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The Weighted Proportional Resource Allocation

Milan VojnovićMicrosoft Research

Joint work with Thành Nguyen

University of Cambridge, Oct 18, 2010

2

Resource Allocation Problem

i

1

n

provider users

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)

3

Resource Allocation Problem (cont’d)

1

providers users

2

m

• Multiple providers competing to provide service to users

• Everyone is selfish

4

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”

5

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

6

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

),,,( nCCC 21

i

jj

ii C

wwx

7

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 = Ci wi

Allocation:

– C = resource capacity

),,,( nCCC 21

Cw

wx

jj

ii

8

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

Px

x

b

bxARxP n

:

PEx. n = 2

9

Ex 1: Network Service

iC

1C

nC

provider users

10

Ex 1: Network Service (cont’d)

iw

1w

nw

provider users

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Ex 1: Network Service (cont’d)

i

jj

ii C

wwx

12

Ex 2: Compute Instance Allocation

• 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

i

1

n

jobs

task

mi

mi

mi ds

x,

,min

• Multi-machine multi-job scheduling

13

Ex 3. Sponsored Search

• Generalized Second Price Auction• Discrimination weights = click-through-rates• Assumes click-through-rates independent of

which ads appear together

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Ex 3: Sponsored Search (cont’d)

1x

• xi = click-through-rate for slot i

• Say $1 per click, so Ui(x) = x

• GSP revenue:

• Max weighted prop. revenue:

(0,0) (6,0)

2x

(0,14)

(5,4)(4,5)

),( 45 for 1

4.952

7

).,.( 9511458)7,7(),( for 222

221

21CC

15

Ex. 3: Sponsored Search (cont’d)• Revenue of weighted allocation auction

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• Applications




• Conclusion

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User’s Objective

• Price-taking: given price pi, user i solves:

• Price-anticipating: given Ci and , user i solves:

ipw

i wUi

i )(max 0 over iw

j

jw

iiwww

i wCUij

ij

i

)(max 0 over iw

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Provider’s Objective

• Choose discrimination weights to maximize own revenue

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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

i

iii xxU )('max Px

over

)(' ii xU

20

Social Optimum

• Social optimum allocation is a solution to

i

ii xU )(max Px

over

x

21

Equilibrium: Price-Taking Users

• Revenue

• Provider chooses discrimination weights

where maximizes over

• Equilibrium bids

• Same revenue as under third-degree price discrimination

ii

ii xxUxR )(')(

)(')(

iii xU

xRC

x

)(xR

Px

iiii xxUw )('

22

Equilibrium: Price-Anticipating Users

• Revenue R given by:

• Provider chooses discrimination weights

where maximizes over

• Equilibrium bids

1

i iii

iii

xRxxUxxU

)()(')('

)(')(

iiii xU

xRxC

x

)(xR

Px

iiiiii

i xxUxRxxU

xRw )(')()('

)(

23

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

Cw

wx

jj

ii

24

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)

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• 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)43

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

27


• Worst-case: serial network of unit capacity links

xxU )(1 xxU )(2xxUn )(

axxU )(0

anna

an

for ,)1(

Efficiency2

an

11

28


• Applications




• Conclusion

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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:

kRk

kR

1

xxU i )('

SiiiiPxknSnSk xxUR )('maxmin

|}:|,,{

1

121

RR

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Proof Key Idea

• Sufficient condition: for every there exists

ki

iiijjiji

iii Rk

kxxUk

kxxUxxUxR

1

)(1

)(max)()( '''

nk 1 :Px

ki

iii RxxU )('

nnnkkk xxUxxUxxU )()()( '11

'111

'1

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)

321

1

32

Worst-Case

• Utilities:

• Nash eq. allocation:

xxU )(1

xxxUxU n 072032 22 .)()()(

nin

ixi

,,21

131

1311

33

Proof Key Ideas• Utilities: 0 iii vxvxU ,)(

P i

ii x 1

)(max)(max xRxRQxPx

i

iiQxiiiPx

xUxU )(max)(max

setcovex a

every for concave(x)x

*

'

RL

iUi

*)(:* RxRxLR

Q

34

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

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• Applications




• Conclusion

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Oligopoly: Multiple Competing Providers

)( miii xxU 1

1ix

1

providers users

2

m

2ix

mix

37

Oligopoly (cont’d)

• User i problem: choose bids that solve

• Provider k problem: choose that maximize the revenue Rk over Pk where

miii www ,,, 21

k

ki

ki

kww

wi wCU

ij

ki

kj

ki )(max

kn

kk xxx ,,, 21

1

ikkk

iki

kk

kji

ki

ki

kk

kji

xRxxxU

xxxU

)()('

)('

'

''

'

38

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 : 0 all for bbaaUaUbU ,]')('[)()( d],[ 00 xa

)(xU

x

L

a

W

b

dWL

39

Examples of d-Utility Functions

),min( bax 0

concave )(' xU 2

0 ccx ),log( 2

0101

1

cxcw ),,[,)(

),()(],[

1101

21

21

1

3612

or .e

0 cc cx ),arctan( 2

“-fair”

)(xU d

40

Social Welfare

• Theorem For price-anticipating users with d-utility functions and oligopoly of competing providers:

(Nash eq. utility) (socially OPT utility)d

321

1

• 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%

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Proof Key Ideas

iii

Pziii

PzzVzU

kk

kk

)(max)(max

k i

iiPzii zva

kmax

0 ,)()( xxvaxVxU iiii

,min kiki vv k

iiiiiki xxUxUv )()( ''

i

kiii

ii

kiPz

xxUzvk

)(max '

i

iiii

i xxUa )('

i

iii

i xUa )( i

ii xU )()( d

42

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

43

To Probe Further

• The Weighted Proportional Allocation Mechanism, Microsoft Research Technical Report, MSR-TR-2010-145