resolving combinatorial markets via posted prices · for every approximation algorithm, the...

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

Resolving Combinatorial Markets via

Posted Prices

Michal Feldman

Tel Aviv University and Microsoft Research


Spectrum AuctionsOnline Ad Auctions

Complex resource allocation

Scheduling Tasks in the Cloud


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


Model: combinatorial markets/auctions

A single seller, selling 𝑚 indivisible goods

𝑛 buyers, each with valuation function 𝑣𝑖 ∶ 2

[𝑚] → 𝑅+

An allocation is a partition of the goods 𝑥 = 𝑥1, … , 𝑥𝑛𝑥𝑖 : bundle allocated to buyer 𝑖

Goal: maximize social welfare

𝑆𝑊 =

𝑖∈[𝑛]

𝑣𝑖(𝑥𝑖)

𝑣1

𝑣2

𝑣3


Algorithmic Mechanism Design

1. Economic efficiency: max social welfare

2. Computational efficiency: poly runtime

3. Incentive compatibility: truth-telling is an equilibrium

approxalgorithms


Algorithmic Mechanism Design



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


For every approximation algorithm, the mechanism:1. (approximately) preserves social welfare of algorithm2. satisfies incentive compatibility

Approximation ALG

MechanismAllocation

PaymentsInput

Black-box reductions


Black-box reductions


Beyond incentive compatibility



3. Additional requirements: incentive compatibility / conflict-freeness / …

Extend the theory of algorithmic mechanism design to additional desiderata






Scenario 2: conflict-freeoutcomes with full

information, general valuations

Scenario 1: dominant strategy incentive compatible (DSIC)

auctions with Bayesian submodular valuations


Scenario 1:DSIC mechanisms for submodular

valuations


Submodular valuations

𝑣 𝑆 ∪ 𝑗 − 𝑣 𝑆 ≤ 𝑣 𝑇 ∪ 𝑗 − 𝑣 𝑇 for 𝑇 ⊆ 𝑆

Decreasing marginal valuations:adding 𝑗 to T is more significant than adding j to S

𝑻

𝒋𝒋𝑺S

T

marginal value of 𝑗given 𝑆

marginal value of 𝑗given 𝑇


Computational models

• A submodular valuation function is an exponential object

• We assume oracle access of two types

Input: a set 𝑺 ⊆ 𝑴Output: 𝒗(𝑺)

Input: item prices 𝒑𝟏, … , 𝒑𝒎Output: a demand set; i.e.,𝒂𝒓𝒈𝒎𝒂𝒙𝑺{𝒗 𝑺 − 𝒋∈𝑺𝒑𝒋}

Value queries Demand queries


Known results (submodular valuations)

• Sub-polynomial approximation requires exponentially many value queries [Dobzinski’11,

Dughmi-Vondrak’11]

Algorithmic DSIC mechanism

• (1 − 1/𝑒) approximation with value queries [Vondrak’08, Feige’09, Dobzinski’07] • poly-time DSIC mechanism

with 𝑂(log𝑚 log log𝑚)approximation under demand queries [Dobzinski’07]

• NP-hard to solve optimally


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 𝑣𝑖 is drawn independently from a known distribution 𝐹𝑖 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]


Posted Price Mechanisms

1. Designer chooses item prices 𝑝 = (𝑝1, … , 𝑝𝑚)

2. For each bidder in an arbitrary order 𝜋:

– Bidder 𝒊’s valuation is realized: 𝒗𝒊 ∼ 𝑭𝒊– 𝒊 chooses a favorite bundle from remaining items

(i.e., a set 𝐒maximizing 𝒖𝒊(𝑺, 𝒑) = 𝒗𝒊(𝑺) − 𝒋∈𝑺𝒑𝒋)

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]


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

2OPT = 1/3.

Expected welfare:

Pr someone buys × 𝐸[𝑣 | 𝑣 > 1/3] =8

9⋅2

3=16

27


Theorem (existential)

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

of the posted price mechanism is ≥1

2𝐸[ Optimal SW ].

⇒ A multi-item extension of prophet inequality

* Our results extend to XOS valuations



Theorem (computational)

Given

• black-box access to a social welfare algorithm 𝐴, and

• sample access to the distributions 𝐹𝑖,

we can compute prices in time 𝑃𝑂𝐿𝑌(𝑛,𝑚, 1/𝜖) such

that the expected SW is ≥1

2𝐸[SW of 𝐴] − 𝜖.




Given

• black-box access to a social welfare algorithm 𝐴, and

• sample access to the distributions 𝐹𝑖,

we can compute prices in time 𝑃𝑂𝐿𝑌(𝑛,𝑚, 1/𝜖) such

that the expected SW is ≥1

2𝐸[SW of 𝐴] − 𝜖.


Corollary [DSIC “for free”]: A DSIC, O(1)-approx, 𝑷𝑶𝑳𝒀(𝒏,𝒎)mechanism for submodular valuations, in the Bayesian setting.


Unit-demand bidders

Choosing prices (unit-demand):

• 𝑖𝑗 : bidder allocated item 𝑗 in the optimal allocation

• 𝑤𝑗 : value of bidder 𝑖𝑗 for item 𝑗

• Choose prices 𝑝𝑗 =1

2𝐸 𝑤𝑗

Claim: These prices generate welfare ≥1

2OPT

To obtain the algorithmic result:

• Replace “optimal allocation” with approx. alloc. 𝐴(𝒗)

• Estimate the value of 𝐸 𝑤𝑗 by sampling


Proof of claim (unit-demand)

Let 𝑖𝑗 be winner of 𝑗 in OPT. Set price 𝑝𝑗 =1

2𝐸[𝑤𝑗]

1. 𝑅𝐸𝑉𝐸𝑁𝑈𝐸 = 𝑗1

2𝐸 𝑤𝑗 ⋅ Pr[𝑗 𝑖𝑠 𝑠𝑜𝑙𝑑]

2. Potential 𝑆𝑈𝑅𝑃𝐿𝑈𝑆 from 𝑗 ≥ 𝐸 𝑤𝑗 − 𝑝𝑗 =1

2𝐸[𝑤𝑗]

3. 𝑆𝑊 ≥ 𝑅𝐸𝑉𝐸𝑁𝑈𝐸 + 𝑗 𝑆𝑈𝑅𝑃𝐿𝑈𝑆𝑗 ⋅ Pr[𝑖𝑗 𝑠𝑒𝑒𝑠 𝑖𝑡𝑒𝑚 𝑗]

4. Pr 𝑖𝑗 𝑠𝑒𝑒𝑠 𝑖𝑡𝑒𝑚 𝑗 ≥ Pr[𝑗 𝑛𝑜𝑡 𝑠𝑜𝑙𝑑]

SW ≥ 𝑗1

2𝐸 𝑤𝑗 ⋅ Pr 𝑗 𝑖𝑠 𝑠𝑜𝑙𝑑 + 𝑗

1

2𝐸 𝑤𝑗 ⋅ Pr[𝑗 𝑛𝑜𝑡 𝑠𝑜𝑙𝑑]


Extension to submodular valuations

Lemma: every submodular function can be expressed as maximum over additive functions

Notation (full information):

𝑥∗ : optimal allocation

𝑣𝑖 : agent 𝑖’s additive function s.t. 𝑣𝑖 𝑥𝑖∗ = 𝑣𝑖(𝑥𝑖

∗)

Prices: 𝑝𝑗 =1

2 𝑣𝑖(𝑗), where 𝑗 ∈ 𝑥𝑖

∗

i.e., half its contribution to optimal SW


Proof idea

Let 𝑆𝑖 be items from 𝑥𝑖∗ sold prior to 𝑖’s arrival

𝑖 can buy 𝑥𝑖∗ ∖ 𝑆𝑖 (leftovers), so:

𝑢𝑖 𝑥𝑖, 𝑝 ≥ 𝑣𝑖 𝑥𝑖∗ ∖ 𝑆𝑖 −

1

2 𝑗∈𝑥𝑖

∗∖𝑆𝑖 𝑣𝑖(𝑗) 𝑖∈𝑁 𝑖∈𝑁 𝑖∈𝑁

𝑖∈𝑁pi ≥1

2 𝑖∈𝑁 𝑗∈𝑥𝑖

∗∩𝑆𝑖 𝑣𝑖(𝑗)

𝑖∈𝑁𝑢𝑖 𝑥𝑖, 𝑝 + 𝑖∈𝑁pi ≥1

2 𝑖∈𝑁 𝑗∈𝑥𝑖

∗ 𝑣𝑖(𝑗)

≥ 𝑗∈𝑥𝑖∗∖𝑆𝑖 𝑣𝑖(𝑗)

=1

2 𝑖∈𝑁𝑣𝑖(𝑥𝑖

∗)𝑆𝑊(𝑥)


Applications of main result


A note on simplicity

[Dobzinski’07]

Simple vs. optimal mechanisms

Obviously Strategy-proof [Li’15]

Posted price mechanisms


Scenario 2:Conflict free outcomes, full information






Scenario 2: conflict-freeoutcomes with full


Scenario 1: dominant strategy incentive compatible (DSIC)

auctions with Bayesian submodular valuations


Background: Walrasian equilibrium

$3

$2

$7

𝑣1

𝑣2

𝑣3

An outcome (𝑥, 𝑝) is a Walrasianequilibrium if:

1. Buyer 𝑖’s allocation, 𝑥𝑖, maximizes 𝑖’s utility (given prices)

2. All items are sold

An outcome is composed of: (1) allocation x = 𝑥1, … , 𝑥𝑛(2) item prices 𝑝 = (𝑝1, … , 𝑝𝑚)


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


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


GS submodular subadditive general

Motivating question

Is there a way to extend the theory of Walrasianequilibrium to combinatorial markets with generalbuyer valuations?


Motivating question

Is there a way to extend the theory of Walrasianequilibrium to combinatorial markets with generalbuyer valuations?

Answer: Yes! Through bundles.


Buyers: Items:

$3

$7

𝑣1

𝑣2

𝑣3

An outcome is conflict free if it maximizes the utility of every buyer

An outcome is composed of:(1) Partition of items into bundlesℬ = (𝐵1, … , 𝐵𝑚′)

(2) Allocation 𝑥 = (𝑥1, … , 𝑥𝑛) over (not necessarily all) bundles

(3) Prices 𝑝𝐵 of bundles

Social WelfareExistence ?

Conflict free outcomes

𝑣𝑖 𝑥𝑖 −

𝐵∈𝑥𝑖

𝑝𝐵 ≥ 𝑣𝑖 𝑆 −

𝐵∈𝑆

𝑝𝐵


OPT can be obtained in a conflict free outcome

4

3

$4

Welfare approximation

𝟑 + ϵ

3

itemsbuyers

1.5

OR

Unavoidable welfare loss: bundling can recover 3 + 𝜖(whereas 𝑂𝑃𝑇 = 4.5)

How much welfare can be preserved in a conflict-free outcome?


Theorem (existential)

Every valuation profile admits a conflict free outcome that preserves at least half of the optimal social welfare




For every valuation profile, given black-box access to a social welfare algorithm 𝑨, we can compute in poly-time* a conflict free outcome

(𝒙, 𝒑) such that 𝑺𝑾 𝒙 ≥𝟏

𝟐(𝑺𝑾 𝒐𝒇 𝑨)

[* assuming demand oracle]



The goal

Given an allocation 𝒀 (returned by approximation algorithm), construct a conflict free outcome (𝑿, 𝒑) that gives at least 𝟏/𝟐 of 𝒀’s social welfare

𝑣𝑌

𝑝𝑋


The construction

• Set initial bundles to be 𝑌1, … , 𝑌𝑛 , with initial “high” prices

• Run a tâtonnement process, in which prices increase and bundles merge (irrevocably)

𝑌5𝑌2 𝑌3𝑌1

𝑝1 𝑝2 𝑝5𝑝3𝑝2 + 𝑝3

𝑝1′

𝑝1′

𝑌4

𝑝4𝑝4 + 𝑝5

Theorem: for EVERY valuation profile, this process terminates,

outcome is conflict free, and 𝑆𝑊 𝑋 ≥1

2𝑆𝑊(𝑌)


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 𝑌𝑖 at half its

contribution to the social welfare (𝒗𝒊 𝒀𝒊

𝟐), then the final

allocation 𝑋 satisfies 𝑆𝑊 𝑋 ≥1

2𝑆𝑊(𝑌)


Proof (𝑆𝑊 𝑋 ≥ 12𝑆𝑊(𝑌))

Observation 1: if 𝑌𝑗 is ever allocated, it remains allocated

throughout

Observation 2: every 𝑌𝑗 that is unallocated is matched in 𝑌 to

one of the “allocated buyers”

𝑋𝑖

𝑖

𝑋1 𝑋2

allocated buyers𝑆𝑊 𝑋 =

𝑖

𝑣𝑖(𝑋𝑖) =

𝑖

𝑝𝑖 +

𝑖

𝑣𝑖 𝑋𝑖 − 𝑝𝑖

≥

𝑗:𝑌𝑗𝑎𝑙𝑙𝑜𝑐𝑎𝑡𝑒𝑑

1

2𝑣𝑗 𝑌𝑗 +

𝑗:𝑌𝑗𝑢𝑛𝑎𝑙𝑙𝑜𝑐𝑎𝑡𝑒𝑑

1

2𝑣𝑗 𝑌𝑗

𝑋𝑘𝑌𝑗, priced at ½ 𝑣𝑗(𝑌𝑗)

j n1

𝑌1 𝑌𝑗 𝑌𝑛𝑣𝑗(𝑌𝑗)

=1

2𝑆𝑊(𝑌)


Summary

• We presented two resource allocation scenarios

Scenario 2: conflict-freeoutcome with full


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


Thank you

resolving combinatorial markets via posted prices · for every approximation algorithm, the...

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