the weighted proportional allocation mechanism milan vojnović microsoft research joint work with...
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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
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
15
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
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
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
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
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
25
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.
26
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
27
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
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
• 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
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