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“Attractors and Repellors: a Game Theoretic interpretation of the cycles in provider-

customer relations”

Advanced Workshop on Internet Topology and Economics Atlanta 12-14 November 2012

Emanuele Giovannetti, Institute for International Management Practice, Anglia

Ruskin University, Cambridge UK Joint with Alessio D’Ignazio

Bank of Italy

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Motivation• Empirical Evidence (Dhamdhere and Dovrolis (2011) shows the presence

of cycles of attractiveness and repellence between in Provider to Customer relations.

• Regular cycles, expressed through the breaking up and reformation of BGP links.(Maybe changes between primary and back links)

• Dhamdhere and Dovrolis (2011) show that, for many providers, strong attractiveness precedes strong repulsiveness by 3-9 months.

• We propose a repeated games interconnection model with asymmetric information, that, in equilibrium, reproduces similar cyclical features.

• We perform a duration analysis, to test through the BGP data, the empirical implications of the model.

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Outline

1. The Static LADU (Link- Abuse-Dewire-Use) Customer- Provider’s Game : disconnection

2. The repeated LADU game under symmetric information: permanent connection

3. Asymmetric information and the temporary dewiring strategy DBT “Dewire with Probability B for T periods” cycles of connectivity

4. The Incentive Compatibility Constraint for the Customer, the role of1. the probability of Breaking up, 2. the duration of the dewiring cycles, and 3. the discount rate.

5. Validation: Duration Analysis from the CAIDA’s Data.

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The The Static LADU (Link- Abuse-Dewire-Use) Customer- Provider’s Game

• P has two period actions: Link (L) or De-wire (D) with C and C’s actions are to Use (U) or Abuse (A ) the link.

• P prefers to link (L) if C does not abuse (U), as it receives access revenues for the agreed capacity K ( charges at 95% centile Bill Norton?Nick this does not consider the cost of different destinations) at a (Blended Rate pricing ) MBps price a.

• C prefers to abuse (A) the link, as this is not observed with certainty, so that the charge is the same aK, but the benefits for the customer is higher.

• If C abuses the link, P prefers to dewire, as the costs are now greater than the benefits from access revenue, aK.

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),(,),(,),(,),(, ALPADPUDPULP ),(,),(,),(,),(, ADCUDCULCALC

Provider\Customer Use (U) Abuse (A)

Link (L) ),(, ULP ),(, ULC

),(, ALP ),(, ALC

Dewrire (D) ),( UDP ),(, UDC

),(, ADP ),(, ADC

The Static LADU (Link- Abuse-Dewire-Use) Customer- Provider Game: inefficiency of the NE

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The repeated LADU under symmetric information: over-cooperation.

1

),(,),(,,

ULCULC

UCICV

0 assuming1

),(,

),(,),(,

),(,,

ADC

ALCADC

ALCA

CICV

• Perfect observability of the customer’s abuse could make cooperation sustainable as an Equilibrium of the infinitely repeated game if the players are sufficiently patient.

• This can be achieved via trigger strategies where• “The provider will start by playing L, and in any

future period it will keep playing L unless in the previous history the customer, played A, in this case P will play D forever”

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Patience, bounded incentives and Cooperation

• The critical discount rate after which cooperation and stable interconnection are observed in equilibrium is a function of the increase in period profits obtained from abusing the link instead than using it.

• Under complete information we should be able to observe permanent and stable interconnection decisions

• but only when the gap in customer’s period profits is not too large.

),(,

),(,),(,

),(,),(,

, .. 1

ALC

ULCALC

ALCULC

CIC eiIC

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Motivation for asymmetric information

• Asymmetric information allows C to abuse the link without P been able to observe this.

• Examples: a random variable observable by P, its profits, revenues, or total traffic, is a function both of an exogenous random shock and of C’s unobserved behaviour, and P is unable to distinguish this.

• If P is never punishing C, then C will abuse the link. • If the Provider decides to punish forever the

customer, then there is no scope for cooperation.

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Need for temporary punishment along the equilibrium path,

• Literature on price wars and on repeated agency.• In each period the provider, P, observes the

realization of its total carried traffic without being able to distinguish between its own and the customer’s traffic components due to uncertainty.(Or the final destination of the customer’s traffic)

• C’s incentives towards deviations require, if P wants to achieve cooperation, to formulate a punishment strategy that will satisfy the customer’s ICC to cooperate.

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The DBT strategy “Dewire with Probability B for T periods”

• Consider “The dewire with probability B for T periods (DBT) strategy” for P:

• “Start by playing the cooperative strategy, L, then switch from L to D, with probability B, If so, play D for T of periods and then revert to L”

• Given the strategy DBT the critical question is:• whether this is sustainable in equilibrium and, if yes, to

determine the appropriate length for the punishment phases to ensure that deviations do not occur.

• The outcome of this strategy produces cycles of wiring and dewiring.

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Customer’s Value from cooperating

• The expected discounted payoffs for C from cooperating, (U ), taking into account that cooperation may break down with probability B, at every date is:

CCTCCULC

CC

UDC

CC

TUDC

TCCULC

CC

VBVBV

VBVBV

i

ii

1**),(,

),(,

1

),(,**

),(,

1

0 that assumingor

...1

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• The deviation Value, Vd ,induced by the DBT strategy following a customer deviation is such that – she will first obtain once the period payoff from

abusing the link, πC (L,A) – this will be followed by T periods of zero

punishment payoffs, from being dewired,– and, starting from the T+1th period, P, will revert

to L, she will receive again the value VC.

cT

ALCd VV 1),(,

Customer’s Value from deviating

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Equilibrium from the customer ICC

• Proposition 1• The DBT strategy is an equilibrium for the

infinitely repeated game of Table (1) if•

• This shows that there is a upper limit for the value of the probability of starting the dewiring phase, B, for the strategy to be an equilibrium.

),(,),(,

),(,),(,1 11

ULCALC

ULCALCT

B

B

and

),(,

),(,

ALC

ULCB

(4)

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The length of the punishment Phase

111

lnln

11 ),(,),(,

),(,),(,

BB

T ALCULCALCULC

We can now focus on solving the ICC for the required time length of the punishment phase. Proposition 2 When the strategy DBT is an equilibrium for the infinitely repeated game of Table (1) Cooperation resumes after T+1 periods where:

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The period incentive to deviate

• In the following we focus on the effects of a change in the customer’s period payoffs from deviation, πC (L,A) on the duration, of the punishment period.

• Proposition 3 • If B< (πC (L,U) / πC (L,A),) the length of the

punishment phase is increasing in the customer’s period profits obtained from abusing the link: πC (L,A)

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Probability of breaking up, length of the punishment phase and type-gap

• The DBT strategy proposed in model has two main elements:

1. The trigger probability of starting a dewiring phase, B.

2. The duration of the punishment phase.

Both these features are affected by the customer’s period incentives to deviate, incentives also usually associated to the size gap.

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Estimate of the Kaplan-Meier curve for the connectivity duration.

• We first focus on the probability of the survival times of the connection for different classes of Ases type gaps,

• We consider three possible types of gaps between ASes.

• These three gaps are defined based on the CAIDA classification of Ases into four different categories.

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Figure 1 The probability of connection survival, conditional to the existing type-gap among the provider and customer.

CAIDA: 1 = Enterprise Customer2 = Small Transit Provider3 = Large Transit Provider4 = Content/Access/Hosting Provider

The green curve plotting the Kaplan-Meier for links between ASes having the higher gap in size lies below (for t<30), the two other curves considered, the red one and the blue one.Hence the probability of breaking up an existing link cumulatively increases with the gap size.

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Probability of links survival for different durations.

• Figure 1 shows three curves plotting the time profile of the probability of links survival for different durations.

• If the Ases’ gap-type is a good proxy for the size of the period incentive to deviate, πC (L,A) given πC (L,U). (If these incentives are larger for the small providers) the strategy punishes more frequently, when this is larger.

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The probability of the length of punishment and type gap

• The next Fig 2 shows the probability of the length of punishment (disconnectivity) again as a function of the typegap difference between the ASes.

• The next graph shows that, the average duration of the punishment period is shorter for type 3 gap .

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Duration of the punishment phase as a function of the type gap between two providers.

The connections with the highest gap require the shortest punishment duration, followed by those with the smallest gap (typegap=1. blue line) and finally the punishment of the =intermediate type (typegap=2) require the longest punishment phases. It emerges a U shaped relation between asymmetry and duration profile of the punishment.

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Extensions

• This last results requires further interpretation given its nonlinear nature. Better metrics should be used as asymmetry in customer cone..

• Refine the Econometric analysis • Using differences in size of traffic in data available• Or differences in customer cones if not…• Assessing the amount of asymmetric information, for example

measuring local complexity metrics of interconnection• Proxies for the discount rate: the probability of AS exiting the

market.• Thanks ..