Gross Substitutes Tutorial Part II: Economic Implications + Pushing - PowerPoint PPT Presentation

Gross Substitutes Tutorial Part II: Economic Implications + Pushing the Boundaries RENATO PAES LEME, GOOGLE RESEARCH INBAL TALGAM-COHEN → TECHNION CS EC 2018

Roadmap Part II-a: Part I-a: Economic Combinatorial properties properties Part I-b: Part II-b: Algorithmic Pushing the properties boundaries 2 EC 2018 GROSS SUBSTITUTES TUTORIAL / PAES LEME & TALGAM-COHEN

Previously, in Part I Remarkable combinatorial + algorithmic properties of GS 1 GS valuation: ◦ Combinatorial exchange properties GS ◦ Optimality of greedy & local search algorithms for DEMAND 𝑜 GS valuations (= market): ◦ Walrasian market equilibrium existence GSGS GS ◦ WELFARE-MAX (and pricing) computationally tractable 3 EC 2018 GROSS SUBSTITUTES TUTORIAL / PAES LEME & TALGAM-COHEN

Plan for Part II 1. Economic implications: Central results in market design that depend on the nice properties of GS Classic theory (and some 2. Pushing the boundaries of GS: recent insights) ◦ Robustness of the algorithmic properties ◦ Extending the economic properties (networks and beyond) State-of-the-art Disclaimer: and open ◦ Literature too big to survey comprehensively challenges 4 EC 2018 GROSS SUBSTITUTES TUTORIAL / PAES LEME & TALGAM-COHEN

Motivation GS assumption fundamental to market design with indivisible items ◦ Sufficient (and in some sense necessary) for the following results: 1. Equilibrium prices exist and have a nice lattice structure 2. VCG outcome is revenue-monotone, stable (in the core) 3. “Invisible hand” – prices coordinate “typical” markets ◦ (GS preserved under economically important transformations) ◦ Interesting connection between economic, algorithmic properties 5 EC 2018 GROSS SUBSTITUTES TUTORIAL / PAES LEME & TALGAM-COHEN

More Motivation: Uncharted Territory General [ABDR’12] Subadditive Subadditive Submodular GS GS ??? [FI’13,FFI+’15, HS’16] 6 EC 2018 GROSS SUBSTITUTES TUTORIAL / PAES LEME & TALGAM-COHEN

GSGS GS Recall Our Market Model 𝑛 buyers 𝑁 (notation follows [Paes Leme’17]) 𝑛 + 1 players in the grand coalition 𝐻 = 𝑁 ∪ {0} ◦ player 𝑗 = 0 is the seller 𝑜 indivisible items 𝑂 Allocation 𝒯 = 𝑇 1 … , 𝑇 𝑛 is a partition of items to 𝑛 bundles Prices: 𝑞 ∈ ℝ 𝑜 is a vector of item prices; let 𝑞 𝑇 = σ 𝑘∈𝑇 𝑞 𝑘 ◦ So 𝑞 𝑂 = seller’s utility (revenue) from clearing the market 7 EC 2018 GROSS SUBSTITUTES TUTORIAL / PAES LEME & TALGAM-COHEN

GS Recall Our Buyer Model Buyer 𝑗 has valuation 𝑤 𝑗 : 2 𝑂 → ℝ Fix item prices 𝑞 ◦ If buyer 𝑗 gets 𝑇 𝑗 , her quasi-linear utility is 𝜌 𝑗 = 𝜌 𝑗 (𝑇 𝑗 , 𝑞) = 𝑤 𝑗 𝑇 𝑗 − 𝑞(𝑇 𝑗 ) ◦ 𝑇 𝑗 is in buyer 𝑗 ’s demand given 𝑞 if 𝑇 𝑗 ∈ arg max 𝜌 𝑗 (𝑇, 𝑞) S 8 EC 2018 GROSS SUBSTITUTES TUTORIAL / PAES LEME & TALGAM-COHEN

Preliminaries 1. THE CORE 2. SUBMODULARITY ON LATTICES 3. FENCHEL DUAL 9 EC 2018 GROSS SUBSTITUTES TUTORIAL / PAES LEME & TALGAM-COHEN

Preliminaries: The Core Consider the cooperative game (𝐻, 𝑥) : ◦ players 𝐻 𝜌 = (2, 3, 3) ◦ coalitional value function 𝑥: 2 𝐻 → ℝ 6 8 𝜌 = utility profile associated with an outcome of the game Coalition 𝐷 ⊆ 𝐻 will not cooperate (“ block ”) if σ 𝑗∈𝐷 𝜌 𝑗 < 𝑥(𝐷) Definition: 𝜌 is in the core if no coalition is blocking, i.e., σ 𝑗∈𝐷 𝜌 𝑗 ≥ 𝑥(𝐷) for every 𝐷 10 EC 2018 GROSS SUBSTITUTES TUTORIAL / PAES LEME & TALGAM-COHEN

Preliminaries: Lattices Lattice = partially ordered elements (𝑌, ≼) with “ join ”s, “ meet ”s ∈ 𝑌 ◦ Join ∨ of 2 elements = smallest element that is ≽ both ◦ Meet ∧ of 2 elements = largest element that is ≼ both 11 EC 2018 GROSS SUBSTITUTES TUTORIAL / PAES LEME & TALGAM-COHEN

Preliminaries: Lattices (2 𝑂 , ⊆) is a lattice: ◦ Join of 𝑇, 𝑈 ∈ 2 𝑂 is 𝑇 ∪ 𝑈 𝑈 𝑇 ◦ Meet of 𝑇, 𝑈 ∈ 2 𝑂 is 𝑇 ∩ 𝑈 (ℝ 𝑜 , ≤) is a lattice: 𝑡 ◦ Join of 𝑡, 𝑢 ∈ ℝ 𝑜 is their component-wise max ◦ Meet of 𝑡, 𝑢 ∈ ℝ 𝑜 is their component-wise min 𝑢 Can naturally define a product lattice ◦ E.g. over 2 𝑂 × ℝ 𝑜 , or ℝ 𝑜 × 2 𝑁 = prices x coalitions 12 EC 2018 GROSS SUBSTITUTES TUTORIAL / PAES LEME & TALGAM-COHEN

Preliminaries: Submodularity on Lattices Definition: 𝑔 is submodular on a lattice if for every 2 elements 𝑡, 𝑢 , 𝑔 𝑡 + 𝑔 𝑢 ≥ 𝑔 𝑡 ∨ 𝑢 + 𝑔 𝑡 ∧ 𝑢 13 EC 2018 GROSS SUBSTITUTES TUTORIAL / PAES LEME & TALGAM-COHEN

GS Preliminaries: Fenchel Dual 𝑤: 2 𝑂 → ℝ = valuation Definition: The Fenchel dual 𝑣: ℝ 𝑂 → ℝ of 𝑤 maps prices to the buyer’s max. utility under these prices 𝑣 𝑞 = max 𝑤 𝑇 − 𝑞(𝑇) = max 𝜌 𝑇, 𝑞 𝑇 𝑇 Theorem [Ausubel- Milgrom’02]: 𝑤 is GS iff its Fenchel dual is submodular 14 EC 2018 GROSS SUBSTITUTES TUTORIAL / PAES LEME & TALGAM-COHEN

GSGS GS Preliminaries: Fenchel Dual & Config. LP 𝜌,𝑞 σ 𝑗 𝜌 𝑗 + 𝑞(𝑂) 𝑦 {σ 𝑗,𝑇 𝑦 𝑗,𝑇 𝑤 𝑗 𝑇 } max min Using Fenchel dual 𝑣 𝑗 ⋅ : s. t. σ 𝑇 𝑦 𝑗,𝑇 ≤ 1 ∀𝑗 s. t. 𝜌 𝑗 ≥ 𝑤 𝑗 𝑇 − 𝑞(𝑇) ∀𝑗, 𝑇 min 𝑞 {෍ 𝑣 𝑗 (𝑞) + 𝑞(𝑂)} 𝑗 σ 𝑗,𝑇:𝑘∈𝑇 𝑦 𝑗,𝑇 ≤ 1∀𝑘 𝜌, 𝑞 ≥ 0 𝑦 ≥ 0 Minimize total utility (including Maximize welfare (sum of values) seller’s) s.t. feasibility of allocation s.t. buyers maximizing their utility 15 EC 2018 GROSS SUBSTITUTES TUTORIAL / PAES LEME & TALGAM-COHEN

Preliminaries: Fenchel Dual From previous slide: For GS, the maximum welfare is equal to min 𝑞 {෍ 𝑣 𝑗 𝑞 + 𝑞(𝑂)} 𝑗∈𝑁 where 𝑣 𝑗 ⋅ = Fenchel dual Applying to buyer 𝑗 and bundle 𝑇 we get the duality between 𝑤 𝑗 , 𝑣 𝑗 : 𝑤 𝑗 𝑇 = min 𝑞 {𝑣 𝑗 (𝑞) + 𝑞(𝑇)} 16 EC 2018 GROSS SUBSTITUTES TUTORIAL / PAES LEME & TALGAM-COHEN

1. Economic Implications of GS 17 EC 2018 GROSS SUBSTITUTES TUTORIAL / PAES LEME & TALGAM-COHEN

Economic Implications of GS 1. Equilibrium prices form a lattice 2. VCG outcome monotone, in the core 3. Prices coordinate “typical” markets Connection between economic, algorithmic properties 18 EC 2018 GROSS SUBSTITUTES TUTORIAL / PAES LEME & TALGAM-COHEN

Structure of Equilibrium Prices for GS Recall: 𝒯, 𝑞 is a Walrasian market equilibrium if: ◦ ∀𝑗 ∶ 𝑇 𝑗 is in 𝑗 ’s demand given 𝑞 ; ◦ the market clears Fix GS market, let 𝑄 be all equil. prices Theorem: [Gul- Stacchetti’99] Equil. prices form a complete lattice ◦ If 𝑞, 𝑞′ are equil. prices then so are 𝑞 ∨ 𝑞 ′ , 𝑞 ∧ 𝑞 ′ ◦ 𝑞 = ⋁𝑄 (component-wise sup ) and 𝑞 = ⋀𝑄 (component-wise inf ) exist in 𝑄 19 EC 2018 GROSS SUBSTITUTES TUTORIAL / PAES LEME & TALGAM-COHEN

Economic Characterization of Extremes 𝑞 = max. equil. price, 𝑞 = min. equil. price Theorem: [Gul- Stacchetti’99] In monotone GS markets, ◦ 𝑞 𝑘 = decrease in welfare if 𝑘 removed from the market ◦ 𝑞 𝑘 = increase in welfare if another copy (perfect substitute) of 𝑘 added to the market 20 EC 2018 GROSS SUBSTITUTES TUTORIAL / PAES LEME & TALGAM-COHEN

Example Max. welfare is 5 ◦ 2 with no pineapple, 3 with no strawberry ◦ 7 with extra pineapple, 5 with extra strawberry 𝑞 ∨ 𝑞 ′ = 𝑞 𝑞′ 3 3 $2 𝑄 2 2 𝑞 $2 2 ADD 0 2 21 EC 2018 GROSS SUBSTITUTES TUTORIAL / PAES LEME & TALGAM-COHEN

A Corollary 𝑞 = min. equil. prices 𝑞 𝑘 = welfare increase if copy of 𝑘 is added to the market [GS’99] In unit-demand markets, 𝑞 coincides with VCG prices ◦ Let 𝑗 be the player allocated 𝑘 in VCG ◦ 𝑗 pays for 𝑘 the difference in welfare buyers 𝑁 ∖ {𝑗} can get from 𝑂 and from 𝑂 ∖ {𝑘} 22 EC 2018 GROSS SUBSTITUTES TUTORIAL / PAES LEME & TALGAM-COHEN

Economic Implications of GS 1. Equilibrium prices form a lattice 2. VCG outcome monotone, in the core 3. Prices coordinate “typical” markets 23 EC 2018 GROSS SUBSTITUTES TUTORIAL / PAES LEME & TALGAM-COHEN

VCG Auction Multi-item generalization of Vickrey (2 nd price) auction The only dominant-strategy truthful, welfare-maximizing auction in which losers do not pay But is it practical? To analyze its properties let’s define the coalitional value function 𝑥 24 EC 2018 GROSS SUBSTITUTES TUTORIAL / PAES LEME & TALGAM-COHEN

GSGS GS Coalitional Value Function 𝑥 Definition: 𝑥 maps any coalition of players 𝐷 ⊆ 𝐻 to the max. welfare from reallocating 𝐷 ’s items among its members ◦ Without the seller (for 𝐷: 0 ∉ 𝐷 ), 𝑥 𝐷 = 0 ◦ For the grand coalition, 𝑥 𝐻 = max. social welfare ( 𝑥 immediately defines a cooperative game among the players – we’ll return to this) 25 EC 2018 GROSS SUBSTITUTES TUTORIAL / PAES LEME & TALGAM-COHEN