The Weighted Proportional Allocation Mechanism Milan Vojnović Microsoft Research Joint work with Thành Nguyen Harvard University, Nov 3, 2009

Resource allocation problem 2 providerusers Resource Provider wants large revenue User wants large surplus (utility – cost) Resource with general constraints –Ex. network service, data centre, sponsored search

Resource allocation problem (cont’d) 3 1 providers users 2 m Oligopoly – multiple providers competing to provide service to users Each provider wants a large revenue

Desiderata Simple auction mechanism –Small amount of information signalled to users –Easy to explain / understand by users Accommodate resources with general constraints High revenue and social welfare –Under strategic providers and strategic users 4

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

The mechanism Provider announces discrimination weights Each user i submits a bid w i Payment by user i = w i Allocation to user i: Discrimination weights so that allocation is feasible 6

Resource constraints An allocation said to be feasible if where P is a polyhedron, i.e. for some matrix A and vector Accommodates complex resources such as network of links, data centres, sponsored search 7 P Ex. n = 2

Ex 1: Network service 8 providerusers

Ex 1: Network service (cont’d) 9 providerusers

Ex 1: Network service (cont’d) 10

Ex 2: data centre resource allocation x i = 1 / (finish time for job i) s i,m = processing speed for job i at machine m d i,m = workload for job i at machine m 11 jobs task Multi-job task scheduling

Ex 3. Sponsored search 12 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) 13 x i = click-through-rate at slot i Say $1 per click, so U i (x) = x GSP revenue: Max weighted prop. revenue: (0,0) (6,0) (0,14) (5,4) (4,5)

Ex. 3: Sponsored Search (cont’d) 14 Revenue of weighted proportional allocation

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

User’s objective Price-taking – given price p i, user i solves: Price-anticipating – given C i and, user i solves: 16

Provider’s objective Choose discrimination weights to maximize the revenue 17

Provider’s objective (cont’d) Maximizing revenue also objective of some pricing schemes Ex. well-known third-degree price discrimination Assumes price taking users = price per unit resource for user i 18

Social optimum Social optimum allocation is a solution to 19

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

Equilibrium: price-anticipating users Revenue R given by: Provider chooses discrimination weights where maximizes over Equilibrium bids 21

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

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) 23

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) 24

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. 25

Related work (cont’d) Worst-case: serial network of unit capacity links 26

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

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 set of k users excluded, i.e. In particular: 28

Example Unit-capacity resource: Symmetric users with utility function U(x) U(x) concave, and U’(x)x concave increasing on [0,1] 29 Ex. revenue under third-degree price discrimination

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

Worst-case Utilities: Nash eq. allocation: 31

Proof key ideas Utilities: 32 P

Summary of properties 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 Unlike to market-clearing where worst-case efficiency is 0 33

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

Oligopoly: multiple competing providers 35 1 providersusers 2 m

Oligopoly (cont’d) User i problem: choose bids that solve Provider k problem: choose that maximize the revenue R k over P k where 36

 -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

Examples of  -utility functions 38 “  -fair”

Social welfare Theorem For price-anticipating users with  -utility functions and oligopoly of competing providers: 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% (Nash eq. utility) (socially OPT utility) 39

Proof key ideas Bounding social welfare by an affine function separates to optimizations for individual providers For provider k consider linear utility functions where 40

Conclusion Proposed weighted proportional allocation mechanism –Simple; applies to general polyhedron constraints Offers competitive revenue and social welfare The revenue at least k/(k+1) times that under third-degree price discrimination with a set of k users excluded Under linear utility functions, efficiency at least 46.41%; tight worst case Efficiency lower bound generalized to an oligopoly of multiple competing providers and a general class of utility functions 41

To Probe Further The Weighted Proportional Allocation Mechanism, Microsoft Research Technical Report, MSR-TR