Upload
others
View
13
Download
0
Embed Size (px)
Citation preview
Differentiated Internet Pricing Using aHierarchical Network Game Model
Hongxia Shen and Tamer Basar
Coordinated Science Laboratory
University of Illinois at Urbana-Champaign
hshen1, [email protected]
ACC 2004, Boston, MA
July 1, 2004
ACC 2004 Differentiated Internet Pricing Using a Hierarchical Network Game Model
OutlineOutlineOutline
➽ General Network [BS’02]
➽ Complete Solution for a Special Single Link Network➽ Uniform Price (UniPri) [BS’02]
➽ Differentiated Prices (DiffPri)
➽ Comparison of the Two Pricing Schemes
➽ General Single Link in a Many-User Regime
➽ Conclusions and Extensions
[BS’02] Basar and Srikant, “Revenue-maximizing pricing and capacity expansion in a many-users
regime,” IEEE INFOCOM 2002.
Hongxia Shen and Tamer Basar, Boston, MA, 07/01/2004 –1–
ACC 2004 Differentiated Internet Pricing Using a Hierarchical Network Game Model
OutlineOutlineOutline
➽ General Network
➽ Complete Solution for a Special Single Link Network➽ Uniform Price (UniPri)
➽ Differentiated Prices (DiffPri)
➽ Comparison of the Two Pricing Schemes
➽ General Single Link in a Many-User Regime
➽ Conclusions and Extensions
Hongxia Shen and Tamer Basar, Boston, MA, 07/01/2004 –1–
ACC 2004 General Network
Problem Formulation
➢ Single Internet Service Provider (ISP)
➢ Set of users, I = {1, . . . , I}; flow of user i, xi, i ∈ I➢ Set of links, L = {1, . . . , L}; capacity of link l, cl, l ∈ L➢ Set of Links xi traverses, Li ⊆ L➢ Unit price charged to user i for using link l, pli, l ∈ Li
➢ Net utility of user i, (xl =∑
i:l∈Lixi; wi, ki, vi: positive scalars)
Fi = wi log(1 + kixi) −∑
l∈Li
1
cl − xl− vixi
∑
l∈Li
pli
➢ Revenue of the ISP,
R =∑
l∈L
∑
i:l∈Li
plixi =∑
i∈I
xi
∑
l∈Li
pli
Hongxia Shen and Tamer Basar, Boston, MA, 07/01/2004 –2–
ACC 2004 General Network
Two-Level Hierarchical Network Game
ISP
max{pli} R
User 1
maxx1F1
User I
maxxIFI
I-player noncooperative game
Stackelberg
game
Leader
Followers
Hongxia Shen and Tamer Basar, Boston, MA, 07/01/2004 –3–
ACC 2004 General Network
Existence of a Unique Nash Equilibrium➢ Suppose that prices are given and fixed.
➢ Add to Fi the quantity not related to xi, [BS’02]
∑
j 6=i
wj log(1 + kjxj) −∑
l/∈Li
1
cl − xl−
∑
j 6=i
vjxj
∑
l∈Lj
plj .
➢ Obtain an equivalent noncooperative game where all the users have acommon objective function (strictly concave),
F =∑
i∈I
wi log(1 + kixi) −∑
i∈I
vixi
∑
l∈Li
pli −∑
l∈L
1
cl − xl.
[BS’02] Basar and Srikant, “Revenue-maximizing pricing and capacity expansion in a many-users
regime,” IEEE INFOCOM 2002.
Hongxia Shen and Tamer Basar, Boston, MA, 07/01/2004 –4–
ACC 2004 Differentiated Internet Pricing Using a Hierarchical Network Game Model
OutlineOutlineOutline
➽ General Network
➽ Complete Solution for a Special Single Link Network➽ Uniform Price (UniPri)
➽ Differentiated Prices (DiffPri)
➽ Comparison of the Two Pricing Schemes
➽ General Single Link in a Many-User Regime
➽ Conclusions and Extensions
Hongxia Shen and Tamer Basar, Boston, MA, 07/01/2004 –1–
ACC 2004 Special Single Link
Special Single Link Network➢ Single link network with a capacity n shared by n users
➢ Net utility of user i, (x :=∑n
j=1xj )
Fi = wi log(1 + xi) −1
n − x− pixi, i ∈ N := {1, · · · , n}
➢ Uniform Price (UniPri): pi = p (complete solution by [BS’02])
➢ Differentiated Prices (DiffPri)
➢ Notations:
xav := xn ; w :=
∑nj=1
wj , wav := wn ; v
1
2 :=∑n
j=1
√wj , v
1
2
av := v1
2
n
[BS’02] Basar and Srikant, “Revenue-maximizing pricing and capacity expansion in a many-users
regime,” IEEE INFOCOM 2002.
Hongxia Shen and Tamer Basar, Boston, MA, 07/01/2004 –5–
ACC 2004 Special Single Link
Positive Solution for UniPri
x∗
av−u = 1 − 2
1 + (n2wav)1
3
, ⇑
d∗
u =1
n − nx∗
av−u
=1 + (n2wav)
1
3
2n, ⇓
x∗
i−u =wi
wav
(x∗
av−u + 1) − 1, i ∈ N, ⇑
p∗
u =wav
2(1 + (n2wav)−
1
3 ) − 1
4n2(1 + (n2wav)
1
3 )2,
r∗u = p∗
ux∗
av−u =wav
2− 3
4n2(n2wav)
2
3 +1
4n2,
if and only if
wi >2(n2wav)
2
3 + 2n2wav
4n2, ∀ i ∈ N
Hongxia Shen and Tamer Basar, Boston, MA, 07/01/2004 –6–
ACC 2004 Special Single Link
Positive Solution for DiffPri
x∗
av−d = 1 − 2
1 + (nv1
2av)
2
3
, ⇑
d∗
d =1 + (nv
1
2av)
2
3
2n, ⇓
x∗
i−d =
√wi
v1
2av
(x∗
av−d + 1) − 1, i ∈ N, ⇑
p∗
i−d =√
wi
v1
2av
2(1 + (nv
1
2av)−
2
3 ) − 1
4n2(1 + (nv
1
2av)
2
3 )2, i ∈ N,
r∗d = wav − 1
2n2(nv
1
2av)2 − 3
4n2(nv
1
2av)
4
3 +1
4n2,
if and only if
wi >2(nv
1
2av)
4
3 + (nv1
2av)2 + (nv
1
2av)
2
3
4n2, ∀ i ∈ N
Hongxia Shen and Tamer Basar, Boston, MA, 07/01/2004 –7–
ACC 2004 Special Single Link
General SolutionIf the necessary and sufficient condition for positive solution is not satisfied:
➊ order the users such that wi > wj only if i < j;
➋ find the largest n∗ ≤ n such that the condition holds for the first n∗ users;
➌ write out the positive solution for the n∗-user problem;
➍ obtain the solution for the n-user problem by appending x∗i = 0, i > n∗.
[BS’02]
[BS’02] Basar and Srikant, “Revenue-maximizing pricing and capacity expansion in a many-users
regime,” IEEE INFOCOM 2002.
Hongxia Shen and Tamer Basar, Boston, MA, 07/01/2004 –8–
ACC 2004 Differentiated Internet Pricing Using a Hierarchical Network Game Model
OutlineOutlineOutline
➽ General Network
➽ Complete Solution for a Special Single Link Network➽ Uniform Price (UniPri)
➽ Differentiated Prices (DiffPri)
➽ Comparison of the Two Pricing Schemes
➽ General Single Link in a Many-User Regime
➽ Conclusions and Extensions
Hongxia Shen and Tamer Basar, Boston, MA, 07/01/2004 –1–
ACC 2004 Comparison of UniPri and DiffPri
Conditions for Positive Solution
n2wav ≥ (nv1
2
av)2
Condition for UniPri => Condition for DiffPri
DiffPri can admit
more users with relatively small wi’s
than UniPri.
Hongxia Shen and Tamer Basar, Boston, MA, 07/01/2004 –9–
ACC 2004 Comparison of UniPri and DiffPri
Same Number of Users Admitted (1)
➢ Throughput: x∗av−u ≥ x∗
av−d
➢ Congestion cost: d∗u ≥ d∗
d
➢ Individual flows:
x∗i−u > x∗
i−d if wi > wx,
x∗i−u = x∗
i−d if wi = wx,
x∗i−u < x∗
i−d if wi < wx
➢ Prices:
p∗u < p∗i−d if wi > wp,
p∗u = p∗i−d if wi = wp,
p∗u > p∗i−d if wi < wp
wmax ≥ wx ≥ wp, wav ≥ (v1
2
av)2 ≥ wmin
Hongxia Shen and Tamer Basar, Boston, MA, 07/01/2004 –10–
ACC 2004 Comparison of UniPri and DiffPri
Same Number of Users Admitted (2)
➢ Individual utilities:
F ∗i−u > F ∗
i−d if wi > wF ,
F ∗i−u = F ∗
i−d if wi = wF ,
F ∗i−u < F ∗
i−d if wi < wF
wmax ≥ wF ≥ wp
➢ Revenue: r∗u ≤ r∗d
Hongxia Shen and Tamer Basar, Boston, MA, 07/01/2004 –11–
ACC 2004 Comparison of UniPri and DiffPri
Same Number of Users Admitted - Example
n = 50
with wi’s evenly distributed
around wav = 0.001
from 0.8775e − 3
through 1.1225e − 3
0.85 0.9 0.95 1 1.05 1.1 1.15
x 10−3
0
0.05
0.1
0.15
0.2
0.25
Users ( wi )
Flo
ws
UniPriDiffPri
0.85 0.9 0.95 1 1.05 1.1 1.15
x 10−3
2.6
2.8
3
3.2
3.4
3.6
x 10−4
Users ( wi )
Pric
es
UniPriDiffPri
0.85 0.9 0.95 1 1.05 1.1 1.15
x 10−3
−0.0235
−0.0235
−0.0234
−0.0234
−0.0233
Users ( wi )
Net
util
ities
UniPriDiffPri
Hongxia Shen and Tamer Basar, Boston, MA, 07/01/2004 –12–
ACC 2004 Comparison of UniPri and DiffPri
More Users Admitted for DiffPriCompare UniPri(n), DiffPri(n), and DiffPri(n), n < n:
➢ Throughput: x∗av−d < x∗
av−d
➢ Congestion cost: d∗u ≥ d∗
d > d∗d
➢ Individual flows: x∗i−d < x∗
i−d
➢ Total revenue: R∗u ≤ R∗
d ≤ R∗d
Hongxia Shen and Tamer Basar, Boston, MA, 07/01/2004 –13–
ACC 2004 Differentiated Internet Pricing Using a Hierarchical Network Game Model
OutlineOutlineOutline
➽ General Network
➽ Complete Solution for a Special Single Link Network➽ Uniform Price (UniPri)
➽ Differentiated Prices (DiffPri)
➽ Comparison of the Two Pricing Schemes
➽ General Single Link in a Many-User Regime
➽ Conclusions and Extensions
Hongxia Shen and Tamer Basar, Boston, MA, 07/01/2004 –1–
ACC 2004 General Single Link
General Single Link Network➢ Single link network with a capacity nc shared by n users
➢ Net utility of user i, (x :=∑n
j=1xj )
Fi = wi log(1 + kixi) −1
nc − x− pixi, i ∈ N := {1, · · · , n}
➢ Uniform Price (UniPri): pi = p
➢ Differentiated Prices (DiffPri)
➢ Notations:
k−1
av := 1
n
∑nj=1
1
kj; z
1
2
av := 1
n
∑nj=1
√
wj
kj
Hongxia Shen and Tamer Basar, Boston, MA, 07/01/2004 –14–
ACC 2004 General Single Link
Asymptotic Solution for UniPri
α =2c(c + k−1
av )2
wavk−1
av
, x∗
av−u(n) ∼ c − α1
3 n−2
3 , ⇑
d∗
u(n) =1
nc − nx∗
av−u(n)
∼ α−1
3 n−1
3 , ⇓
x∗
i−u(n) ∼ wi
wav
(c + k−1
av ) − 1
ki
− wi
wav
α1
3 n−2
3 , i ∈ N, ⇑
p∗
u(n) ∼ wav
c + k−1
av
+ (2c
k−1
av
− 1)α−2
3 n−2
3 , ⇓
r∗u(n) =p∗
u(n)x∗
av−u(n)
c∼ wav
c + k−1
av
− 3α−2
3 n−2
3 , ⇑
if and only if
wiki >wav
c + k−1
av
+2c
k−1
av
α−2
3 n−2
3 , ∀ i ∈ N
Hongxia Shen and Tamer Basar, Boston, MA, 07/01/2004 –15–
ACC 2004 General Single Link
Asymptotic Solution for DiffPri
β =2c(c + k−1
av )2
(z1
2av)2
, x∗
av−d(n) ∼ c − β1
3 n−2
3 , ⇑
d∗
d(n) =1
nc − nx∗
av−d(n)
∼ β−1
3 n−1
3 , ⇓
x∗
i−d(n) ∼√
wi/ki
z1
2av
(c + k−1
av ) − 1
ki
−√
wi/ki
z1
2av
β1
3 n−2
3 , i ∈ N, ⇑
p∗
i−d(n) ∼ z1
2av
√wiki
c + k−1
av
+ (2c√
wiki
z1
2av
− 1)β−2
3 n−2
3 , i ∈ N, ⇓
r∗d(n) ∼ wav
c− (z
1
2av)2
c(c + k−1
av )− 3β−
2
3 n−2
3 , ⇑
if and only if√
wiki >z
1
2av
k−1
av + x∗
av−d(n)
∼ z1
2av
c + k−1
av
+2c
z1
2av
β−2
3 n−2
3 , ∀ i ∈ N
Hongxia Shen and Tamer Basar, Boston, MA, 07/01/2004 –16–
ACC 2004 General Single Link
Asymptotic Comparison (1)
➢ Same conclusion: DiffPri admits more users
➢ Throughput: x∗av−u
∼= x∗av−d, ˜x
∗u ≤ ˜x
∗d
➢ Congestion cost: d∗u ≥ d∗d
➢ Individual flows:
x∗i−u > x∗
i−d if wiki > wk,
x∗i−u = x∗
i−d if wiki = wk,
x∗i−u < x∗
i−d if wiki < wk
➢ Prices:
p∗u < p∗i−d if wiki > wk,
p∗u = p∗i−d if wiki = wk,
p∗u > p∗i−d if wiki < wk
Hongxia Shen and Tamer Basar, Boston, MA, 07/01/2004 –17–
ACC 2004 General Single Link
Asymptotic Comparison (2)
➢ Individual utilities:
F ∗i−u > F ∗
i−d if wiki > wkF ,
F ∗i−u = F ∗
i−d if wiki = wkF ,
F ∗i−u < F ∗
i−d if wiki < wkF
(wiki)max ≥ wkF ≥ wk :=(
wav/z1
2
av
)2 ≥ (wiki)min
➢ Revenue: r∗u ≤ r∗d
Hongxia Shen and Tamer Basar, Boston, MA, 07/01/2004 –18–
ACC 2004 General Single Link
How DiffPri Affects Users
Users
wiki ≤ wk
Users
wk < wiki ≤ wkF
Users
wkF < wiki
DiffPri
flows ⇑, prices ⇓,
net utilities ⇑
DiffPri
flows ⇓, prices ⇑,
net utilities ⇑
DiffPri
flows ⇓, prices ⇑,
net utilities ⇓
DiffPri for all users: total flow ⇑, congestion cost ⇓
Hongxia Shen and Tamer Basar, Boston, MA, 07/01/2004 –19–
ACC 2004 Differentiated Internet Pricing Using a Hierarchical Network Game Model
OutlineOutlineOutline
➽ General Network
➽ Complete Solution for a Special Single Link Network➽ Uniform Price (UniPri)
➽ Differentiated Prices (DiffPri)
➽ Comparison of the Two Pricing Schemes
➽ General Single Link in a Many-User Regime
➽ Conclusions and Extensions
Hongxia Shen and Tamer Basar, Boston, MA, 07/01/2004 –1–
ACC 2004 Conclusions and Extensions
ConclusionsPrice differentiation leads to a more egalitarian resourcedistribution at fairer prices:
➢ more users admitted
➢ higher total flow, alleviated congestion
➢ beneficial to the ISP: improved revenue
➢ beneficial to users with relatively small utility parameters: reducedprices, increased flows and utilities
➢ disadvantageous to other users: decreased utilities
ISP tends to have more users admitted (UniPri or DiffPri):
➢ increased throughput and flows, reduced congestion, decreased pricesand improved revenue
➢ incentive for the ISP to increase the capacity
Hongxia Shen and Tamer Basar, Boston, MA, 07/01/2004 –20–
ACC 2004 Conclusions and Extensions
Extensions➢ Linear network [BS’02A] and other general networks
➢ Incomplete information
➢ Multiple ISPs
[BS’02A] Basar and Srikant, “A Stackelberg network game with a large number of followers,” J.
Optimization Theory and Applications, Dec. 2002.
End of the Talk
Hongxia Shen and Tamer Basar, Boston, MA, 07/01/2004 –21–