Resolving Combinatorial Markets via Posted Prices · For every approximation algorithm, the...
Transcript of 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
Michal Feldman – Tel Aviv University and Microsoft ResearchConference on Web & Internet Economics – December 2015
Spectrum AuctionsOnline Ad Auctions
Complex resource allocation
Scheduling Tasks in the Cloud
Michal Feldman – Tel Aviv University and Microsoft ResearchConference 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 ResearchConference on Web & Internet Economics – December 2015
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
Michal Feldman – Tel Aviv University and Microsoft ResearchConference 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
approxalgorithms
Michal Feldman – Tel Aviv University and Microsoft ResearchConference 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 ResearchConference on Web & Internet Economics – December 2015
For every approximation algorithm, the mechanism:1. (approximately) preserves social welfare of algorithm2. satisfies incentive compatibility
Approximation ALG
MechanismAllocation
PaymentsInput
Black-box reductions
Michal Feldman – Tel Aviv University and Microsoft ResearchConference on Web & Internet Economics – December 2015
Black-box reductions
Michal Feldman – Tel Aviv University and Microsoft ResearchConference 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 ResearchConference 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-freeoutcomes with full
information, general valuations
Scenario 1: dominant strategy incentive compatible (DSIC)
auctions with Bayesian submodular valuations
Michal Feldman – Tel Aviv University and Microsoft ResearchConference on Web & Internet Economics – December 2015
Scenario 1:DSIC mechanisms for submodular
valuations
Michal Feldman – Tel Aviv University and Microsoft ResearchConference on Web & Internet Economics – December 2015
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
Michal Feldman – Tel Aviv University and Microsoft ResearchConference 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 𝑺 ⊆ 𝑴Output: 𝒗(𝑺)
Input: item prices 𝒑𝟏, … , 𝒑𝒎Output: a demand set; i.e.,𝒂𝒓𝒈𝒎𝒂𝒙𝑺{𝒗 𝑺 − 𝒋∈𝑺𝒑𝒋}
Value queries Demand queries
Michal Feldman – Tel Aviv University and Microsoft ResearchConference 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/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
Michal Feldman – Tel Aviv University and Microsoft ResearchConference 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 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]
Michal Feldman – Tel Aviv University and Microsoft ResearchConference on Web & Internet Economics – December 2015
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]
Michal Feldman – Tel Aviv University and Microsoft ResearchConference 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
2OPT = 1/3.
Expected welfare:
Pr someone buys × 𝐸[𝑣 | 𝑣 > 1/3] =8
9⋅2
3=16
27
Michal Feldman – Tel Aviv University and Microsoft ResearchConference on Web & Internet Economics – December 2015
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
[F-Gravin-Lucier’15]
Michal Feldman – Tel Aviv University and Microsoft ResearchConference on Web & Internet Economics – December 2015
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.
[F-Gravin-Lucier’15]
Michal Feldman – Tel Aviv University and Microsoft ResearchConference on Web & Internet Economics – December 2015
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.
[F-Gravin-Lucier’15]
Corollary [DSIC “for free”]: A DSIC, O(1)-approx, 𝑷𝑶𝑳𝒀(𝒏,𝒎)mechanism for submodular valuations, in the Bayesian setting.
Michal Feldman – Tel Aviv University and Microsoft ResearchConference on Web & Internet Economics – December 2015
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
Michal Feldman – Tel Aviv University and Microsoft ResearchConference on Web & Internet Economics – December 2015
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]
Michal Feldman – Tel Aviv University and Microsoft ResearchConference 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):
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
Michal Feldman – Tel Aviv University and Microsoft ResearchConference on Web & Internet Economics – December 2015
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)
Michal Feldman – Tel Aviv University and Microsoft ResearchConference on Web & Internet Economics – December 2015
Applications of main result
Michal Feldman – Tel Aviv University and Microsoft ResearchConference 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 ResearchConference on Web & Internet Economics – December 2015
Scenario 2:Conflict free outcomes, full information
Michal Feldman – Tel Aviv University and Microsoft ResearchConference 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-freeoutcomes with full
information, general valuations
Scenario 1: dominant strategy incentive compatible (DSIC)
auctions with Bayesian submodular valuations
Michal Feldman – Tel Aviv University and Microsoft ResearchConference on Web & Internet Economics – December 2015
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)
Michal Feldman – Tel Aviv University and Microsoft ResearchConference 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 ResearchConference 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 ResearchConference on Web & Internet Economics – December 2015
GS submodular subadditive general
Motivating question
Is there a way to extend the theory of Walrasianequilibrium to combinatorial markets with generalbuyer valuations?
Michal Feldman – Tel Aviv University and Microsoft ResearchConference on Web & Internet Economics – December 2015
Motivating question
Is there a way to extend the theory of Walrasianequilibrium to combinatorial markets with generalbuyer valuations?
Answer: Yes! Through bundles.
Michal Feldman – Tel Aviv University and Microsoft ResearchConference on Web & Internet Economics – December 2015
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
Michal Feldman – Tel Aviv University and Microsoft ResearchConference on Web & Internet Economics – December 2015
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?
Michal Feldman – Tel Aviv University and Microsoft ResearchConference on Web & Internet Economics – December 2015
Theorem (existential)
Every valuation profile admits a conflict free outcome that preserves at least half of the optimal social welfare
[F-Gravin-Lucier’13]
Michal Feldman – Tel Aviv University and Microsoft ResearchConference on Web & Internet Economics – December 2015
[F-Gravin-Lucier’13]
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]
Theorem (computational)
Michal Feldman – Tel Aviv University and Microsoft ResearchConference on Web & Internet Economics – December 2015
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
Michal Feldman – Tel Aviv University and Microsoft ResearchConference on Web & Internet Economics – December 2015
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)
Michal Feldman – Tel Aviv University and Microsoft ResearchConference 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 𝑌𝑖 at half its
contribution to the social welfare (𝒗𝒊 𝒀𝒊
𝟐), then the final
allocation 𝑋 satisfies 𝑆𝑊 𝑋 ≥1
2𝑆𝑊(𝑌)
Michal Feldman – Tel Aviv University and Microsoft ResearchConference on Web & Internet Economics – December 2015
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)
Michal Feldman – Tel Aviv University and Microsoft ResearchConference on Web & Internet Economics – December 2015
Summary
• We presented two resource allocation scenarios
Scenario 2: conflict-freeoutcome 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 ResearchConference on Web & Internet Economics – December 2015
Thank you