Inentive to ooperate

Herewepresentasingle-shotgameofuserloyaltywhihwasintroduedin[11℄. Later,weextend

thisgametoaninnitelyrepeatedgame,andshowhowaooperativemaximumanbeenfored,

where the long-term prot of ISPs are higher than that of playing the equilibrium strategy of

thestage gamein eah round.

The stage game

Consider a market with two loal ISPs ompeting in pries for a xed number of ustomers.

Customers aresplit into three partitionsupon their brand loyalty: the rst group onsistsof

l ₁

ustomerswho areallloyal to ISP

1

^{inthe sense thatifISP}

1

^'sprie

p ₁

^{islessthan or equal toa}

reservationvalue

α

^{, they hoose ISP}

₁

^{astheir servieprovider, otherwisethey do not purhase}

Internet aess. The seond grouponsists of

l ₂

^{loyalustomers of ISP}

₂

^{,while the third group}

ontains

n

^{swithers,whobuyserviefromtheheapestprovider,ifitsprieisnotgreaterthan}

α

^{. Ifthe providersannoune thesame prie}

(p 1 = p 2 < α)

^{,thenhalfof theswithers hooses}

ISP

1

^{and the other half hooses ISP}

2

^{. The ow of thegame is that ISPs announe their pries}

simultaneously,thenustomersmaketheir hoies. Thisgameisreferred toas

G ₀

^.

Note, that though values

l ₁ > 0

^,

l ₂ > 0

^and

α > 0

^{are ommon knowledge, group}

mem-bership of a given ustomer annot bedetermined, so there is no prie disrimination possible.

Furthermore, forsimpliitywe assumeaonstant unitostofzeroforbothrms,and thatISP

1

hasthelargerloyal userbase,

l ₁ > l ₂

^.

Given the above andthat

p ₁ ≤ α

^and

p ₂ ≤ α

^,ISP

₁

^{'s payoan beexpressedas}

π ₁ (p ₁ , p ₂ ) =



 



 



(l ₁ + n)p ₁ p ₁ < p ₂ (l 1 + 0.5n)p 1 p 1 = p 2

l ₁ p ₁ p ₁ > p ₂

(2.1)

Itanbeshown(see[11℄and[12℄)thatthisgamehasauniqueNashequilibriuminmixed

strate-gies. In this ase, equilibrium prots are

π 1 = l 1 α

^and

π 2 = ^l _l ² ₁ ⁺ⁿ _+n l 1 α.

^{Asit an be notied, the}

equilibriumhasshiftedompared tothe simpleBertrandgamewithoutonsumerloyalty(whih

is (0,0)inase ofzero produtionosts), both partieshaving apositivepayoinequilibrium.

The repeated game

Now, we extend the previous model, and show that the innitely repeated

G ₀

^{hasa sub-game}

perfet equilibrium, whih an be enfored by a threat strategy, namely the Nash equilibrium

strategy of the stagegame

G ₀

^.

In the following we onstrut

G _r

^{as the innitely repeated extension of}

G ₀

^{. Payo is}

dis-ounted at step

k

^{with a disount fator}

Θ < 1

^{. The game is ontinuous at innity sine the}

disountedpayoinanystepisboundedby

α(l ₁ +n)

^{. Thiswayweanusetheone-stepdeviation}

priniple to prove sub-gameperfetionof agiven strategy set.

Now,ifthetwoprovidersooperateandsettheirpriesequaltothereservationvalue

α

^,they

willshare"swithers"equally,inadditiontokeepingtheirownloyalusers. Thiswaytheirpayos

(

π

^{oop)wouldbehigherthaninthe}equilibriumase(

π

^eq),sine

π ₁

^oop

= (l ₁ +0.5n)α > π ₁

^eq

= l ₁ α

^,

and

π

^oop

₂ = (l ₂ + 0.5n)α > π ₂

^eq

= ^l _l ^{2 +n}

1 +n l ₁ α

^if

n > l ₁ − 2l ₂

^{. Intheooperativeasethejoint prot}

of thetwo ISPsis themaximumahievable

(n + l 1 + l 2 )α

^{. This ooperation is highlybeneial}

for both parties. Ifsomehow one ISPtries to grab thewholefree marketina singlestep

k

^,the

otherISPan ounterat fromstep

k + 1

^{byhargingtheNash}equilibriumpriefrom

G ₀

^further

on,whihresults ina dereasedpayofor thetraitor. We showthatthisNashreversionassures

sub-game perfetion forthe following strategy prole under thestated onditions.

Proposition 1 The strategy prole Cooperate until the other player deviates and then play

aordingto theequilibrium in

G ₀

^{isa sub-gameperfet Nash} equilibrium for therepeated game

G _r

^{, if}

n > l ₁ − 2l ₂

^and

Θ > ¹ ₂ + ^{l 1 −}

n+l 1 n+l 2 l 2

2n

^.

Proof: A strategy prole issub-gameperfet,ifthefollowing holds:

π _i

^nodev

(k, ∞ ) > π _i

^dev

(k) + π _i

^dev

(k + 1, ∞ ),

^(2.2)

meaning that the sum prot is greater if there is no deviation from theagreed strategy. If we

assumethat ISP

2

^{deviates, thistranslates to}

X ∞

i=k

Θ ⁱ π

^oop

₂ > Θ ^k π ₂

^dev

+ X ∞

i=k+1

Θ ⁱ π ₂

^eq

.

^(2.3)

After solving (2.3)for

Θ

^weget

Θ > π

^dev

₂ − π

^oop

₂

π ₂

^dev

− π ₂

^eq

.

^(2.4)

The one-stepdeviation at step

k

^{isrealized by} underuttingtheother ISPbya marginal

ǫ > 0

^,

and harging a prie of

α − ǫ

^{to the users. This way}

π

^dev

₂ = (l ₂ + n)(α − ǫ)

^{. Further on, we}

substitutedierent payos for ISP

2

^{,andsimplify the} expression:

Θ ₂ > 1

2 + l ₁ − ^n+l _n+l ¹ ₂ l ₂

2n .

^(2.5)

If ISP

1

^{deviates, by following the same steps we get}

Θ ₁ > ¹ ₂

^{. Sine}

l ₁ > l ₂

^{by denition, also}

Θ ₂ > Θ ₁

^{,soiftheatual}

Θ > ¹ ₂ + ^{l 1 −}

n+l 1 n+l 2 l 2

2n

^,the proposition holds.

The optimal deision of ISPs lies in evaluating the assumption inequalities. For Player 1

(larger loyal userbase) it isalways better to ooperateand try to ahieve sub-game perfetion.

For Player 2it is a matterof loyal user basesize: if

n > l ₁ − 2l ₂

^{, Player 2will alsoooperate.}

If not, thenshe will play the

G ₀

equilibrium strategy,while Player 1 will play theooperative strategy for one round. Then from round 2(beause of Nash reversion), player 1 will also play

the

G ₀

equilibriumstrategy.

While expliit ooperation may be illegal, this inentive may lead to disussions between

servie providers. Note, that a two-ISP setting may seem artiial, it is ertainly not, e.g., a

large fration of Internet users in the US an only hoose between the loal able and phone

ompany. Moreover, there is some speulation about a artel-like ooperation among large

players in the US Internet market [43℄. Wu mentions that in the United States and in most

of the world, a monopoly or duopoly ontrols the pipes that supply homes with information.

These ompanies, primarily phone and able ompanies, have a natural interest in ontrolling

supply to maintain prie levels and extrat maximum prot from their investments, similar to

how OPEC sets prodution quotas to guarantee high pries. While this phenomena is rooted

innetneutralityandprotetinginfrastrutural investment,sine itinvolvessettingmanipulated

In document B daeUiveiyfTe h (Pldal 32-35)