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

The Weighted Proportional Allocation Mechanism

Milan Vojnović

Microsoft Research

Joint work with Thành Nguyen

Harvard University, Nov 3, 2009

2

Resource allocation problem

i

1

n

provider users

Resource

• Provider wants large revenue• User wants large surplus (utility – cost)• Resource with general constraints

– Ex. network service, data centre, sponsored search

3

Resource allocation problem (cont’d)

1

providers users

2

m

• Oligopoly – multiple providers competing to provide service to users

• Each provider wants a large revenue

4

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

5

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

6

The mechanism

• Provider announces discrimination weights

• Each user i submits a bid wi

Payment by user i = wi

Allocation to user i:

• Discrimination weights so that allocation is feasible

),,,( nCCC 21

i

jj

ii C

w

wx

7

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

Px

x

b

bxARxP n

:

PEx. n = 2

8

Ex 1: Network service

iC

1C

nC

provider users

9

Ex 1: Network service (cont’d)

iw

1w

nw

provider users

10

Ex 1: Network service (cont’d)

i

jj

ii C

w

wx

11

Ex 2: data centre resource 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 d

sx

,

,min

• Multi-job task scheduling

12

Ex 3. Sponsored search

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

which ads appear together

13

Ex 3: Sponsored search (cont’d)

1x

• xi = click-through-rate at 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

),(),( 222

221

21 77 for 4.952

7 CC

).,.( 9511458

14

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

15

Outline

• The mechanism

• Applications




• Conclusion

16

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

iiww

wi wCU

ijij

i

)(max 0 over iw

17

Provider’s objective

• Choose discrimination weights to maximize the revenue

18

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

i

iii xxU )('max Px

over

)(' ii xU

19

Social optimum

• Social optimum allocation is a solution to

i

ii xU )(max Px

over

x

20

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 )('

21

Equilibrium: price-anticipating users

• Revenue R given by:

• Provider chooses discrimination weights

where maximizes over

• Equilibrium bids

1

i iii

iii

xRxxU

xxU

)()('

)('

)('

)(

iiii xU

xRxC

x

)(xR

Px

iiiiii

i xxUxRxxU

xRw )('

)()('

)(

22

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

23

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)

24


• 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)4

3

25


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

26


• Worst-case: serial network of unit capacity links

xxU )(1 xxU )(2xxUn )(

axxU )(0

anna

an

for 1

efficiency2

,)( an

1

1

27

Outline

• The mechanism

• Applications




• Conclusion

28

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:

kRk

kR

1

xxU i )('

Siiii

PxknSnSk xxUR )('maxmin

|}:|,,{

1

12

1 RR

29

Example

• Unit-capacity resource:• Symmetric users with utility function U(x)• U(x) concave, and U’(x)x concave increasing on [0,1]

1i

ix

)(')( nn UR 111 )(' knk UR 1

an naR 111 )( a

k knaR 1)(

ankn

kR

R

11

111 ))(( /

Ex. (0,1)a ,)( axxU

)(nokn

for 1

0R revenue underthird-degree price discrimination

30

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)

3

21

1

31

Worst-case

• Utilities:

• Nash eq. allocation:

xxU )(1

xxxUxU n 072032 22 .)()()(

nin

ixi

,,21

3

1

13

11

32

Proof key ideas

• Utilities: 0 iii vxvxU ,)(

*)(:* RxRxLR

P i

ii x 1

iii xxQ 1:

)(max)(max xRxRQxPx

i

iiQx

iii

PxxUxU )(max)(max

setcovex a

every for concave(x)x

*R

i

L

iU

33

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

34

Outline

• The mechanism

• Applications




• Conclusion

35

Oligopoly: multiple competing providers

)( miii xxU 1

1ix

1

providers users

2

m

2ix

mix

36

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

)()('

)('

'

''

'

37

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 ,]')('[)()( ],[ 00 xa

)(xU

x

L

a

W

b

W

L

38

Examples of d-utility functions

),min( bax 0

concave )(' xU 2

0 ccx ),log( 2

0101

1

cxcw

),,[,)(

),()(

],[

11

01

21

21

1

3612

or .e

0 cc cx ),arctan( 2

“a-fair”

)(xU

39

Social welfare

• Theorem For price-anticipating users with d-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 d = 1, 2, worst-case efficiency at least 31, 24%

(Nash eq. utility) (socially OPT utility)

3

21

1

40

Proof key ideas• Bounding social welfare by an affine function separates to

optimizations for individual providers

• For provider k consider linear utility functions where

iki

ki axvxV min)(

)(xU

x

ia

ix

W

iii

Pziii

PzzVzU

kk

kk

)(max)(max

k i

iiPz

ii zva

kmax

kiiiii

ki xxUxUv )('')('

)( iii xUa i

ii xU )()(3

21

xv ki

41

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

42

To Probe Further

• The Weighted Proportional Allocation Mechanism, Microsoft Research Technical Report, MSR-TR-2009-123

Documents

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