺 𝑛
Talgam-Cohen Robust Market Design: Information & Computation 19 Related Work • Information-robustness as a goal • Scarf’58, Wilson’87, Bertsimas - Thiele’14, … • Prior-independent mechanism design • Bulow- Klemperer’96, Segal’03, Bergemann - Morris’05, Dhangwatnotai -et- al.’11, Devanur’11, Chawla-et- al.’13, Bandi - Bertsimas’14, … • Simple versus optimal mechanisms • Hartline- Roughgarden’09, Chawla -et- al.’10, Hart - Nisan’12, Babioff -et- al.’15, …
Talgam-Cohen Robust Market Design: Information & Computation 20 How to Limit Supply • Limiting supply of homogeneous items is a standard marketing method • How to limit supply of heterogeneous items? Arbitrary Best half matching • Instead, let the market decide of size 2 • Definition: VCG with supply limit ℓ • Allocation: Welfare-maximizing matching limited to ℓ items • Pricing : As usual (“externalities”)
Talgam-Cohen Robust Market Design: Information & Computation 21 𝔽 revenue of VCG with 𝑜 + 𝑛 buyers ≥ OPT Proof Idea 𝔽 revenue of VCG with supply limit 𝑜/2 ≥ α ⋅ OPT 1. VCG with supply limit approximates the revenue of VCG with added buyers • Idea: Limiting the supply creates the same supply-demand ratio as adding buyers 2. To show that VCG with added buyers is comparable to OPT : VCG + revenue from item 𝒌 OPT revenue from item 𝒌 At least what the 𝑜 unmatched At most what 𝑜 buyers would pay for 𝑘 [Chawla-et- al.’10] buyers would pay for 𝑘 • The challenge: Showing that the unmatched buyers lead to high prices for sold items • Use “principle of deferred decision” and stability of matching
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