Resolving Combinatorial Markets via Posted Prices Michal Feldman - - PowerPoint PPT Presentation

Michal Feldman – Tel Aviv University and Microsoft Research Conference on Web & Internet Economics – December 2015

Resolving Combinatorial Markets via Posted Prices

Michal Feldman

Tel Aviv University and Microsoft Research

Michal Feldman – Tel Aviv University and Microsoft Research Conference on Web & Internet Economics – December 2015

Spectrum Auctions Online Ad Auctions

Complex resource allocation

Scheduling Tasks in the Cloud

Michal Feldman – Tel Aviv University and Microsoft Research Conference on Web & Internet Economics – December 2015

Talk outline

Model: combinatorial markets / auctions Black-box reductions: from algorithms to mechanisms Applications

1. Scenario 1: DSIC mechanism for submodular buyers 2. Scenario 2: conflict-free outcomes for general buyers

Michal Feldman – Tel Aviv University and Microsoft Research Conference on Web & Internet Economics – December 2015

Model: combinatorial markets/auctions

A single seller, selling 𝑛 indivisible goods 𝑜 buyers, each with valuation function 𝑤𝑗 ∶ 2[𝑛] → 𝑆+ An allocation is a partition of the goods 𝑦 = 𝑦1, … , 𝑦𝑜 𝑦𝑗 : bundle allocated to buyer 𝑗 Goal: maximize social welfare

𝑇𝑋 =

𝑗∈[𝑜]

𝑤𝑗(𝑦𝑗)

𝑤1 𝑤2 𝑤3

Michal Feldman – Tel Aviv University and Microsoft Research Conference on Web & Internet Economics – December 2015

Algorithmic Mechanism Design

• 1. Economic efficiency: max social welfare
• 2. Computational efficiency: poly runtime
• 3. Incentive compatibility: truth-telling is an

equilibrium

appro pprox alg lgor

• rith

thms

Michal Feldman – Tel Aviv University and Microsoft Research Conference on Web & Internet Economics – December 2015

Algorithmic Mechanism Design

• 1. Economic efficiency: max social welfare
• 2. Computational efficiency: poly runtime
• 3. Incentive compatibility: truth-telling is an

equilibrium

Goal: we wish incentive compatibility to cause no (or small) additional welfare loss beyond loss already incurred due to computational constraints

Michal Feldman – Tel Aviv University and Microsoft Research Conference on Web & Internet Economics – December 2015

For every approximation algorithm, the mechanism:

• 1. (approximately) preserves social welfare of algorithm
• 2. satisfies incentive compatibility

Approximation ALG

Mechanism Allocation Payments Input

Black-box reductions

Michal Feldman – Tel Aviv University and Microsoft Research Conference on Web & Internet Economics – December 2015

Black-box reductions

Michal Feldman – Tel Aviv University and Microsoft Research Conference on Web & Internet Economics – December 2015

Beyond incentive compatibility

• 1. Economic efficiency: max social welfare
• 2. Computational efficiency: poly runtime
• 3. Additional requirements:

incentive compatibility / conflict-freeness / …

Extend the theory of algorithmic mechanism design to additional desiderata

Michal Feldman – Tel Aviv University and Microsoft Research Conference on Web & Internet Economics – December 2015

Beyond incentive compatibility

• 1. Economic efficiency: max social welfare
• 2. Computational efficiency: poly runtime
• 3. Additional requirements:

incentive compatibility / conflict-freeness / …

Scenario 2: conflict-free

• utcomes with full

information, general valuations Scenario 1: dominant strategy incentive compatible (DSIC) auctions with Bayesian submodular valuations

Michal Feldman – Tel Aviv University and Microsoft Research Conference on Web & Internet Economics – December 2015

Scenario 1:

DSIC mechanisms for submodular valuations

Michal Feldman – Tel Aviv University and Microsoft Research Conference on Web & Internet Economics – December 2015

Submodular valuations

𝑤 𝑇 ∪ 𝑘 − 𝑤 𝑇 ≤ 𝑤 𝑈 ∪ 𝑘 − 𝑤 𝑈 for 𝑈 ⊆ 𝑇 Decreasing marginal valuations: adding 𝑘 to T is more significant than adding j to S

𝑼

𝒌𝒌

𝑻

S T

marginal value of 𝑘 given 𝑇 marginal value of 𝑘 given 𝑈

Michal Feldman – Tel Aviv University and Microsoft Research Conference on Web & Internet Economics – December 2015

Computational models

• A submodular valuation function is an

exponential object

• We assume oracle access of two types

Input: a set 𝑻 ⊆ 𝑵 Output: 𝒘(𝑻) Input: item prices 𝒒𝟐, … , 𝒒𝒏 Output: a demand set; i.e., 𝒃𝒔𝒉𝒏𝒃𝒚𝑻{𝒘 𝑻 − 𝒌∈𝑻 𝒒𝒌}

Value queries Demand queries

Michal Feldman – Tel Aviv University and Microsoft Research Conference on Web & Internet Economics – December 2015

Known results (submodular valuations)

• Sub-polynomial approximation

requires exponentially many value queries [Dobzinski’11,

Dughmi-Vondrak’11]

Algorithmic DSIC mechanism

• (1 − 1/𝑓) approximation

with value queries

[Vondrak’08, Feige’09, Dobzinski’07]

• poly-time DSIC mechanism

with 𝑃(log 𝑛 log log 𝑛) approximation under demand queries [Dobzinski’07]

• NP-hard to solve optimally

Michal Feldman – Tel Aviv University and Microsoft Research Conference on Web & Internet Economics – December 2015

Major open problem

Is there a poly-time incentive compatible mechanism that achieves a constant-factor approximation for submodular valuations, under demand oracle? Theorem: YES for Bayesian settings (i.e., each 𝑤𝑗 is drawn independently from a known distribution 𝐺

𝑗 over

submodular valuations on [0,1]]) Moreover, our mechanism is:

• 1. simple (based on posted prices)
• 2. truly poly-time (independent of support size)
• 3. dominant strategy IC (stronger than Bayesain IC)

[F-Gravin-Lucier’15]

Michal Feldman – Tel Aviv University and Microsoft Research Conference on Web & Internet Economics – December 2015

Posted Price Mechanisms

• 1. Designer chooses item prices 𝑞 = (𝑞1, … , 𝑞𝑛)
• 2. For each bidder in an arbitrary order 𝜌:

– Bidder 𝒋’s valuation is realized: 𝒘𝒋 ∼ 𝑮𝒋 – 𝒋 chooses a favorite bundle from remaining items (i.e., a set 𝐓 maximizing 𝒗𝒋(𝑻, 𝒒) = 𝒘𝒋(𝑻) − 𝒌∈𝑻 𝒒𝒌) Remarks:

• Arrival order & tie-breaking can be arbitrary
• Prices are static (set once and for all)
• Mechanism is obviously strategy proof [Li’15]
• Sequential posted pricing [Chawla-Hartline-Kleinberg’07, Chawla-Malek-

Sivan’10, Chawla-Hartline-Malek-Sivan’10,Kleinberg-Weinberg’12]

Michal Feldman – Tel Aviv University and Microsoft Research Conference on Web & Internet Economics – December 2015

Posted Price Mechanisms

Example:

One item, two bidders, values uniform on [0,1]. Expected optimal social welfare is 2/3. Post a price of

1 2 OPT = 1/3.

Expected welfare:

Pr someone buys × 𝐹[𝑤 | 𝑤 > 1/3] = 8 9 ⋅ 2 3 = 16 27

Michal Feldman – Tel Aviv University and Microsoft Research Conference on Web & Internet Economics – December 2015

Theorem (existential)

For distributions over submodular* valuations, there always exists a price vector such that the expected SW

• f the posted price mechanism is ≥

1 2 𝐹[ Optimal SW ].

⇒ A multi-item extension of prophet inequality

* Our results extend to XOS valuations

[F-Gravin-Lucier’15]

Michal Feldman – Tel Aviv University and Microsoft Research Conference on Web & Internet Economics – December 2015

Theorem (computational)

Given

• black-box access to a social welfare algorithm 𝐵, and
• sample access to the distributions 𝐺

𝑗,

we can compute prices in time 𝑄𝑃𝑀𝑍(𝑜, 𝑛, 1/𝜗) such that the expected SW is ≥

1 2 𝐹[SW of 𝐵] − 𝜗.

[F-Gravin-Lucier’15]

Michal Feldman – Tel Aviv University and Microsoft Research Conference on Web & Internet Economics – December 2015

Theorem (computational)

Given

• black-box access to a social welfare algorithm 𝐵, and
• sample access to the distributions 𝐺

𝑗,

we can compute prices in time 𝑄𝑃𝑀𝑍(𝑜, 𝑛, 1/𝜗) such that the expected SW is ≥

1 2 𝐹[SW of 𝐵] − 𝜗.

[F-Gravin-Lucier’15] Corollary [DSIC “for free”]: A DSIC, O(1)-approx, 𝑸𝑷𝑴𝒁(𝒐, 𝒏) mechanism for submodular valuations, in the Bayesian setting.

Michal Feldman – Tel Aviv University and Microsoft Research Conference on Web & Internet Economics – December 2015

Unit-demand bidders

Choosing prices (unit-demand):

• 𝑗𝑘 : bidder allocated item 𝑘 in the optimal allocation
• 𝑥

𝑘 : value of bidder 𝑗𝑘 for item 𝑘

• Choose prices 𝑞𝑘 = 1

2 𝐹 𝑥 𝑘

Claim: These prices generate welfare ≥ 1

2 OPT

To obtain the algorithmic result:

• Replace “optimal allocation” with approx. alloc. 𝐵(𝒘)
• Estimate the value of 𝐹 𝑥

𝑘 by sampling

Michal Feldman – Tel Aviv University and Microsoft Research Conference on Web & Internet Economics – December 2015

Proof of claim (unit-demand)

Let 𝑗𝑘 be winner of 𝑘 in OPT. Set price 𝑞𝑘 =

1 2 𝐹[𝑥 𝑘]

• 1. 𝑆𝐹𝑊𝐹𝑂𝑉𝐹 = 𝑘

1 2 𝐹 𝑥 𝑘 ⋅ Pr[𝑘 𝑗𝑡 𝑡𝑝𝑚𝑒]

• 2. Potential 𝑇𝑉𝑆𝑄𝑀𝑉𝑇 from 𝑘 ≥ 𝐹 𝑥

𝑘 − 𝑞𝑘 = 1 2 𝐹[𝑥 𝑘]

• 3. 𝑇𝑋 ≥ 𝑆𝐹𝑊𝐹𝑂𝑉𝐹 + 𝑘 𝑇𝑉𝑆𝑄𝑀𝑉𝑇

𝑘 ⋅ Pr[𝑗𝑘 𝑡𝑓𝑓𝑡 𝑗𝑢𝑓𝑛 𝑘]

• 4. Pr 𝑗𝑘 𝑡𝑓𝑓𝑡 𝑗𝑢𝑓𝑛 𝑘 ≥ Pr[𝑘 𝑜𝑝𝑢 𝑡𝑝𝑚𝑒]

SW ≥ 𝑘

1 2 𝐹 𝑥 𝑘 ⋅ Pr 𝑘 𝑗𝑡 𝑡𝑝𝑚𝑒 + 𝑘 1 2 𝐹 𝑥 𝑘 ⋅ Pr[𝑘 𝑜𝑝𝑢 𝑡𝑝𝑚𝑒]

Michal Feldman – Tel Aviv University and Microsoft Research Conference on Web & Internet Economics – December 2015

Extension to submodular valuations

Lemma: every submodular function can be expressed as maximum over additive functions Notation (full information): 𝑦∗ : optimal allocation 𝑤𝑗 : agent 𝑗’s additive function s.t. 𝑤𝑗 𝑦𝑗

∗ =

𝑤𝑗(𝑦𝑗

∗)

Prices: 𝑞𝑘 =

1 2

𝑤𝑗(𝑘), where 𝑘 ∈ 𝑦𝑗

∗

i.e., half its contribution to optimal SW

Michal Feldman – Tel Aviv University and Microsoft Research Conference on Web & Internet Economics – December 2015

Proof idea

Let 𝑇𝑗 be items from 𝑦𝑗

∗ sold prior to 𝑗’s arrival

𝑗 can buy 𝑦𝑗

∗ ∖ 𝑇𝑗 (leftovers), so:

𝑣𝑗 𝑦𝑗, 𝑞 ≥ 𝑤𝑗 𝑦𝑗

∗ ∖ 𝑇𝑗 − 1 2

𝑘∈𝑦𝑗

∗∖𝑇𝑗

𝑤𝑗(𝑘) 𝑗∈𝑂 𝑗∈𝑂 𝑗∈𝑂 𝑗∈𝑂pi ≥ 1 2 𝑗∈𝑂 𝑘∈𝑦𝑗

∗∩𝑇𝑗

𝑤𝑗(𝑘) 𝑗∈𝑂𝑣𝑗 𝑦𝑗, 𝑞 + 𝑗∈𝑂pi ≥ 1 2 𝑗∈𝑂 𝑘∈𝑦𝑗

∗

𝑤𝑗(𝑘) ≥ 𝑘∈𝑦𝑗

∗∖𝑇𝑗

𝑤𝑗(𝑘) = 1 2 𝑗∈𝑂𝑤𝑗(𝑦𝑗

∗)

𝑇𝑋(𝑦)

Michal Feldman – Tel Aviv University and Microsoft Research Conference on Web & Internet Economics – December 2015

Applications of main result

Michal Feldman – Tel Aviv University and Microsoft Research Conference on Web & Internet Economics – December 2015

A note on simplicity

[Dobzinski’07]

Simple vs. optimal mechanisms Obviously Strategy-proof [Li’15] Posted price mechanisms

Michal Feldman – Tel Aviv University and Microsoft Research Conference on Web & Internet Economics – December 2015

Scenario 2:

Conflict free outcomes, full information

Michal Feldman – Tel Aviv University and Microsoft Research Conference on Web & Internet Economics – December 2015

Beyond incentive compatibility

• 1. Economic efficiency: max social welfare
• 2. Computational efficiency: poly runtime
• 3. Additional requirements:

incentive compatibility / conflict-freeness / …

Scenario 2: conflict-free

• utcomes with full

information, general valuations Scenario 1: dominant strategy incentive compatible (DSIC) auctions with Bayesian submodular valuations

Michal Feldman – Tel Aviv University and Microsoft Research Conference on Web & Internet Economics – December 2015

Background: Walrasian equilibrium

$3 $2 $7

𝑤1 𝑤2 𝑤3

An outcome (𝑦, 𝑞) is a Walrasian equilibrium if:

• 1. Buyer 𝑗’s allocation, 𝑦𝑗,

maximizes 𝑗’s utility (given prices)

• 2. All items are sold

An outcome is composed of: (1) allocation x = 𝑦1, … , 𝑦𝑜 (2) item prices 𝑞 = (𝑞1, … , 𝑞𝑛)

Michal Feldman – Tel Aviv University and Microsoft Research Conference on Web & Internet Economics – December 2015

Walrasian equilibrium (WE)

Bright side

• Simple: succinct item prices
• Conflict free: no buyer prefers

a different bundle

• Maximizes social welfare

(first welfare theorem) Dark side

• Existence is extremely

restricted [Kelso-Crawford’82, Gul-Stachetti’99]

4 3 WE doesn’t exist

Michal Feldman – Tel Aviv University and Microsoft Research Conference on Web & Internet Economics – December 2015

Walrasian equilibrium (WE)

Bright side

• Simple: succinct item prices
• Conflict free: no buyer prefers

a different bundle

• Maximizes social welfare

(first welfare theorem) Dark side

• Existence is extremely

restricted [Kelso-Crawford’82, Gul-Stachetti’99]

Gross substitutes

Michal Feldman – Tel Aviv University and Microsoft Research Conference on Web & Internet Economics – December 2015

GS submodular subadditive general

Motivating question

Is there a way to extend the theory of Walrasian equilibrium to combinatorial markets with general buyer valuations?

Michal Feldman – Tel Aviv University and Microsoft Research Conference on Web & Internet Economics – December 2015

Motivating question

Is there a way to extend the theory of Walrasian equilibrium to combinatorial markets with general buyer valuations?

Answer: Yes! Through bundles.

Michal Feldman – Tel Aviv University and Microsoft Research Conference on Web & Internet Economics – December 2015

Buyers: Items:

$3 $7

𝑤1 𝑤2 𝑤3 An outcome is conflict free if it maximizes the utility of every buyer An outcome is composed of: (1) Partition of items into bundles ℬ = (𝐶1, … , 𝐶𝑛′) (2) Allocation 𝑦 = (𝑦1, … , 𝑦𝑜) over (not necessarily all) bundles (3) Prices 𝑞𝐶 of bundles

Social Welfare Existence

?

Conflict free outcomes

𝑤𝑗 𝑦𝑗 −

𝐶∈𝑦𝑗

𝑞𝐶 ≥ 𝑤𝑗 𝑇 −

𝐶∈𝑇

𝑞𝐶

Michal Feldman – Tel Aviv University and Microsoft Research Conference on Web & Internet Economics – December 2015

OPT can be obtained in a conflict free outcome

4 3 $4

Welfare approximation

𝟒 + ϵ 3

items buyers

1.5

OR Unavoidable welfare loss: bundling can recover 3 + 𝜗 (whereas 𝑃𝑄𝑈 = 4.5)

How much welfare can be preserved in a conflict-free

• utcome?

Michal Feldman – Tel Aviv University and Microsoft Research Conference on Web & Internet Economics – December 2015

Theorem (existential)

Every valuation profile admits a conflict free

• utcome that preserves at least half of the
• ptimal social welfare

[F-Gravin-Lucier’13]

Michal Feldman – Tel Aviv University and Microsoft Research Conference on Web & Internet Economics – December 2015

[F-Gravin-Lucier’13]

For every valuation profile, given black-box access to a social welfare algorithm 𝑩, we can compute in poly-time* a conflict free outcome (𝒚, 𝒒) such that 𝑻𝑿 𝒚 ≥

𝟐 𝟑 (𝑻𝑿 𝒑𝒈 𝑩)

[* assuming demand oracle]

Theorem (computational)

Michal Feldman – Tel Aviv University and Microsoft Research Conference on Web & Internet Economics – December 2015

The goal

Given an allocation 𝒁 (returned by approximation algorithm), construct a conflict free outcome (𝒀, 𝒒) that gives at least 𝟐/𝟑 of 𝒁’s social welfare

𝑤 𝑍 𝑞 𝑌

Michal Feldman – Tel Aviv University and Microsoft Research Conference on Web & Internet Economics – December 2015

The construction

• Set initial bundles to be 𝑍

1, … , 𝑍 𝑜 , with initial “high” prices

• Run a tâtonnement process, in which prices increase and bundles

merge (irrevocably)

𝑍

5

𝑍

2

𝑍

3

𝑍

1

𝑞1 𝑞2 𝑞5 𝑞3 𝑞2 + 𝑞3 𝑞1

′

𝑞1

′ 𝑍

4

𝑞4 𝑞4 + 𝑞5 Theorem: for EVERY valuation profile, this process terminates,

• utcome is conflict free, and 𝑇𝑋 𝑌 ≥

1 2 𝑇𝑋(𝑍)

Michal Feldman – Tel Aviv University and Microsoft Research Conference on Web & Internet Economics – December 2015

Analysis

• Process terminates: prices only increase and bundles never

split (if we are careful, terminates in poly time).

• Upon termination, final allocation is conflict free (by

construction)

• Claim: if we (initially) price every bundle 𝑍

𝑗 at half its

contribution to the social welfare (

𝒘𝒋 𝒁𝒋 𝟑 ), then the final

allocation 𝑌 satisfies 𝑇𝑋 𝑌 ≥

1 2 𝑇𝑋(𝑍)

Michal Feldman – Tel Aviv University and Microsoft Research Conference on Web & Internet Economics – December 2015

Proof (𝑇𝑋 𝑌 ≥

1 2 𝑇𝑋(𝑍))

Observation 1: if 𝑍

𝑘 is ever allocated, it remains allocated

throughout Observation 2: every 𝑍

𝑘 that is unallocated is matched in 𝑍 to

• ne of the “allocated buyers”

𝑌𝑗

𝑗

𝑌1 𝑌2 allocated buyers

𝑇𝑋 𝑌 =

𝑗

𝑤𝑗(𝑌𝑗) =

𝑗

𝑞𝑗 +

𝑗

𝑤𝑗 𝑌𝑗 − 𝑞𝑗 ≥

𝑘:𝑍𝑘𝑏𝑚𝑚𝑝𝑑𝑏𝑢𝑓𝑒

1 2 𝑤𝑘 𝑍

𝑘 + 𝑘:𝑍𝑘𝑣𝑜𝑏𝑚𝑚𝑝𝑑𝑏𝑢𝑓𝑒

1 2 𝑤𝑘 𝑍

𝑘

𝑌𝑙

𝑍

𝑘, priced at ½ 𝑤𝑘(𝑍 𝑘) j n 1 𝑍

1

𝑍

𝑘

𝑍

𝑜

𝑤𝑘(𝑍

𝑘)

= 1 2 𝑇𝑋(𝑍)

Michal Feldman – Tel Aviv University and Microsoft Research Conference on Web & Internet Economics – December 2015

Summary

• We presented two resource allocation scenarios

Scenario 2: conflict-free

• utcome with full

information, general valuations Scenario 1: DSIC auctions with Bayesian submodular valuations

• We showed that in both cases a constant fraction of

the optimal welfare can be preserved

• Both results follow the black-box paradigm
• Posted price mechanisms is an interesting class of

mechanisms

Michal Feldman – Tel Aviv University and Microsoft Research Conference on Web & Internet Economics – December 2015

Thank you