Equilibrium solutions

and

I ₂ =

( 1

^if

s 2 = F 0

^if

s ₂ = U

Thepayoof light users (Player 3) is:

Π ₃ (s ₁ , s ₂ , s ₃ ) = (w ₃ − t ₃ )R(1 − I ₁ ) − s ₃ (1 − I ₂ )

^(2.14)

Note thatindiator variables areomplementeddue to opposingonditions.

Player2's payois the following:

Π ₂ (s ₁ , s ₂ , s ₃ ) =

( s ₁

^if

s ₂ = F s ₃

^if

s ₂ = U

(2.15)

Now we formulate the harateristi funtion usingthestandard approah,keeping in mind

that ertain oalitions of players are not reasonable beause of quarreling. Those oalitions

reeive zeroutility,formally:

ν (H) = 0 | C / ∈ {{ 1 } , { 3 } , { 12 } , { 23 }}

^and

H ∈ 2 ^N .

^(2.16)

For the reasonableoalitionsthe orrespondingutilities are:

ν( { 1 } ) = max

s 1

min s 2 ,s 3

Π ₁ (s ₁ , s ₂ , s ₃ )

ν( { 3 } ) = max

s 3

min s 1 ,s 2

Π ₃ (s ₁ , s ₂ , s ₃ )

ν( { 12 } ) = max

s 1 ,s 2

min s 3

[Π ₁ (s ₁ , s ₂ , s ₃ ) + Π ₂ (s ₁ , s ₂ , s ₃ )]

ν( { 23 } ) = max

s 2 ,s 3

min s 1

[Π ₂ (s ₁ , s ₂ , s ₃ ) + Π ₃ (s ₁ , s ₂ , s ₃ )]

(2.17)

UsingEquations2.16and2.17weompiletheharateristifuntionspresentedinTable2.3.

Dierent olumns represent dierent user distributions in thepopulation. If heavy users area

majority(

w 1 > 1/2

^{) theywilldominatevoting(heavyuserregime). Iflight usersareamajority}

(

w ₃ > 1/2

^{)theywill be thedominant player(lightuser regime). Ifneitheroftheabove aretrue}

(

w 1 < 1/2, w 3 < 1/2

^{, but due to onstraints of}

w i

^,

w 1 + w 2 > 1/2, w 2 + w 3 > 1/2

^{), theplayers}

enter a balaned regime, where the outome of the priinggame will be deided by theoered

side-payments.

Table2.3: Charateristifuntionfortheuser-inuenedpriinggame(

w ₁

^and

t ₁

^arethe

popula-tionratioandtraratioofheavyusers,while

w ₃

^and

t ₃

^{arethoseofthelightusers,}respetively) Charateristi funtion Heavyuser regime Balanedregime Light user regime

(

w ₁ > 1/2

^{) (}

w ₁ < 1/2, w ₃ < 1/2

^{) (}

w ₃ > 1/2

⁾

ν( { 1 } ) (t 1 − w 1 )R 0 0

ν( { 3 } ) 0 0 (w 3 − t 3 )R

ν ( { 12 } ) (t 1 − w 1 )R (t 1 − w 1 )R 0

ν ( { 23 } ) 0 (w 3 − t 3 )R (w 3 − t 3 )R

ν(H) 0 0 0

for all other

H ∈ 2 ^N

{{ 1 } , { 2 } , { 3 }} , {{ 12 } , { 3 }} , {{ 1 } ,

{ 23 }} , {{ 123 }}

^{. Thenwedenethesetofallowable oalitionalstrutures (}

ψ(P )

^{)thatsatisfythe}

onstraintsimposedby quarreling. For

G

ψ(P) = ψ = { ( { 12 } , { 3 } ), ( { 1 } , { 23 } ) } .

^(2.18)

For agiven

P ∈ ψ

^,let

X(P )

^{betheset of}imputations asfollows:

X(P ) = { (x ₁ , x ₂ , x ₃ ) ∈ R ³ | X

i ∈ H

x _i = ν(H)

^{for all}

H ∈ P

^and

x _i ≥ ν( { i } )

for

i = 1, 2, 3 }

^(2.19)

Intuitively an imputation is a distribution of the maximum side-payment suh that eah

playerreeivesat leastthesame amountofmoney thattheyangetiftheyhooseto stayalone

(individual rationality), and eah oalition in the struture

P

^{reeives the total} side-payment they an ahieve (group rationality).

Now, werestritthesetofimputationsto theore

C(P )

^{. Theoreisdenedto bethesetof}

undominated imputations. To put it dierently, theore is theset of imputations under whih

no oalition hasavaluegreater than thesum of itsmembers' payos. Formally:

C(P ) = (

(x ₁ , x ₂ , x ₃ ) ∈ X(P ) | X

i∈H

x _i ≥ ν(H)

^{for all}

H ∈ [

{ J ∈ P | P ∈ ψ } )

(2.20)

Considering our game

G

^{, Equation 2.20 is equivalent to the standard ore (sine}

ν(H) = 0

for unreasonable oalitions),so

C(P ) = (

(x ₁ , x ₂ , x ₃ ) ∈ X(P) | X

i ∈ H

x _i ≥ ν(H)

^{for all}

H ∈ 2 ^N )

(2.21)

As it an be notied the ore is dependent on a ertain oalitional struture

P

^{. For us to}

determine whih struture willemergewhen playing thegame, we dene a

ψ

^-stablepair

[x, P ]

^:

[x, P ] | x ∈ C(P), P ∈ ψ

^(2.22)

Now applying this solution to the harateristi funtion

ν(H)

^{in Table 2.3, we have three}

dierent asesdepending onuser regimes.

Heavy user regime

In this ase heavy users are dominant in the population, thus

w ₁ > 1/2

^{. The only possible}

imputation is

x = ((t ₁ − w ₁ )R, 0, 0)

^{henethere aretwo}

ψ

^{-stablepairs:}

[((t ₁ − w ₁ )R, 0, 0), {{ 12 } , { 3 }} ]

^and

[((t ₁ − w ₁ )R, 0, 0), {{ 1 } , { 23 }} ]

Note that both oalitional strutures are possible, sine it does not matter whih side medium

users take.

Inwords, this meansheavy users dominatethe voting, no side-payment istransferred.

Con-sidering the individual user's point of view, let

c _i

^{denote the ost of a single user i. Flat-rate}

priingisimplementedbythe ISP,Internet aessostsaresharedperapita,henetheostfor

a singleuser isindependent of histra and equal for every useris

c _i = R

n

^{for all}

i ∈ N

^(2.23)

Light user regime

Here light users have the absolute majority aross the population

(w ₃ > 1/2)

^{. Following the}

same line ofthought asinthe heavy userregime, we derive the

ψ

^{-stable pairs for thisase:}

[(0, 0, (w ₃ − t ₃ )R), {{ 1 } , { 23 }} ]

^and

[(0, 0, (w ₃ − t ₃ )R), {{ 12 } , { 3 }} ]

Asexpetedlightusersdominatethevoting,noside-paymentismadetomediumusers. From

a singleuser's perspetive, let

τ _i

^{denote the tra volume of user}

i

^{. Sine} usage-based priing is implemented by the ISP, Internet aess osts are shared proportionally to tra volume.

Therefore theaessost for user

i

^is

c _i = τ _i

T R

^{for all}

i ∈ N.

^(2.24)

Table 2.4:

ψ

^{-stable pairs inthebalaned regime}

Side-payment parameters Core solution Coalitional struture

0 < s

^max

₃ < s

^max

₁ (s

^max

₁ − s ₁ , s ₁ , 0) ( { 12 } , { 3 } )

0 < s

^max

₃ = s

^max

₁ (0, s

^max

₁ , 0) ( { 12 } , { 3 } )

^or

( { 1 } , { 23 } ) 0 < s

^max

₁ < s

^max

₃ (0, s ₃ , s

^max

₃ − s ₃ ) ( { 1 } , { 23 } )

Balaned regime

Inthisaseooperationisexpliitlyneededtoformawinningoalition,sine

w ₁ < 1/2, w ₃ < 1/2

^,

and

w ₁ + w ₂ > 1/2, w ₃ + w ₂ > 1/2

^. Side-payments determine theoutome of thevoting game.

For easier analysis let

s

^max

₁ = (t ₁ − w ₁ )R

^and

s

^max

₃ = (w ₃ − t ₃ )R

^{be the maximum reasonable}

side-payment possibly oered byPlayer 1 and Player 3, respetively. The imputations and the

ore forany

s

^max

₁ , s

^max

₃

^are:

X( { 1 } , { 23 } ) = { (x 1 , x 2 , x 3 ) ∈ R ³ | x 1 = 0, x 2 ≥ 0, x 3 ≥ 0, x 2 + x 3 = s

^max

₃ } C( { 1 } , { 23 } ) =

( ∅ ,

^if

s

^max

₁ > s

^max

₃

(0, s

^max

₁ + ǫ, s

^max

₃ − s

^max

₁ − ǫ),

^if

s

^max

₃ ≥ s

^max

₁

where

0 ≤ ǫ ≤ s

^max

₃ − s

^max

₁

^. Furthermore:

X( { 12 } , { 3 } ) = { (x ₁ , x ₂ , x ₃ ) ∈ R ³ | x ₁ ≥ 0, x ₂ ≥ 0, x ₃ = 0, x ₁ + x ₂ = s

^max

₁ }

C( { 12 } , { 3 } ) =

( ∅ ,

^if

s

^max

₁ < s

^max

₃

(s

^max

₁ − s

^max

₃ − ǫ, s

^max

₃ + ǫ, 0),

^if

s

^max

₁ ≥ s

^max

₃

where

0 ≤ ǫ ≤ s

^max

₁ − s

^max

₃

^.

Let us rst study theoalitional struture

( { 1 } , { 23 } )

^{. The ore is empty if the maximum}

side-payment of Player 3 is smaller than that of Player 1. This is due to the fat that Player

2 wants to form a oalition with Player 1 and get more money than

s

^max

₃

^{, but the onstraint}

on imputations prevents this. Onthe other hand, if themaximum side-payment of Player 3 is

greaterthan Player1's,than the oreisnon-emptywithPlayer 3(thelightusers)winning, and

thegame

G

^{isbalaned. Player3pays}

s

^max

₁ + ǫ

^{toPlayer2andretains}

s

^max

₃ − s

^max

₁ − ǫ

^{. A similar}

(butopposing) explanation appliesfor the oalitional struture

{{ 12 } , { 3 }}

^.

The solution of the user-inuened priing game is given as

ψ

^{-stable pairs in Table 2.4.}

Note that the

ψ

^{-stable onept does not restrit the} possibilities. In the rst row of the table heavy userswin (at-ratepriingis hosen), buta side-payment of at least

s

^max

₃

^hastobepaid.

Aording to the third row, light users win by paying at least

s

^max

₁

^{to medium users. If the}

maximumside-payments areequal,theoutome is indeterminate.

Now, letustakealookat howindividualusersan sharetheburdenofside-paymentsinthe

balaned regime. Let

H, M, L ⊂ N

^{be the setof heavy,medium andlight users.}

Flat-rate priing. Suppose that

s

^max

₃ < s

^max

₁

^{,hene heavy and medium users team up to}

implement at-rate priing. A suitable division of side-payments among heavy users would be

to sharetheadditionalost equally,resultinginapayment of

s 1

|H|

^{foreahheavy user}

i

^{. Alsoby}

hoosing theat-rate approah, medium users share the prot fromthe side-payments equally,

eah mediumuser getting

s 1

| M |

^.

Now we an give the monthly ost ofa singleuser:

c _i =



 

 



 

 



R

N + _|H| ^{s 1}

^if

i ∈ H

R

N − _|M| ^{s 1}

^if

i ∈ M

R

N

^if

i ∈ L

(2.25)

Usage-based priing. Suppose that

s

^max

₁ < s

^max

₃

^{, therefore light and medium users join}

fores to ahieve usage-based priing. A suitable division of side-payments among light users

would be to sharethe the additionalost proportional totra volume,resulting ina payment

of

s 3 τ i

t 3 T

^{for eahlight user}

i

^{. Alsobyhoosingthe}usage-basedapproah,mediumusersan agree to benetfrom theside-payments proportionally to their tra volume, so eahmedium user j

user gets

s 3 τ j

t 2 T

^.

Now we an give the monthly ost ofa singleuser:

c _i =



 

 



 

 



Rτ i

T

^if

i ∈ H

Rτ i

T − ^s _{t 2} ^{3 τ} _T ⁱ

^if

i ∈ M

Rτ i

T + ^s _t ^{3 τ i}

3 T

^if

i ∈ M

(2.26)

In document B daeUiveiyfTe h (Pldal 56-60)