Transit price negotiation: repeated game approach

Sogea 23 Mai 2007 Nancy, France

D.Barth, J.Cohen, L.Echabbi and C.Hamlaouichah@prism.uvsq.fr

Interdomain Routing : example

3b

1d

3a

1c2aAS3

AS1

AS21a

2c2b

1b

3c

source

4b4a

4c

AS45b

5a5c

AS56a

AS6

6c6b

7a

AS7

7c7bDestination

Interdomain routing : BGP

AS3

AS1

AS2

source

AS4 AS5

AS7

AS6

Shortest Vs cheapest

Price Routing informations Destination

Interdomain routing : economic model

AS3Provider1

AS1

AS2Provider2

source

The rest of the internet

Pay the first provider on the selected route

Bilateral nature of economic contracts

Problem:

How AS should set

their transit prices ?

Game : AS =

Players

P3>P2

∑ p

rice

s of

AS

on th

e ro

ute

Definitions

Nash equilibrium of a game : is a choice of strategies by the player where each player’s strategy is the best response to other’s strategies.

Subgame perfect equilibrium : the player strategies represent a Nash equilibrium in each subgame (given any history of the game given by past plays, the adopted strategies still represent a nash equilibrium trough the rest of the game)

Mathematical model

The network is given by a graph where the nodes are the AS.

Constant per packet price proposed by each node

No traffic splitting

AS 1

AS2

AS5 AS4

AS3p1

p3

p5p4

p2

A particular case 1 source , 1 destination , N providers (Identical Quality) Discret prices, pricemin = Ci, pricemax = pmax

Game with complete information (AS is aware of the game history)

Repeated game: step = all providers announcing price + source choosing the cheapest provider.

Source can switch from a provider to another (cheapest route)

Provider objective : to maximize benefit.

Source

Provider 1

DestinationProvider 2

Provider N

Bertrand game with two players: equal costs

p1=p2

p1

p2

pmax

pmax p*1= f (p2)p*2= f (p1)

The only one Nash equilibrium is to propose a price= pricemin

When costs are different, the lowest cost provider should propose the cost of the other provider minus one in order to get the market

pmin

pmin

Two providers: equal costs (minimum price)

Share the market while maintaining higher prices Alternate pmax as in the following table

This strategy is proved to be a subgame perfect equilibrium (due to the one deviation principle). Intuition --> If the game have a long duration, punishment will introduce lower benefit. (http://wwwex.prism.uvsq.fr/rapports/2006/document_2006_104.pdf)

Optimal strategy based on cooperation

pmaxPmax+1Player 2

pmax+1pmaxPlayer 1

odd stages

If one player deviates then the other one punishes him by indefinitely playing the NE i.e announcing c

even stages

N providers: different costs

Source

Provider 1

DestinationProvider 2

Provider N

Cost of provider i = ci with c1< c2 < …< cn

Provider 1 has to make a choice :

Take all the market by announcing c2-1 Share the market with provider 2 by announcing c3-1 each 2 stages (we talk about coalition with provider 2) …

We prove that the other providers have an incentive to match provider 1 optimal strategy and thus form a coalition in order to share the market

Provider 1 chooses the best strategy.

Different disjoint routes: equal costs

Source

Provider 1

DestinationProvider 2

Provider N

Ultimatum game between providers on the same route : direct providers propose a route at price they want. (set the max price such that they attract source and predecessor remain interested)

Bertrand game with different costs between the different routes where the cost of provider is the length of the path from him to the destination

The same analysis used in simple model: The shortest path is the most interesting route ( it can be proposed at the minimum possible price)

Price announced by AS i = price paid by AS i to its provider+ transit price of AS i

More powerful to decide the strategy

General case : sketch idea

Source

Provider 1

DestinationProvider 2

Provider 3

x

1

Pmax=8

Get all the market

General case : sketch idea

Source

Provider 1

DestinationProvider 2

Provider 3

x5

6

1Pmax=8

Why 6?3rd route can not be proposed at this priceProvider 1 will gets 6 each 2 steps -> more interesting then to get all the market with benefit = 1

Share the market Alternate their announced price

General case : sketch idea

Source

Provider 1

DestinationProvider 2

Provider 3

x

8

8Pmax=8

Share the market

Compute successive coalitions as long as that does not call into question the preceding coalitionsThe average benefit of each node is maximum considering the strategy chosen by each node more powerful then him

5

8

3

Dynamic distributed game

Objectives :

Stabilizing behaviour of the distributed system ?

Whether theoretical results match results in distributed framework ?

Nodes have local view of game

Price announcing follows an asynchronous model

Distributed algorithmic model

Pi: local price per unit of traffic.

Provider(i) : One of node's neighbors that can reach destination . Proposes the best route. (cheapest route)

State(i): O node is crossed by transit traffic N otherwise

Local information at node i

Node is informed of all the variables of his neighbors.

Protocol for communicating state variablesN

N

N N N

N

NN

1. At the beginning : routes are not established .

N

O

N N N

N

NNState Update msg

O O O

O

State Update msg State Update msg

N

O

N N N

N

NNState Update msg

O O O

O

State Update msg State Update msg

N N N

OOO

State Update msg

State Update msgState Update msg

2. Source chooses acceptable route->state=ONode's state is updated when it receives « state update message »

3. Source switch on a new received route -> State of node on new route (better price) is updated iteratively into O

if state (i) = O then pi pi+1 else if (pi > pmin) then pi pi-1

Provider with no transit trafic decrease price

Provider that have transit trafic increase price

To attract trafic

To reach the maximum possible benefit

Price adjustment strategy

Intuition:

Can some specific local strategies lead to a similar state that the one expected by theoretical analysis ?

Simulation analysis

• Omnet simulator (discrete event simulator ) .• Different topologies.• Same propagation delay .• Neither queueing nor scheduling delay are considered.• Same stage game duration.

Simulation analysis

Direct provider start with pmax

Simulation results:

•When transit price starts from

pmax, prices are adjusted until t

= 150 ms where routes

proposed to the source become

acceptable•Coalition between providers(41 and 44 share the market at high price).

Simulation analysis

Direct provider 41 starts with pmax.Direct provider 44 starts with price=1

Simulation results:•When one provider choose to start with price< pmax, then he takes the market during few step.•Prices are adjusted until a situation where both routes share the market.•Benefit when starting with

pmax is better

Conclusion

Strategy allows providers to maintain average transit price highest possible. Generalized strategy to a more complex situation (In progress)

Strategy lead to a flip flop routing interesting issues is to investigate How can we avoid such behaviour?

Collusion is largely illegal in the United States (as well as Canada and most of the EU) due to antitrust law, but implicit collusion in the form of price leadership and tacit understandings still takes place. Several recent examples of collusion in the United States include:

• Price fixing and market division among manufacturers of heavy electrical equipment in the 1960s.

• An attempt by Major League Baseball owners to restrict players' salaries in the mid-1980s.

• Price fixing within food manufacturers providing cafeteria food to schools and the military in 1993.

• Market division and output determination of livestock feed additive by companies in the US, Japan and South Korea in 1996.