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

SLIDE 1

Gross Substitutes Tutorial Part II: Economic Implications + Pushing the Boundaries

RENATO PAES LEME, GOOGLE RESEARCH INBAL TALGAM-COHEN → TECHNION CS EC 2018

SLIDE 2

Roadmap

Part II-a: Economic properties

Part II-b: Pushing the boundaries Part I-b: Algorithmic properties

Part I-a: Combinatorial properties

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Previously, in Part I

Remarkable combinatorial + algorithmic properties of GS 1 GS valuation:

Combinatorial exchange properties
Optimality of greedy & local search algorithms for DEMAND

𝑜 GS valuations (= market):

Walrasian market equilibrium existence
WELFARE-MAX (and pricing) computationally tractable

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GS GSGS GS

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Plan for Part II

1. Economic implications: Central results in market design that

depend on the nice properties of GS

2. Pushing the boundaries of GS:
Robustness of the algorithmic properties
Extending the economic properties (networks and beyond)

Disclaimer:

Literature too big to survey comprehensively

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Classic theory (and some recent insights) State-of-the-art and open challenges

SLIDE 5

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

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SLIDE 6

General Subadditive Subadditive Submodular

More Motivation: Uncharted Territory

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GS GS [ABDR’12] [FI’13,FFI+’15, HS’16] ???

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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 = 𝑇1 … , 𝑇𝑛 is a partition of items to 𝑛 bundles Prices: 𝑞 ∈ ℝ𝑜 is a vector of item prices; let 𝑞 𝑇 = σ𝑘∈𝑇 𝑞𝑘

So 𝑞 𝑂 = seller’s utility (revenue) from clearing the market

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GSGS GS

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Recall Our Buyer Model

Buyer 𝑗 has valuation 𝑤𝑗: 2𝑂 → ℝ Fix item prices 𝑞

If buyer 𝑗 gets 𝑇𝑗, her quasi-linear utility is

𝜌𝑗 = 𝜌𝑗(𝑇𝑗, 𝑞) = 𝑤𝑗 𝑇𝑗 − 𝑞(𝑇𝑗)

𝑇𝑗 is in buyer 𝑗’s demand given 𝑞 if

𝑇𝑗 ∈ arg max

S

𝜌𝑗(𝑇, 𝑞)

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GS

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Preliminaries

1. THE CORE 2. SUBMODULARITY ON LATTICES 3. FENCHEL DUAL

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Preliminaries: The Core

Consider the cooperative game (𝐻, 𝑥):

players 𝐻
coalitional value function 𝑥: 2𝐻 → ℝ

𝜌 = utility profile associated with an outcome of the game Coalition 𝐷 ⊆ 𝐻 will not cooperate (“block”) if σ𝑗∈𝐷𝜌𝑗 < 𝑥(𝐷) Definition: 𝜌 is in the core if no coalition is blocking, i.e., σ𝑗∈𝐷𝜌𝑗 ≥ 𝑥(𝐷) for every 𝐷

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6 8 𝜌 = (2, 3, 3)

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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

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Preliminaries: Lattices

(2𝑂, ⊆) is a lattice:

Join of 𝑇, 𝑈 ∈ 2𝑂 is 𝑇 ∪ 𝑈
Meet of 𝑇, 𝑈 ∈ 2𝑂 is 𝑇 ∩ 𝑈

(ℝ𝑜, ≤) is a lattice:

Join of 𝑡, 𝑢 ∈ ℝ𝑜 is their component-wise max
Meet of 𝑡, 𝑢 ∈ ℝ𝑜 is their component-wise min

Can naturally define a product lattice

E.g. over 2𝑂 × ℝ𝑜, or ℝ𝑜 × 2𝑁 = prices x coalitions

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𝑇 𝑈 𝑡 𝑢

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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

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Preliminaries: Fenchel Dual

𝑤: 2𝑂 → ℝ = valuation Definition: The Fenchel dual 𝑣: ℝ𝑂 → ℝ of 𝑤 maps prices to the buyer’s max. utility under these prices 𝑣 𝑞 = max

𝑇

𝑤 𝑇 − 𝑞(𝑇) = max

𝑇

𝜌 𝑇, 𝑞 Theorem [Ausubel-Milgrom’02]: 𝑤 is GS iff its Fenchel dual is submodular

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GS

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Preliminaries: Fenchel Dual & Config. LP

max

𝑦 {σ𝑗,𝑇 𝑦𝑗,𝑇𝑤𝑗 𝑇 }

s. t. σ𝑇 𝑦𝑗,𝑇 ≤ 1 ∀𝑗

σ𝑗,𝑇:𝑘∈𝑇 𝑦𝑗,𝑇 ≤ 1∀𝑘 𝑦 ≥ 0 Maximize welfare (sum of values) s.t. feasibility of allocation min

𝜌,𝑞 σ𝑗 𝜌𝑗 + 𝑞(𝑂)

s. t. 𝜌𝑗 ≥ 𝑤𝑗 𝑇 − 𝑞(𝑇) ∀𝑗, 𝑇

𝜌, 𝑞 ≥ 0 Minimize total utility (including seller’s) s.t. buyers maximizing their utility

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GSGS GS

Using Fenchel dual 𝑣𝑗 ⋅ : min

𝑞 {෍ 𝑗

𝑣𝑗(𝑞) + 𝑞(𝑂)}

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Preliminaries: Fenchel Dual

From previous slide: For GS, the maximum welfare is equal to min

𝑞 {෍ 𝑗∈𝑁

𝑣𝑗 𝑞 + 𝑞(𝑂)} where 𝑣𝑗 ⋅ = Fenchel dual Applying to buyer 𝑗 and bundle 𝑇 we get the duality between 𝑤𝑗, 𝑣𝑗: 𝑤𝑗 𝑇 = min

𝑞 {𝑣𝑗(𝑞) + 𝑞(𝑇)}

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1. Economic Implications
f GS

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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

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Structure of Equilibrium Prices for GS

Recall: 𝒯, 𝑞 is a Walrasian market equilibrium if:

∀𝑗 ∶ 𝑇𝑗 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

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Economic Characterization of Extremes

𝑞 = max. equil. price, 𝑞 = min. equil. price Theorem: [Gul-Stacchetti’99] In monotone GS markets,

𝑞𝑘 = decrease in welfare if 𝑘 removed from the market
𝑞𝑘 = increase in welfare if another copy (perfect substitute) of 𝑘 added to the

market

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SLIDE 21

Example

Max. welfare is 5
2 with no pineapple, 3 with no strawberry
7 with extra pineapple, 5 with extra strawberry

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ADD 3 2 2 2 3 2 𝑄 𝑞 𝑞′ 𝑞 ∨ 𝑞′ = 𝑞 $2 $2

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A Corollary

𝑞 = min. equil. prices 𝑞𝑘 = 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}

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Economic Implications of GS

1. Equilibrium prices form a lattice
2. VCG outcome monotone, in the core
3. Prices coordinate “typical” markets

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VCG Auction

Multi-item generalization of Vickrey (2nd 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

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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)

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GSGS GS

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VCG Auction in Terms of 𝑥

𝑥 = coalitional value function VCG allocation: Welfare-maximizing VCG utilities: For every buyer 𝑗 > 0, 𝜌𝑗 = 𝑥 𝐻 − 𝑥(𝐻 ∖ {𝑗}) (a buyer’s utility is her marginal contribution to the social welfare; seller’s utility is the welfare minus the marginals)

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When VCG Goes Wrong

Example: 2 items

Buyer 1: All-or-nothing with value 1
Buyers 2 and 3: Unit-demand with value 1

VCG:

Allocation: Buyers 2, 3 each get an item
Utilities of players 0 to 3: (0, 0, 1, 1)

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VCG outcome blocked by coalition of players 0 and 1!

UD

1 1

UD ALL

1

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When VCG Goes Wrong

Example: 2 items

Buyer 1: All-or-nothing with value 1
Buyers 2 and 3: Unit-demand with value 1

VCG:

Allocation: Buyers 2, 3 each get an item
Utilities of players 0 to 3: (0, 0, 1, 1)

VCG without buyer 3:

Allocation: Buyer 2 gets as item (or buyer 1 gets both)
Utilities of players 0 to 2: (1, 0, 0)

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Non-monotone revenue!

UD

1 1

UD ALL

1

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What Goes Wrong

VCG:

Utilities of players 0 to 3: (0, 0, 1, 1)

VCG without buyer 3:

Utilities of players 0 to 2: (1, 0, 0)

Buyers 2’s marginal contribution to the welfare increases when the coalition includes buyer 3 → coalitional value function 𝑥 is not submodular

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UD

1 1

UD ALL

1

SLIDE 30

Characterization of Good VCG

𝑥 = coalitional value function 𝜌(𝐷) = utility profile from applying VCG to coalition 𝐷 Theorem [Ausubel-Milgrom’02]: Equivalence among -

1. For every 𝐷, 𝜌(𝐷) is in the core (not blocked by any coalition)
2. For every 𝐷, 𝜌(𝐷) is monotone in buyers
in particular, revenue-monotone
3. Function 𝑥 is buyer-submodular
(= submodular when restricted to coalitions including the seller)

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SLIDE 31

Buyer-Submodularity and GS

𝑥 = coalitional value function 𝒲 = class of valuations that contains additive valuations Theorem [Ausubel-Milgrom’02]: For 𝑥 to be buyer-submodular for every market with valuations ⊆ 𝒲, a necessary and sufficient condition is that 𝒲 ⊆ GS “Maximal domain” result

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GSGS GS

SLIDE 32

Recall: For GS markets, the maximum welfare is equal to min

𝑞 {෍ 𝑗∈𝑁

𝑣𝑗 𝑞 + 𝑞(𝑂)} where 𝑣𝑗 ⋅ = Fenchel dual Applied to buyer coalition 𝐷 ⊆ 𝑁, 𝑥 𝐷 ∪ {0} = min

𝑞 {෍ 𝑗∈𝐷

𝑣𝑗 𝑞 + 𝑞(𝑂)}

Proof Sketch: Sufficiency

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SLIDE 33

Proof Sketch: Sufficiency

𝑥 𝐷 ∪ {0} = min

𝑞 {σ𝑗∈𝐷 𝑣𝑗 𝑞 + 𝑞(𝑂)}

Since Fenchel duals {𝑣𝑗} are submodular on ℝ𝑜 for GS → 𝑔 is submodular on the product lattice ℝ𝑜 × 2𝑁 A result by [Topkis’78] shows min

𝑞 {𝑔 𝑞, 𝐷 } is submodular on 2𝑁.

QED

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*Based on slides by Paul Milgrom

Denote by 𝑔 𝑞, 𝐷

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Proof Sketch: Necessity

Let 𝑤 be non-GS Consider a coalition of 𝑤 with additive valuation 𝑞′:

𝑥 {𝑤, 𝑞′} = min

𝑞 {𝑣 𝑞 + ෍ 𝑘

max 0, 𝑞𝑘

′ − 𝑞𝑘 + 𝑞(𝑂)} =

𝑣 𝑞′ + 𝑞′(𝑂)

Generalizes to coalitions with several additive valuations by observing their

join is the minimizer

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*Based on slides by Paul Milgrom

SLIDE 35

Proof Sketch: Necessity

Let 𝑤 be non-GS → Fenchel dual 𝑣 non-submodular ∃𝑞, 𝑞′: 𝑣 𝑞∨ + 𝑣 𝑞∧ > 𝑣 𝑞 + 𝑣(𝑞′) Add 3 additive buyers with valuations 𝑞, 𝑞′, 𝑞∧

𝑥 {𝑤, 𝑞∧} = 𝑣 𝑞∧ + 𝑞∧(𝑂) 𝑥 {𝑤, 𝑞∧, 𝑞, 𝑞′} = 𝑣 𝑞∨ + 𝑞∨(𝑂) > 𝑥 {𝑤, 𝑞∧, 𝑞} = 𝑣 𝑞 + 𝑞(𝑂) 𝑥 {𝑤, 𝑞∧, 𝑞′} = 𝑣 𝑞′ + 𝑞′(𝑂) → 𝑥 not buyer-submodular. QED

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𝑞 𝑞′ 𝑞∧ 𝑞∨

*Based on slides by Paul Milgrom

SLIDE 36

Economic Implications

1. Equilibrium prices form a lattice
2. VCG outcome monotone, in the core
3. Prices coordinate “typical” markets

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SLIDE 37

Breather

Riddle: How is Fenchel connected to the building below?

German-born Jewish mathematician who emigrated following Nazi

suppression and settled in Denmark

His younger brother Heinz immigrated to Israel and became a renowned

architect, designing this Tel-Aviv landmark

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SLIDE 38

Do Equil. Prices Coordinate Markets?

Question posed by [Hsu+’16], following [Hayek’45]:

“Fundamentally, in a system in which the knowledge of the relevant facts is

dispersed among many people, prices can act to coordinate the separate actions of different people…”

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SLIDE 39

Bad Example with GS Valuations

[Cohen-Addad-et-al’16]: Wlog 𝑞1 ≤ 𝑞2 ≤ 𝑞3

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*Based on slides by Alon Eden

Item 1 Item 2 Item 3 1 1 1 1 1 1

ver-

demand!

SLIDE 40

What Goes Wrong

Welfare-maximizing allocation is not unique

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*Based on slides by Alon Eden

Item 1 Item 2 Item 3 1 1 1 1 1 1

SLIDE 41

What Goes Wrong

Welfare-maximizing allocation is not unique

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*Based on slides by Alon Eden

Item 1 Item 2 Item 3 1 1 1 1 1 1

SLIDE 42

Uniqueness Necessary for Coordination

By 2nd Welfare Theorem: Equilibrium prices support any max-welfare allocation

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𝑞2 = ൗ 1 2

Item 1 Item 2 Item 3 1 1 1 1 1 1

𝑞3 = ൗ 1 2 𝑞1 = ൗ 1 2 Demand= {{1}, {3}}

SLIDE 43

Coordinating Prices

Definition: Walrasian equilibrium prices 𝑞 are robust if every buyer has a single bundle in demand given 𝑞

Robust prices are market-coordinating

Theorem: [Cohen-Added-et-al’16, Paes Leme-Wong’17] For a GS market, uniqueness of max-welfare allocation is sufficient for existence of robust equil. prices

Moreover, almost all equil. prices are robust

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SLIDE 44

Plan: Assume GS + uniqueness of max-welfare allocation (and integral values for simplicity); show a ball of equilibrium prices exists This establishes robust pricing: Assume for contradiction both 𝑇∗, 𝑈 in player’s demand given 𝑞

Pf: Uniqueness is Sufficient

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𝑞 𝑇 𝑈 𝑘

SLIDE 45

Plan: Assume GS + uniqueness of max-welfare allocation (and integral values for simplicity); show a ball of equilibrium prices exists This establishes robust pricing: Assume for contradiction both 𝑇∗, 𝑈 in player’s demand given 𝑞 Let 𝑞′ = 𝑞 with 𝑞𝑘 decreased; should also support 𝑇∗, contradiction

Pf: Uniqueness is Sufficient

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𝑞𝑞′ 𝑇 𝑈 𝑘

SLIDE 46

Pf: Exchange Graph [Murota]

Exchange graph for the unique max-welfare allocation:

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Edge weights 𝑥 = how much buyer would lose from exchanging

range with strawberry

(or giving up orange)

SLIDE 47

Pf: Cycles and Equilibrium Prices

A function 𝜚 on the nodes is a potential if 𝑥

𝑘,𝑙 ≥ 𝜚 𝑙 − 𝜚 𝑘

Theorem: ∃ potential 𝜚 ⟺ no negative cycle ⟺ −𝜚 = equil. prices

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Edge weights 𝑥 = how much buyer would lose from exchanging

range with strawberry

(or giving up orange)

𝑥

𝑘,𝑙

𝑘 𝑙 Theorem: ∃ ball of potentials / equil. prices ⟺ all cycles strictly positive

SLIDE 48

Pf: Ball of Equilibrium Prices

0-weight cycle = alternative max-welfare allocation. QED

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Edge weights 𝑥 = how much buyer would lose from exchanging

range with strawberry

(or giving up orange)

𝑥

𝑘,𝑙

𝑘 𝑙 Theorem: ∃ ball of equil. prices ⟺ all cycles strictly positive

SLIDE 49

Do Prices Coordinate Typical Markets?

I.e., do GS markets typically have a unique max-welfare allocation? We say a GS market typically satisfies a condition if it holds whp under a tiny random perturbation of arbitrary GS valuations Challenge: Find a perturbation model that maintains GS

(Ideally one in which the perturbation can be drawn from a discrete set)

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SLIDE 50

2 GS-Preserving Perturbation Models

For simplicity, unit-demand 𝑤𝑗 The perturbation: additive valuation 𝑏𝑗

1. 𝑤𝑗

′(𝑇) = 𝑤𝑗 𝑇 + 𝑏𝑗 𝑇 [P-LW’17]

𝑤𝑗

′ not unit-demand

2. 𝑤𝑗

′(𝑘) = 𝑤𝑗 𝑘 + 𝑏𝑗 𝑘 [Hsu+’16]

𝑤𝑗

′ unit-demand

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1. 𝑤𝑗

′ 𝑂 = 𝑤1 + 𝑏1 + 𝑏2

2. 𝑤𝑗

′ 𝑂 = 𝑤1 + 𝑏1

GS

ADD UD 𝑤1 𝑤2 𝑏2 𝑏1

≥

SLIDE 51

Unique Max-Welfare Allocation is Typical

Lemma: [P-LW’17,Hsu+’16] For sufficiently small perturbation, whp the perturbed market has a unique max-welfare allocation

(Also max-welfare in the original market)
Perturbation can be from sufficiently large discrete range [cf. MVV’87

Isolation Lemma]

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SLIDE 52

Market Coordination: Additional Results

[Cohen-Addad-et-al’16]: “Necessity” of GS for market coordination

∃ non-GS market with:
1. unique max-welfare allocation
2. Walrasian equilibrium
3. no coordinating prices (not even dynamic!)

[Hsu-et-al’16]: Robustness of min. equilibrium prices (not in ball)

For perturbed markets such prices induce little overdemand

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SLIDE 53

Economic Implications

1. Equilibrium prices form a lattice
2. VCG outcome monotone, in the core
3. Prices coordinate “typical” markets

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SLIDE 54

Recap

GS plays central role in the following:

1. Equilibrium prices exist and form a lattice
2. VCG outcome monotone, in the core
A GS market is characterized by a submodular coalitional value function 𝑥
Buyers’ utilities in VCG are their marginal contribution to 𝑥
3. Prices coordinate “typical” markets
For GS, prices coordinate iff max-welfare allocation is unique
Perturbed GS markets have a unique max-welfare allocation

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SLIDE 55

Necessity of GS Algorithmic Properties

Part I: Algorithmic properties of GS

Frontier of tractability for DEMAND and WELFARE-MAX

Part II: Economic implications of GS

Including existence of equil. prices

[RoughgardenT’15]: A direct connection between market equilibrium (non)existence and computational complexity of DEMAND, WELFARE-MAX

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Is there a direct connection?

SLIDE 56

Market Equilibrium & Related Problems

Recall: 𝒯, 𝑞 is a Walrasian market equilibrium if:

∀𝑗 ∶ 𝑇𝑗 is in 𝑗’s demand given 𝑞;
the market clears

Related computational problems: 𝒲 = class of valuations DEMAND: On input 𝑤 ∈ 𝒲 and 𝑞, output a bundle 𝑇 in demand given 𝑞 WELFARE-MAX: On input 𝑤1, … , 𝑤𝑛 ∈ 𝒲, output a max-welfare allocation 𝒯 REVENUE-MAX: On input 𝑞, output a max-revenue allocation 𝒯

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∀𝑗 ∶ 𝑇𝑗 solves DEMAND(𝑤𝑗, 𝑞); 𝒯 solves REVENUE-MAX(𝑞)

SLIDE 57

From Complexity to Equil. Nonexistence

𝒲 = class of valuations Theorem: [RoughgardenT’15]

A necessary condition for guaranteed existence of Walrasian equil. for 𝒲:

DEMAND is at least as computationally hard as WELFARE-MAX for 𝒲

→ If under P ≠ NP WELFARE-MAX is harder than DEMAND, equil. existence

not guaranteed for 𝒲

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SLIDE 58

Example

𝒲 = capped additive valuations DEMAND = KNAPSACK → pseudo-polynomial time algo. WELFARE-MAX = BIN-PACKING → strongly NP-hard If P ≠ NP then WELFARE-MAX is harder than DEMAND Conclusion: equil. existence not guaranteed for capped additive

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SLIDE 59

Complexity Approach: Some Pros & Cons

Con: Need P ≠ NP (or similar) assumption Pros: Alternative to “maximal domain” results

Case in point: Equil. existence not guaranteed for 𝒲 ∶ unit-demand ⊆ 𝒲 unless

𝒲 = GS [GulStacchetti’99]

Misses many 𝒲s that do not contain unit-demand

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Gross substitutes and complements [Sun-Yang’06, Teytelboym’13], 𝑙-gross substitutes [Ben-Zwi’13], superadditive [Parkes-Ungar’00, Sun-Yang’14], tree, graphical or feature-based valuations [Candogan’14, Candogan’15, Candogan- Pekec’14], …

SLIDE 60

Complexity Approach: Some Pros & Cons

Con: Need P ≠ NP (or similar) assumption Pros: Alternative to “maximal domain” results

Case in point: Equil. existence not guaranteed for 𝒲 ∶ unit-demand ⊆ 𝒲 unless

𝒲 = GS [GulStacchetti’99]

Misses many 𝒲s that do not contain unit-demand

The complexity approach generalizes to show nonguaranteed existence of relaxed equilibrium notions in typical markets

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Open direction: Apply the complexity approach to other economic properties of GS

SLIDE 61

2. Pushing the Boundaries
f GS
ROBUSTNESS OF THE ALGORITHMIC PROPERTIES
EXTENDING THE ECONOMIC PROPERTIES

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SLIDE 62

Motivation: Incentive Auction Mystery

“Few FCC policies have generated more attention than the Incentive Auction. ‘Groundbreaking,’ ‘revolutionary,’ and ‘first-in-the-world’ are just a few common descriptions of this innovative approach to making efficient, market-driven use of our spectrum resources.”

$20 billion auction
Freed up 84 MHz of spectrum
2018 Franz Edelman Award

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SLIDE 63

Incentive Auction Model

TV broadcasters with values 𝑤1, … , 𝑤𝑛 for staying on the air

Auction outcome = on-air broadcaster set 𝐵
𝐵 repacked into a reduced band of spectrum

Feasibility constraint:

ℱ ⊆ 2𝑁 = sets of broadcasters that can be feasibly repacked
Outcome is feasible if 𝐵 ∈ ℱ
ℱ downward-closed

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SLIDE 64

Incentive Auction Model

Broadcasters going off the air: 𝐵 = on-air broadcasters : Goal: Minimize total value that goes off the air = maximize 𝐵’s total value, subject to feasibility of repacking max

𝐵∈ℱ σ𝑗∈𝐵 𝑤𝑗

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𝑤1 𝑤4 𝑤3 𝑤2 𝑤5

packing constraint (e.g. knapsack)

SLIDE 65

The Mystery

Fact 1: max

𝐵∈ℱ σ𝑗∈𝐵 𝑤𝑗 greedily solvable iff ℱ defines a matroid over the

broadcasters Fact 2: In the Incentive Auction ℱ is not a matroid Fact 3: In simulations Greedy achieves > 95% of OPT on average

Over values sampled according to FCC predictions
[Newman, Leyton-Brown, Milgrom & Segal 2017]

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Equivalently, if 𝑤 is GS where 𝑤 𝐵 = max

𝐵⊇𝐵′∈ℱ σ𝑗∈𝐵′ 𝑤𝑗

SLIDE 66

The Mystery

Fact 1: max

𝐵∈ℱ σ𝑗∈𝐵 𝑤𝑗 greedily solvable iff ℱ defines a matroid over the

broadcasters Fact 2: In the Incentive Auction ℱ is not a matroid Fact 3: In simulations Greedy achieves > 95% of OPT on average

Over values sampled according to FCC predictions
[Newman, Leyton-Brown, Milgrom & Segal 2017]

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Equivalently, if 𝑤 is GS where 𝑤 𝐵 = max

𝐵⊇𝐵′∈ℱ σ𝑗∈𝐵′ 𝑤𝑗

Is ℱ “95% a matroid”? Is 𝑤 “95% GS”?

SLIDE 67

Research Agenda

In theory: Only GS markets guaranteed to work Folklore belief:

Many markets work well in practice since they’re “approximately GS”
I.e. good properties are robust

Agenda: We need theory predicting when markets actually work well

Starting with good models of “approximately GS”
Cf. “beyond worst case” agenda (replace markets with algorithms…)

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Good algorithmic, economic properties

SLIDE 68

Plan

2 recent approaches to “approximate GS”

1. Start from good performance of greedy
2. Start from approximating a very basic GS subclass: linear valuations

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SLIDE 69

𝐵′ 𝐵′′ 𝐵

Approach 1: Matroids

ℱ defines a matroid over 𝑁 if:

1. Rank quotient of ℱ is 1

min

𝐵⊆𝑁

min

𝐵′,𝐵′′⊆𝐵 𝑛𝑏𝑦𝑗𝑛𝑏𝑚 𝑗𝑜 ℱ

|𝐵′| |𝐵′′| = 1

2. [Equivalently] The exchange property holds:
For every 2 feasible sets 𝐵′, 𝐵′′, if 𝐵′ < |𝐵′′| then there’s an element we

can add from 𝐵′′ to 𝐵′ while maintaining feasibility

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𝐵′ 𝐵′′ 𝐵 𝑘

SLIDE 70

Approximate Matroids

ℱ defines a 𝜍-matroid over 𝑁 for ANY 𝜍 ≤ 1 if:

1. Rank quotient of ℱ is 𝜍

min

𝐵⊆𝑁

min

𝐵′,𝐵′′⊆𝐵 𝑛𝑏𝑦𝑗𝑛𝑏𝑚 𝑗𝑜 ℱ

|𝐵′| |𝐵′′| = 𝜍

2. [Equivalently] The 𝜍-exchange property holds:
For every 2 feasible sets 𝐵′, 𝐵′′, if 𝐵′ < 𝜍|𝐵′′| then there’s an element we

can add from 𝐵′′ to 𝐵′ while maintaining feasibility

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𝐵′ 𝐵′′ 𝐵 𝑘

SLIDE 71

Approximate Matroids

Theorem [Korte-Hausmann’78]: max

𝐵∈ℱ σ𝑗∈𝐵 𝑤𝑗 greedily 𝜍-approximable for any values 𝑤1, … , 𝑤𝑛 iff ℱ

defines a 𝜍-matroid over 𝑁 Note: Recent alternative notion of approx. matroids [Milgrom’17]

ℱ is 𝜍-close to a matroid ℳ if feasible sets in ℱ 𝜍-covered by sets in ℳ
Greedily optimizing wrt ℳ gives 𝜍-approximation wrt ℱ

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SLIDE 72

Open Questions

1. Does GS theory (approx.) extend to approx. matroid valuations?
Rank functions 𝑤 𝐵 = max

𝐵⊇𝐵′∈ℱ σ𝑗∈𝐵′ 𝑤𝑗, and their closure under mergers etc.

2. Alternative approximation notions
E.g., which notion ensures that greedy approximately minimizes the total

value going off the air

3. Empirical study
Is ℱ in the Incentive Auction an approx. matroid?

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SLIDE 73

Approach 2:

Study natural approximations of linear valuations 𝑤 𝑇 = 𝑤 ∅ + σ𝑘∈𝑇 𝑤(𝑘) for all 𝑇 Why linear?

Fundamental but still many open questions
Equivalent to modular

𝑤 𝑇 + 𝑤(𝑈) = 𝑤 𝑇 ∪ 𝑈 + 𝑤(𝑇 ∩ 𝑈) for all 𝑇, 𝑈

Additive valuations (𝑤 ∅ = 0) “too easy”

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SLIDE 74

Natural Approximations of Linear

Pointwise approximation of linear 𝑤′:

Multiplicatively: 𝑤′ 𝑇 ≤ 𝑤 𝑇 ≤ 1 + 𝜗 𝑤′(𝑇) for every 𝑇
Additively: |𝑤′ 𝑇 − 𝑤(𝑇)| ≤ 𝜗 for every 𝑇

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𝑤′

𝜗

𝑤 |𝑇| 𝑤(𝑇)

SLIDE 75

Natural Approximations of Linear

Pointwise approximation of linear 𝑤′:

Multiplicatively: 𝑤′ 𝑇 ≤ 𝑤 𝑇 ≤ 1 + 𝜗 𝑤′(𝑇) for every 𝑇
Additively: |𝑤′ 𝑇 − 𝑤(𝑇)| ≤ 𝜗 for every 𝑇

Approximate modularity: |𝑤 𝑇 + 𝑤 𝑈 − 𝑤 𝑇 ∪ 𝑈 + 𝑤 𝑇 ∩ 𝑈 | ≤ 𝜗

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SLIDE 76

What’s Known

𝑤 = pointwise (multiplicative) (1 + 𝜗)-approximation of linear 𝑤′ Theorem: [Roughgarden-T.-Vondrak’17]

Without querying 𝑤(𝑇) exponentially many times, there is no const.-factor

approximation of max. welfare

Unless 𝑤 is also (1 + 𝛽)-approximately submodular

Can get a (1 − 3𝜗)/(1 + 𝛽)-approximation
A la valuation hierarchies like ℳ𝒬ℋ [FFIILS’15]

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SLIDE 77

General Subadditive Subadditive Submodular

What’s Known

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GS GS Positive result: can approximate welfare Negative result

SLIDE 78

What’s Known

What about approximate modularity? |𝑤 𝑇 + 𝑤 𝑈 − 𝑤 𝑇 ∪ 𝑈 + 𝑤 𝑇 ∩ 𝑈 | ≤ 𝜗 Theorem: [Feige-Feldman-T.’17]

If 𝑤 is 𝜗-approximately modular then 𝑤 is a pointwise (additive) 13𝜗-

approximation of a linear 𝑤′

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SLIDE 79

Summary

Incentive Auction mystery: Greedy works surprisingly well Approaches:

1. Approx. matroids – needs more research
2. Approx. linear valuations – algorithmic properties not robust to

natural approximation notions

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SLIDE 80

Alternative Approaches

Other reasons why worst-case instances wouldn’t appear in practice Stable welfare-maximization [Chatziafratis et al.’17]

Small changes in the valuations do not change max-welfare allocation
Analog of “large margin” assumption in ML

Revealed preference approach [Echenique et al.’11]

Data: (prices, demanded bundle) pairs
For rationalizable data, there always exists a consistent tractable valuation

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SLIDE 81

2. Pushing the Boundaries
f GS
ROBUSTNESS OF THE ALGORITHMIC PROPERTIES
EXTENDING THE ECONOMIC PROPERTIES

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SLIDE 82

Matching with Contracts

[Roth-Sotomayor’90] “Two-Sided Matching” book

Separates models with and without money but shows similar results

[Hatfield-Milgrom’95] “Matching with Contracts”

Unifies the models (e.g., doctors and hospitals with combinatorial auctions)
Bilateral “contracts” specify the matching and its conditions (like wages)
Substitutability of the preferences plays an important role

[HKNOW’18] The most recent (?) in a long line of research

Unifying different substitutability concepts for an individual agent
Unifying stability and equilibrium concepts for markets

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SLIDE 83

A General Model: Trading Networks

A multi-sided setting with:

Nodes = agents (a buyer in some trades can be a seller in others)
Directed edges = trades
Valuations over set of trades, prices, quasi-linear utilities

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1 2 4 3 𝑞1,2 𝑞2,3 𝑞2,4 𝑞4,2 seller buyer 𝜌4 = 𝑤4 { 2,4 , (4,2)} − 𝑞2,4 + 𝑞4,2

SLIDE 84

Trading Networks

Main results: Under substitutability of the valuations,

Market equilibrium exists
Equilibria equivalent to stable outcomes (i.e., cannot be blocked by coalitions
f trades, where sufficient to consider paths/cycle)

[Candogan-Epitropou-Vohra’16] show equivalence to network flow

Equilibria correspond to optimal flow and its dual
Stability corresponds to no improving cycle
Algorithmic implications

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SLIDE 85

Demand Types [Baldwin-Klemperer’18]

New way of describing valuation classes

Possible ways in which demand can change in response to small price change

Yields new characterization theorem for market equil. existence Example:

2 items
Class of unit-demand valuations
Demand type:

±{ 1, −1 , 0,1 , (1,0)}

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SLIDE 86

Characterization Theorem

Theorem: [BK’18] A market equilibrium exists for any market with concave valuations

f demand type 𝒠 iff 𝒠 is unimodular
Unimodular = every set of 𝑜 vectors has a determinant 0, 1 or -1

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SLIDE 87

Main Take Away

Much more to study in the realm of GS:

1. Recent fundamental results (like unique max-welfare allocation →

price coordination)

2. Strong ties to algorithms (like trading networks vs. network flow,
equil. existence vs. computational complexity) and math (like

unimodularity thm)

3. Open crucial puzzles (like beyond worst case performance of

greedy)

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SLIDE 88

Some Related EC Talks

Tuesday@2:25PM Combinatorial auctions with endowment effect Moshe Babaioff, Shahar Dobzinski and Sigal Oren Tuesday@2:25PM Designing core-selecting payment rules: A computational search approach Benjamin Lubin, Benedikt Bunz and Sven Seuken Tuesday@2:25PM Fast core pricing for rich advertising auctions Jason Hartline, Nicole Immorlica, Mohammad Reza Khani, Brendan Lucier and Rad Niazadeh Thursday@2:10PM Trading networks with frictions Tamas Fleiner, Ravi Jagadeesan, Zsuzsanna Janko and Alexander Teytelboym Thursday@2:10PM Chain stability in trading networks John Hatfield, Scott Kominers, Alexandru Nichifor, Michael Ostrovsky and Alexander Westkamp Thursday@4PM On the construction of substitutes Eric Balkanski and Renato Paes Leme And more…

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