Differentiated Internet Pricing Using a Hierarchical Network Game Model

Differentiated Internet Pricing Using a Hierarchical Network Game Model Hongxia Shen and Tamer Bas¸ar 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

Outline ➽ 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] Bas¸ar and Srikant, “Revenue-maximizing pricing and capacity expansion in a many-users regime,” IEEE INFOCOM 2002.

Hongxia Shen and Tamer Bas¸ar, Boston, MA, 07/01/2004

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Outline ➽ 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


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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 P ➢ Net utility of user i, (x ¯l = i:l∈Li xi ; wi , ki , vi : positive scalars) Fi = wi log(1 + ki xi ) −

X

l∈Li

X 1 − v i xi pli cl − x ¯l l∈Li

➢ Revenue of the ISP, R=

X X

l∈L i:l∈Li

pli xi =

X i∈I

xi

X

pli

l∈Li


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General Network

Two-Level Hierarchical Network Game ISP

Leader

max{pli } R Stackelberg game

User 1

User I

maxx1 F1

maxxI FI

Followers

I -player noncooperative game


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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] X X X X 1 wj log(1 + kj xj ) − − v j xj plj . cl − x ¯l j6=i

l∈L / i

j6=i

l∈Lj

➢ Obtain an equivalent noncooperative game where all the users have a common objective function (strictly concave), X X X 1 X F = wi log(1 + ki xi ) − v i xi pli − . cl − x ¯l i∈I

i∈I

l∈Li

l∈L



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Special Single Link

Special Single Link Network ➢ Single link network with a capacity n shared by n users Pn ➢ Net utility of user i, (x ¯ := j=1 xj ) 1 − pi xi , i ∈ N := {1, · · · , n} Fi = wi log(1 + xi ) − n−x ¯ ➢ Uniform Price (UniPri): pi = p (complete solution by [BS’02]) ➢ Differentiated Prices (DiffPri) ➢ Notations: xav :=

x ¯ n;

w ¯ :=

Pn

j=1

wj , wav :=

w ¯ n;

1 2

v¯ :=

Pn

j=1

√

1 2

wj , vav :=

1

v ¯2 n



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Special Single Link

Positive Solution for UniPri x∗av−u = 1 −

2 1+

(n2 w

av )

1 3

, ⇑ 2

1 3

1 + (n wav ) 1 = , ⇓ n − nx∗av−u 2n wi ∗ x∗i−u = (xav−u + 1) − 1, i ∈ N, ⇑ wav 1 2 wav 1 ∗ 2 −1 2 3 pu = (1 + (n wav ) ) − 2 (1 + (n wav ) 3 ) , 2 4n 2 wav 1 3 ru∗ = p∗u x∗av−u = − 2 (n2 wav ) 3 + 2 , 2 4n 4n d∗u =

if and only if 2

2 3

2(n wav ) + 2n2 wav wi > , ∀i∈N 4n2


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Special Single Link

Positive Solution for DiffPri x∗av−d = 1 −

2 1 2

1 + (nvav ) 1 2

2 3

, ⇑

2

1 + (nvav ) 3 d∗d = , ⇓ 2n √ wi ∗ ∗ xi−d = 1 (xav−d + 1) − 1, i ∈ N, ⇑ 2 vav 1 2 1 1 2 2 2 √ vav 1 ∗ 2 −3 2 pi−d = wi (1 + (nvav ) ) − 2 (1 + (nvav ) 3 ) , i ∈ N, 2 4n 1 1 4 1 3 1 ∗ 2 2 2 rd = wav − 2 (nvav ) − 2 (nvav )3 + 2 , 2n 4n 4n

if and only if 1 2

wi >

4 3

1 2

2

1 2

2 3

2(nvav ) + (nvav ) + (nvav ) , ∀i∈N 4n2


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Special Single Link

General Solution If 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]



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Comparison of UniPri and DiffPri

Conditions for Positive Solution 2

1 2

n wav ≥ (nvav )2

Condition for UniPri = > Condition for DiffPri

DiffPri can admit more users with relatively small wi ’s than UniPri.


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Same Number of Users Admitted (1) ➢ Throughput: x∗av−u ≥ x∗av−d ➢ Congestion cost: d∗u ≥ d∗d ➢ Individual flows:

 ∗ ∗  > x x  i−u i−d if wi > wx ,  x∗i−u = x∗i−d if wi = wx ,

➢ Prices:

  

x∗i−u < x∗i−d if wi < wx

 ∗ ∗  < p p  u i−d if wi > wp ,  p∗u = p∗i−d if wi = wp ,    ∗ pu > p∗i−d if wi < wp 1 2

wmax ≥ wx ≥ wp , wav ≥ (vav )2 ≥ wmin Hongxia Shen and Tamer Bas¸ar, Boston, MA, 07/01/2004

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Same Number of Users Admitted (2) ➢ Individual utilities:  ∗ ∗  F > F if wi > wF ,  i−u i−d  ∗ ∗ = Fi−d if wi = wF , Fi−u

  

∗ ∗ < Fi−d if wi < wF Fi−u

wmax ≥ wF ≥ wp ➢ Revenue: ru∗ ≤ rd∗


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Same Number of Users Admitted - Example UniPri DiffPri

n = 50

0.25

with wi ’s evenly distributed

Flows

0.2

around wav

0.15

from 0.8775e − 3

0.1

through 1.1225e − 3

0.05

0 0.85

0.9

0.95

1 Users ( w )

1.05

1.1

1.15 −3

i

x 10

−0.0233

−4

x 10 3.6

= 0.001

UniPri DiffPri

UniPri DiffPri

−0.0234

Net utilities

3.4

Prices

3.2

3

−0.0234

−0.0235

2.8

−0.0235 2.6 0.85

0.9

0.95

1 Users ( w ) i

1.05

1.1

1.15 −3

0.85

0.9

0.95

x 10


1 Users ( wi )

1.05

1.1

1.15 −3

x 10

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More Users Admitted for DiffPri Compare UniPri(n), DiffPri(n), and DiffPri(n ˆ ), n

dˆ∗d ➢ Individual flows: x∗i−d < x ˆ∗i−d

ˆ∗ ➢ Total revenue: Ru∗ ≤ Rd∗ ≤ R d


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General Single Link

General Single Link Network ➢ Single link network with a capacity nc shared by n users Pn ➢ Net utility of user i, (x ¯ := j=1 xj ) Fi = wi log(1 + ki xi ) −

1 − pi xi , i ∈ N := {1, · · · , n} nc − x ¯

➢ Uniform Price (UniPri): pi = p ➢ Differentiated Prices (DiffPri) ➢ Notations: −1 kav :=

1 n

Pn

1

1 2 j=1 kj ; zav

:=

1 n

Pn

j=1


q

wj kj

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General Single Link

Asymptotic Solution for UniPri −1 2 1 −2 ) 2c(c + kav ∗ 3n 3, ⇑ α= , x (n) ∼ c − α av−u −1 wav kav 1 ∗ −1 −1 3 du (n) = ∼ α n 3, ⇓ ∗ nc − nxav−u (n) wi 1 wi 13 − 23 ∗ −1 xi−u (n) ∼ (c + kav ) − − α n , i ∈ N, ⇑ wav ki wav wav 2c ∗ −2 −2 3 pu (n) ∼ + ( −1 − 1)α n 3 , ⇓ −1 c + kav kav p∗u (n)x∗av−u (n) wav ∗ −2 −2 3 ru (n) = − 3α n 3 , ⇑ ∼ −1 c c + kav

if and only if

wav 2c − 23 − 23 w i ki > + −1 α n , ∀ i ∈ N −1 c + kav kav


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General Single Link

Asymptotic Solution for DiffPri β=

−1 2 ) 2c(c + kav 1 2

,

x∗av−d (n)

(zav )2 d∗d (n)

1 3

∼c−β n

−2 3

, ⇑

1 −1 −1 3 = ∼ β n 3, ⇓ ∗ nc − nxav−d (n)

x∗i−d (n) ∼

p

wi /ki 1 2

−1 (c + kav )−

1 − ki

p

wi /ki 1 2

1

2

β 3 n− 3 , i ∈ N, ⇑

zav zav 1 √ √ 2 w k w i ki z 2c i i av −2 −2 3 n 3 , i ∈ N, ⇓ p∗i−d (n) ∼ + ( − 1)β 1 −1 c + kav z2 av

1 2

rd∗ (n) if and only if

p

(zav )2 wav −2 −2 3n 3, ⇑ ∼ − − 3β −1 c c(c + kav ) 1 2

1 2

zav 2c − 23 − 23 zav wi ki > −1 + 1 β n , ∀i∈N ∼ −1 kav + x∗av−d (n) c + kav 2 zav Hongxia Shen and Tamer Bas¸ar, Boston, MA, 07/01/2004

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General Single Link

Asymptotic Comparison (1) ➢ Same conclusion: DiffPri admits more users ∼ ˜¯∗ ≤ x ˜¯∗ ➢ Throughput: x ˜∗ ,x ˜∗ =x av−u

av−d

u

d

➢ Congestion cost: d˜∗u ≥ d˜∗d

➢ Individual flows:  ∗ ∗ ˜  > x ˜ x ˜  i−u i−d if wi ki > wk,  ˜ ˜∗i−d if wi ki = wk, x ˜∗i−u = x    ∗ ˜ ˜∗i−d if wi ki < wk x ˜i−u < x

➢ Prices:

 ∗ ∗ ˜  < p ˜ if wi ki > wk, p ˜  u i−d  ˜ p˜∗u = p˜∗i−d if wi ki = wk,    ∗ ˜ p˜u > p˜∗i−d if wi ki < wk


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General Single Link

Asymptotic Comparison (2) ➢ Individual utilities:  ˜ F, ˜ ∗ > F˜ ∗ if wi ki > wk  F  i−u i−d  ∗ ∗ ˜ F, = F˜i−d if wi ki = wk F˜i−u    ˜∗ ∗ ˜F if wi ki < wk Fi−u < F˜i−d 1

(wi ki )max

2 ˜ F ≥ wk ˜ := wav /zav ≥ wk

2

≥ (wi ki )min

➢ Revenue: r˜u∗ ≤ r˜d∗


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General Single Link

How DiffPri Affects Users DiffPri for all users: total flow ⇑, congestion cost ⇓

DiffPri

DiffPri

DiffPri

flows ⇑, prices ⇓,

flows ⇓, prices ⇑,

flows ⇓, prices ⇑,

Users

Users

Users

˜ wi ki ≤ wk

˜F ˜ < wi ki ≤ wk wk

˜ F < w i ki wk

net utilities ⇑

net utilities ⇑


net utilities ⇓

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Conclusions and Extensions

Conclusions Price differentiation leads to a more egalitarian resource distribution 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: reduced prices, 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 prices and improved revenue ➢ incentive for the ISP to increase the capacity


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Conclusions and Extensions

Extensions ➢ Linear network [BS’02A] and other general networks ➢ Incomplete information ➢ Multiple ISPs

[BS’02A] Bas¸ar and Srikant, “A Stackelberg network game with a large number of followers,” J. Optimization Theory and Applications, Dec. 2002.

End of the Talk


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