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

Resolving Combinatorial Markets via Posted Prices Michal Feldman Tel Aviv University and Microsoft Research Conference on Web & Internet Economics – December 2015 Michal Feldman – Tel Aviv University and Microsoft Research

Complex resource allocation Online Ad Auctions Spectrum Auctions Scheduling Tasks in the Cloud Conference on Web & Internet Economics – December 2015 Michal Feldman – Tel Aviv University and Microsoft Research

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

Model: combinatorial markets/auctions A single seller, selling 𝑛 indivisible goods 𝑤 1 𝑜 buyers, each with valuation function 𝑤 𝑗 ∶ 2 [𝑛] → 𝑆 + 𝑤 2 An allocation is a partition of the goods 𝑦 = 𝑦 1 , … , 𝑦 𝑜 𝑦 𝑗 : bundle allocated to buyer 𝑗 𝑤 3 Goal: maximize social welfare 𝑇𝑋 = 𝑤 𝑗 (𝑦 𝑗 ) 𝑗∈[𝑜] Conference on Web & Internet Economics – December 2015 Michal Feldman – Tel Aviv University and Microsoft Research

Algorithmic Mechanism Design 1. Economic efficiency: max social welfare appro pprox alg lgor orith thms 2. Computational efficiency: poly runtime 3. Incentive compatibility: truth-telling is an equilibrium Conference on Web & Internet Economics – December 2015 Michal Feldman – Tel Aviv University and Microsoft Research

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

Black-box reductions Approximation ALG Allocation Input Mechanism Payments For every approximation algorithm, the mechanism: 1. (approximately) preserves social welfare of algorithm 2. satisfies incentive compatibility Conference on Web & Internet Economics – December 2015 Michal Feldman – Tel Aviv University and Microsoft Research

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

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

Beyond incentive compatibility 1. Economic efficiency: max social welfare 2. Computational efficiency: poly runtime 3. Additional requirements: incentive compatibility / conflict-freeness / … Scenario 1: dominant Scenario 2: conflict-free strategy incentive outcomes with full compatible (DSIC) information, general auctions with Bayesian valuations submodular valuations Conference on Web & Internet Economics – December 2015 Michal Feldman – Tel Aviv University and Microsoft Research

Scenario 1: DSIC mechanisms for submodular valuations Conference on Web & Internet Economics – December 2015 Michal Feldman – Tel Aviv University and Microsoft Research

Submodular valuations Decreasing marginal valuations: adding 𝑘 to T is more significant than adding j to S 𝑤 𝑇 ∪ 𝑘 − 𝑤 𝑇 ≤ 𝑤 𝑈 ∪ 𝑘 − 𝑤 𝑈 for 𝑈 ⊆ 𝑇 marginal value of 𝑘 marginal value of 𝑘 given 𝑇 given 𝑈 𝒌𝒌 S 𝑻 T 𝑼 Conference on Web & Internet Economics – December 2015 Michal Feldman – Tel Aviv University and Microsoft Research

Computational models • A submodular valuation function is an exponential object • We assume oracle access of two types Value queries Demand queries Input: item prices 𝒒 𝟐 , … , 𝒒 𝒏 Input: a set 𝑻 ⊆ 𝑵 Output: a demand set; i.e., Output: 𝒘(𝑻) 𝒃𝒔𝒉𝒏𝒃𝒚 𝑻 {𝒘 𝑻 − 𝒌∈𝑻 𝒒 𝒌 } Conference on Web & Internet Economics – December 2015 Michal Feldman – Tel Aviv University and Microsoft Research

Known results (submodular valuations) Algorithmic DSIC mechanism • NP-hard to solve optimally • Sub-polynomial approximation requires exponentially many • (1 − 1/𝑓) approximation value queries [Dobzinski ’ 11, with value queries Dughmi-Vondrak ’ 11] [Vondrak ’ 08, Feige ’ 09, • poly-time DSIC mechanism Dobzinski ’ 07] with 𝑃(log 𝑛 log log 𝑛) approximation under demand queries [Dobzinski ’ 07] Conference on Web & Internet Economics – December 2015 Michal Feldman – Tel Aviv University and Microsoft Research

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 𝑤 𝑗 is drawn independently from a known distribution 𝐺 𝑗 over submodular valuations on [0,1]] ) [F-Gravin-Lucier ’ 15] 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) Conference on Web & Internet Economics – December 2015 Michal Feldman – Tel Aviv University and Microsoft Research

Posted Price Mechanisms 1. Designer chooses item prices 𝑞 = (𝑞 1 , … , 𝑞 𝑛 ) 2. For each bidder in an arbitrary order 𝜌 : – Bidder 𝒋 ’ s valuation is realized: 𝒘 𝒋 ∼ 𝑮 𝒋 – 𝒋 chooses a favorite bundle from remaining items (i.e., a set 𝐓 maximizing 𝒗 𝒋 (𝑻, 𝒒) = 𝒘 𝒋 (𝑻) − 𝒌∈𝑻 𝒒 𝒌 ) 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] Conference on Web & Internet Economics – December 2015 Michal Feldman – Tel Aviv University and Microsoft Research

Posted Price Mechanisms Example: One item, two bidders, values uniform on [0,1] . Expected optimal social welfare is 2/3 . 1 2 OPT = 1/3 . Post a price of Expected welfare: 8 9 ⋅ 2 = 16 Pr someone buys × 𝐹[𝑤 | 𝑤 > 1/3] = 3 27 Conference on Web & Internet Economics – December 2015 Michal Feldman – Tel Aviv University and Microsoft Research

Theorem (existential) For distributions over submodular* valuations, there always exists a price vector such that the expected SW 1 of the posted price mechanism is ≥ 2 𝐹[ Optimal SW ] . [F-Gravin-Lucier ’ 15] ⇒ A multi-item extension of prophet inequality * Our results extend to XOS valuations Conference on Web & Internet Economics – December 2015 Michal Feldman – Tel Aviv University and Microsoft Research

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 1 that the expected SW is ≥ 2 𝐹[ SW of 𝐵] − 𝜗 . [F-Gravin-Lucier ’ 15] Conference on Web & Internet Economics – December 2015 Michal Feldman – Tel Aviv University and Microsoft Research

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 1 that the expected SW is ≥ 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. Conference on Web & Internet Economics – December 2015 Michal Feldman – Tel Aviv University and Microsoft Research

Unit-demand bidders Choosing prices (unit-demand): • 𝑗 𝑘 : bidder allocated item 𝑘 in the optimal allocation • 𝑥 𝑘 : value of bidder 𝑗 𝑘 for item 𝑘 • Choose prices 𝑞 𝑘 = 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 Conference on Web & Internet Economics – December 2015 Michal Feldman – Tel Aviv University and Microsoft Research