The Weighted Proportional Allocation Mechanism Milan Vojnović Microsoft Research Joint work with Thành Nguyen rvard 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
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