[email protected] nobel wp2 2004. szept. 2-3. stockholm game theory in inter-domain routing lÓja...

NOBEL WP2 2004. Szept. 2-3. [email protected]

Game Theory in Inter-domain Routing

LÓJA Krisztina - SZIGETI János - CINKLER Tibor

BME TMIT

Budapest, Hungary


Contribution to D15

• The research is a part of WP2 A.2.3

• Contributed to D15 Chapter 6.5

The Influence of Advertised Costs / Weights on the Utilization and Incomes of Operators

– Multiprovider network – competition


Game theory: decision theory with at least two players where the decision of a players has an effect on the payoff of other players.

Non-cooperative game theory: the players can not make binding agreements.

What we need for a game theoretic model:

-the set of players

-the set of strategies

-the payoffs (that depends on all the decision)

-the information they have when they decide

Game Theory – Overview


Mapping of Game Theory Objects

Player

Domain

Strategy

Connection

Cost Multiplier

2.0×

Payoff

Income /

Revenue

Information

Various

Information

connection


Inter-domain Routing

There are more domains (Autonomous Systems, AS)

•The traffic goes through them

•They decide the price and get a profit

•The traffic travels on the cheapest route

•The payoff of a domain depends on

the decision of other domains as well.

AS 1

AS 2 AS 3


Game Theoretic Model for Inter-domain Routing

Players: domains (autonomous systems)

Decision: price of the advertised connections

Set of strategies: the integers or the real numbers or an interval if the price has an upper and a lower bound or a set of prices

Payoff: the profit of the domains, the price multiplied with the congestion

Information: the players know the network and in our first models they also know the traffic demand


Path Changing Situation

S T1 1

1

1

1

3

33

Path A Cost: 5Path B Cost: 9

1.S T

1 11

1

1

4

33


2.S T

1 11

1

1

5

33


3.

S T1 1

1

1

1

6

33


4.S T

1 11

1

1

7

33


5.S T

1 11

1

1

8

33


6.


We can determine for the game:

- global optimum (when the summarized payoff of all the players is maximal)

- Nash equilibrium (the stable point of the game, when no player has an incentive to deviate unilaterally, it is not always optimal)

- Pareto optimum (when the players cannot change strategies so that no player will have lower payoff and at least one of them higher payoff)

What can we do?


We can determine for every domain:

- the maximal payoff

- the minimal payoff

- the reservation cost (the payoff that a domain can guarantee for itself, the maximum of the minimal payoffs)


Example 1

AS 1

AS 2 AS 3

S 1

S 2S 3

T 1

T 2 T 3

Let we assume that

•all the connections in a domain have the same price

•one unit of traffic is transferred from Si to Ti (i=1,2,3)

•the traffic travels on the cheapest route, if it is not unique, the traffic is divided uniquely

•the price is a positive integer not higher than an upper bound, the payoff = price *congestion

•it is a one-shot game and the players decide simultaneously Global optimum: the price is the highest possible price in every domain

Reservation cost can be reached when the price is the highest

Pareto optimum: the price is the highest possible price in every domain

Although it is not a Nash equilibrium, if a domain unilaterally deviates

and wants lower price, it gets higher profit


AS 1

AS 2 AS 3

Let we assume, that the prices of different connections of the domain can be different, and there is an upper bound on every path, if the price is higher, the users will not use that path.

In this case there are more Nash equilibria, all of them are Pareto optimal but in the case of simultaneous decisions the players cannot choose the same Nash equilibrium without previous discussion.

Example 2


AS 1

AS 2 AS 3

S 1

S 2S 3

Let we assume that the users do not use a path if its price is higher than an upper bound, and the connections have an upper bound in the capacities.

The global optimum is not Nash equilibrium. It would be a Nash equilibrium, however, if we could assume that a traffic do not choose the cheaper path if it is longer.

Nash equilibrium is useful when we model dynamic pricing with repeated games.

Example 3


Possibilities of refining the model

- many domains, many source-destination pairs

- instead of one-shot game we can use repeated games

- we can take into consideration:

•the dependence of the traffic demand on the price of the route

•the finite capacities of the connections

•that the domains do not know the traffic demands

Simulations will help us when the model is too difficult.


Simulation

•Network topology:•3 domains•3 sources•6 wavelengths

•Traffic:•Symmetric between sources•Average load is 4 (66%)

•Strategy:•The domain calculates the allocation cost for each advertised connection and multiplies it by its constant factor {1.0, 1.2, 1.4, 1.6, 1.8, 2.0}, this means we have 6×6×6 = 216 scenarios

A

BC

S1S2

S3


Results for Domain A

11.2

1.41.6

1.82

1

1.2

1.4

1.61.8

20

200

400

600

800

1000

1200

C multiplier

B multiplier

Income of domain A

11.2

1.41.6

1.82

1

1.2

1.4

1.61.8

20

200

400

600

800

1000

1200

1400

1600

C multiplier

B multiplier

Income of domain A

11.2

1.41.6

1.82

1

1.2

1.4

1.61.8

20

200

400

600

800

1000

1200

1400

1600

1800

C multiplier

B multiplier

Income of domain A

11.2

1.41.6

1.82

11.2

1.41.6

1.82 0

1

2

3

4

5

6

C multiplier

B multiplier

Unserved demandsA multiplier: 1.0 A multiplier: 1.4 A multiplier: 2.0

11.2

1.41.6

1.82

1

1.2

1.4

1.6

1.82

0

0.5

1

1.5

2

2.5

3

3.5

4

C multiplier

B multiplier

Unserved demands

11.2

1.41.6

1.82

1

1.2

1.4

1.6

1.82

0

1

2

3

4

5

6

C multiplier

B multiplier

Unserved demands


Results for Domain B

11.2

1.41.6

1.82

11.2

1.41.6

1.82 0

1

2

3

4

5

6

C multiplier

B multiplier


11.2

1.41.6

1.82

1

1.2

1.4

1.6

1.82

0

0.5

1

1.5

2

2.5

3

3.5

4

C multiplier

B multiplier

Unserved demands

11.2

1.41.6

1.82

1

1.2

1.4

1.6

1.82

0

1

2

3

4

5

6

C multiplier

B multiplier

Unserved demands

11.2

1.41.6

1.82

1

1.2

1.4

1.61.8

20

200

400

600

800

1000

1200

1400

C multiplier

B multiplier

Income of domain B

11.2

1.41.6

1.82

1

1.2

1.4

1.6

1.82

0

200

400

600

800

1000

1200

1400

C multiplier

B multiplier

Income of domain B

11.2

1.41.6

1.82

1

1.2

1.4

1.6

1.82

0

200

400

600

800

1000

1200

1400

1600

C multiplier

B multiplier

Income of domain B


Results for Domain C

11.2

1.41.6

1.82

11.2

1.41.6

1.82 0

1

2

3

4

5

6

C multiplier

B multiplier


11.2

1.41.6

1.82

1

1.2

1.4

1.6

1.82

0

0.5

1

1.5

2

2.5

3

3.5

4

C multiplier

B multiplier

Unserved demands

11.2

1.41.6

1.82

1

1.2

1.4

1.6

1.82

0

1

2

3

4

5

6

C multiplier

B multiplier

Unserved demands

11.2

1.41.6

1.82

B: 1.0B: 1.2

B:1.4B:1.6

B:1.8B:2.0

0

200

400

600

800

1000

1200

1400

C multiplierB multiplier

Income of domain C

11.2

1.41.6

1.82

1

1.2

1.4

1.6

1.82

0

200

400

600

800

1000

1200

1400

C multiplier

B multiplier

Income of domain C

11.2

1.41.6

1.82

1

1.2

1.4

1.6

1.82

0

200

400

600

800

1000

1200

1400

1600

1800

C multiplier

B multiplier

Income of domain C


Conclusions

• Small variance of the advertised connection cost may have a great impact on the traffic routing (significant steps in the revenue figure)

• Even in our small test there are results that require explanation

• The simulation results depend on the traffic pattern (order of demands) – various patterns and finding average is needed


Thank you for your attention!

[email protected] nobel wp2 2004. szept. 2-3. stockholm game theory in inter-domain routing lÓja...

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