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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
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)
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Resource Allocation Problem (cont’d)
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”
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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
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The Weighted Resource Allocation
• Weighted Allocation Auction:
– Provider announces discrimination weights
– Each user i submits bid wi
Payment = wi
Allocation:
– Discrimination weights so that allocation is feasible
),,,( nCCC 21
i
jj
ii C
w
wx
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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 bid wi
Payment = Ci wi
Allocation:
– C = resource capacity
),,,( nCCC 21
Cw
wx
jj
ii
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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
Px
x
b
bxARxP n
:
PEx. n = 2
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Ex 1: Network Service
iC
1C
nC
provider users
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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
w
wx
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Ex 2. 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 2: Sponsored Search (cont’d)
1x
• xi = click-through-rate for slot i
• Say $1 per click, so Ui(x) = x
• Max weighted prop. revenue:
• GSP revenue 1(0,0) (6,0)
2x
(0,14)
(5,4)
(4,5) 4.95
2
7
).,.( 9511458)7,7(),( for 222
221
21CC
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Ex. 3: Sponsored Search (cont’d)• 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
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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
iiww
wi wCU
ijij
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
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Social Optimum
• Social optimum allocation is a solution to
i
ii xU )(max Px
over
x
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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
ii
ii xxUxR )(')(
)('
)(
iii xU
xRC
x
)(xR
Px
iiii xxUw )('
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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 )('
)()('
)(
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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
Cw
wx
jj
ii
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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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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)4
3
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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.
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Related Work (cont’d)
• Worst-case: serial network of unit capacity links
xxU )(1 xxU )(2xxUn )(
axxU )(0
anna
an
for ,)1(
Efficiency2
an
1
1
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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
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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 )('
Siiii
PxknSnSk xxUR )('maxmin
|}:|,,{
1
12
1 RR
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Proof Key Idea
• Sufficient condition: for every there exists
ki
iiijjiji
iii Rk
kxxU
k
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)
3
21
1
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Worst-Case
• Utilities:
• Nash eq. allocation:
xxU )(1
xxxUxU n 072032 22 .)()()(
nin
ixi
,,21
1
3
1
13
11
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Proof Key Ideas
• Utilities: 0 iii vxvxU ,)(
P i
ii x 1
)(max)(max xRxRQxPx
i
iiQx
iii
PxxUxU )(max)(max
setcovex a
every for concave(x)x
*
'
RL
iUi
*)(:* RxRxLR
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
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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
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Multiple Competing Providers
)( miii xxU 1
1ix
1
providers users
2
m
2ix
mix
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Multiple Competing Providers (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
)()('
)('
'
''
'
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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
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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
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Social Welfare
• Theorem For price-anticipating users with d-utility functions and multiple competing providers:
(Nash eq. utility) (socially OPT utility)
3
21
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
iiPz
ii zva
kmax
0 ,)()( xxvaxVxU iiii
,min kiki vv k
iiiiiki xxUxUv )()( ''
i
kiii
ii
ki
PzxxUzv
k
)(max '
i
iiii
i xxUa )('
i
iii
i xUa )( i
ii xU )()(
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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
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To Probe Further
• The Weighted Proportional Allocation Mechanism– Conference paper, ACM Sigmetrics 2011 – Microsoft Research Technical Report, MSR-TR-2010-145