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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 - PowerPoint PPT Presentation
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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
11
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
14
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
16
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
17
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
18
Provider’s Objective
• Choose discrimination weights to maximize own revenue
19
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)
25
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)43
26
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.
27
Related Work (cont’d)
• Worst-case: serial network of unit capacity links
xxU )(1 xxU )(2xxUn )(
axxU )(0
anna
an
for ,)1(
Efficiency2
an
11
28
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
29
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
30
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
31
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
35
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
36
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%
41
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