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The Weighted Proportional Resource Allocation. Milan Vojnović Microsoft Research Joint work with Thành Nguyen. University of Cambridge, Oct 18, 2010. Resource Allocation Problem. provider. users. Resource with general constraints Ex. network service, data centre, sponsored search
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The Weighted Proportional Resource Allocation Milan Vojnović Microsoft Research Joint work with Thành Nguyen University of Cambridge, Oct 18, 2010
Resource Allocation Problem provider users • 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) Resource
Resource Allocation Problem (cont’d) providers users • Multiple providers competing to provide service to users • Everyone is selfish 1 2 m
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”
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
The Weighted Resource Allocation • Weighted Allocation Auction: • Provider announces discrimination weights • Each user i submits a bid wi Payment = wi Allocation: • Discrimination weights so that allocation is feasible
The Weighted Resource Allocation (cont’d) • 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 a bid wi Payment = Ciwi Allocation: • C = resource capacity
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 Ex. n = 2 P
Ex 1: Network Service provider users
Ex 1: Network Service (cont’d) provider users
Ex 2: Compute Instance Allocation • Multi-machine multi-job scheduling • xi = 1 / (finish time for job i) • si,m = processing speed for job i at machine m • di,m = workload for job i at machine m task jobs
Ex 3. Sponsored Search • Generalized Second Price Auction • Discrimination weights = click-through-rates • Assumes click-through-rates independent of which ads appear together
Ex 3: Sponsored Search (cont’d) • xi = click-through-rate for slot i • Say $1 per click, so Ui(x) = x • GSP revenue: • Max weighted prop. revenue: (0,14) (4,5) (5,4) (6,0) (0,0)
Ex. 3: Sponsored Search (cont’d) • Revenue of weighted allocation auction
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
User’s Objective • Price-taking: given price pi, user i solves: • Price-anticipating: given Ci and , user i solves:
Provider’s Objective • Choose discrimination weights to maximize own revenue
Provider’s Objective (cont’d) • 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
Social Optimum • Social optimum allocation is a solution to
Equilibrium: Price-Taking Users • Revenue • Provider chooses discrimination weights where maximizes over • Equilibrium bids • Same revenue as under third-degree price discrimination
Equilibrium: Price-Anticipating Users • Revenue R given by: • Provider chooses discrimination weights where maximizes over • Equilibrium bids
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
Related Work (cont’d) • 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)
Related Work (cont’d) • 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)
Related Work (cont’d) • Theorem (Hajek & Yang, 2004) For price-anticipating users with concave, non-negative utility functions and scalar bids, worst-case efficiency is 0.
Related Work (cont’d) • Worst-case: serial network of unit capacity links
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
Revenue • Theorem For price-anticipating users, if for every user i, is a concave function, thenwhere R-k is the revenue under third-degree price discrimination with a worst-case set of k users excluded, i.e.In particular:
Proof Key Idea • Sufficient condition: for every there exists and
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)
Worst-Case • Utilities: • Nash eq. allocation:
Proof Key Ideas • Utilities: Q P
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
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
Oligopoly: Multiple Competing Providers 1 2 m providers users
Oligopoly (cont’d) • User i problem: choose bids that solve • Provider k problem: choose that maximize the revenue Rk over Pk where
d-Utility Functions • Def. U(x) a d-utility function: • Non-negative, non-decreasing, concave • U’(x)x concave over [0,x0]; U’(x)x maximum at x0 • For every :
Examples of d-Utility Functions “a-fair”
Social Welfare • Theorem For price-anticipating users with d-utility functions and oligopoly of competing providers: (Nash eq. utility) (socially OPT utility) • 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 d = 1, 2, worst-case efficiency at least 31, 24%
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
To Probe Further • The Weighted Proportional Allocation Mechanism, Microsoft Research Technical Report, MSR-TR-2010-145