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The Weighted Proportional Resource Allocation Milan Vojnović Microsoft Research Joint work with Thành Nguyen Microsoft Research Asia, Beijing, April, 2011

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Resource Allocation Problem 2 providerusers Resource Resource with general constraints –Ex. network service, data centre, sponsored search Everyone is selfish: –Provider wants large revenue –Each user wants large surplus (utility – cost)

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Resource Allocation Problem (contd) 3 1 providers users 2 m Multiple providers competing to provide service to users Everyone is selfish

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Desiderata Simple auction mechanisms –Small amount of information signalled to users –Easy to explain to users Accommodate resources with general constraints High revenue and social welfare –Under everyone is selfish 4

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Outline The mechanism Applications Game-theory framework and related work Revenue and social welfare –Monopoly under linear utility functions –Generalization to multiple providers and more general utility functions Conclusion 5

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The Weighted Resource Allocation Weighted Allocation Auction: –Provider announces discrimination weights –Each user i submits bid w i Payment = w i Allocation: –Discrimination weights so that allocation is feasible 6

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The Weighted Resource Allocation (contd) Similar results hold also for weighted payment auction (Ma et al, 2010); an auction for specific resource constraints; results not presented in this slide deck Weighted Payment Auction: –Provider announces discrimination weights –Each user i submits bid w i Payment = C i w i Allocation: –C = resource capacity 7

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Resource Constraints An allocation is feasible if where P is a polyhedron, i.e. for some matrix A and vector Accommodates complex resources such as networks of links, data centres, sponsored search 8 P Ex. n = 2

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Ex 1: Network Service 9 providerusers

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Ex 1: Network Service (contd) 10 providerusers

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Ex 1: Network Service (contd) 11

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Ex 2. Sponsored Search 12 Generalized Second Price Auction Discrimination weights = click-through-rates Assumes click-through-rates independent of which ads appear together

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Ex 2: Sponsored Search (contd) 13 (0,0) (6,0) (0,14) (5,4) (4,5)

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Ex. 3: Sponsored Search (contd) 14 Revenue of weighted allocation auction

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Outline The mechanism Applications Game-theory framework and related work Revenue and social welfare –Monopoly under linear utility functions –Generalization to multiple providers and more general utility functions Conclusion 15

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Users Objective Price-taking: given price p i, user i solves: Price-anticipating: given C i and, user i solves: 16

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Providers Objective Choose discrimination weights to maximize own revenue 17

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Providers Objective (contd) Maximizing revenue standard objective of pricing schemes Ex. well-known third-degree price discrimination Assumes price taking users = price per unit resource for user i 18

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Social Optimum Social optimum allocation is a solution to 19

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Equilibrium: Price-Taking Users Revenue Provider chooses discrimination weights where maximizes over Equilibrium bids Same revenue as under third-degree price discrimination 20

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Equilibrium: Price-Anticipating Users Revenue R given by: Provider chooses discrimination weights where maximizes over Equilibrium bids 21

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Related Work Proportional resource sharing – ex. generalized proportional sharing (Parkeh & Gallager, 1993) Proportional allocation for network resources (Kelly, 1997) where for each infinitely-divisible resource of capacity C –No price discrimination –Charging market-clearing prices 22

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Related Work (contd) Theorem (Kelly, 1997) For price-taking users with concave, utility functions, efficiency is 100%. Assumes scalar bids = each user submits a single bid for a subset of resources (ex. single bid per path) 23

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Related Work (contd) Theorem (Johari & Tsitsiklis, 2004) For price-anticipating users with concave, non-negative utility functions and vector bids, efficiency is at least 75%: The worst-case achieved for linear utility functions. Vector bids = each user submits individual bid per each resource (ex. single bid for each link of a path) (Nash eq. utility) (socially OPT utility) 24

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Related Work (contd) Theorem (Hajek & Yang, 2004) For price-anticipating users with concave, non-negative utility functions and scalar bids, worst-case efficiency is 0. 25

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Related Work (contd) Worst-case: serial network of unit capacity links 26

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Outline The mechanism Applications Game-theory framework and related work Revenue and social welfare –Monopoly under linear utility functions –Generalization to multiple providers and more general utility functions Conclusion 27

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Revenue Theorem For price-anticipating users, if for every user i, is a concave function, then where R -k is the revenue under third-degree price discrimination with a worst-case set of k users excluded, i.e. In particular: 28

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Proof Key Idea Sufficient condition: for every there exists 29 and

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Social Welfare Theorem For price-anticipating users with linear utility functions, efficiency > 46.41%: This bound is tight. Worst-case: many users with one dominant user. (Nash eq. utility) (socially OPT utility) 30

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Worst-Case Utilities: Nash eq. allocation: 31

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Proof Key Ideas Utilities: 32 P Q

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Summary of Results Competitive revenue and social welfare under linear utility functions and monopoly of a single provider –Revenue at least k/(k+1) times the revenue under third-degree price discrimination with a set of k users excluded –Efficiency at least 46.41%; tight worst case In contrast to market-clearing where worst-case efficiency is 0 33

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Outline The mechanism Applications Game-theory framework and related work Revenue and social welfare –Monopoly under linear utility functions –Generalization to multiple providers and more general utility functions Conclusion 34

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Multiple Competing Providers 35 1 providersusers 2 m

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Multiple Competing Providers (contd) User i problem: choose bids that solve Provider k problem: choose that maximize the revenue R k over P k where 36

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-Utility Functions Def. U(x) a -utility function: –Non-negative, non-decreasing, concave –U(x)x concave over [0,x 0 ]; U(x)x maximum at x 0 –For every : 37

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Examples of -Utility Functions 38 -fair

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Social Welfare Theorem For price-anticipating users with -utility functions and multiple competing providers: (Nash eq. utility) (socially OPT utility) 39 The worst-case achieved for linear utility functions. The bound holds for any number of users n and any number of providers m. Ex. for = 1, 2, worst-case efficiency at least 31, 24%

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Proof Key Ideas 40

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Conclusion Established revenue and social welfare properties of weighted proportional resource allocation in competitive settings where everyone is selfish Identified cases with competitive revenue and social welfare The revenue is at least k/(k+1) times the revenue under third- degree price discrimination with a set of k users excluded Under linear utility functions, efficiency is at least 46.41%; tight worst case Efficiency lower bound generalized to multiple competing providers and a general class of utility functions 41

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To Probe Further The Weighted Proportional Allocation Mechanism –Conference paper, ACM Sigmetrics 2011 –Microsoft Research Technical Report, MSR-TR

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