Transit price negotiation:
repeated game approach
Sogea 23 Mai 2007 Nancy, France
D.Barth, J.Cohen, L.Echabbi and
C.Hamlaoui
[email protected]
Interdomain Routing : example
7c Destination
7a
7b
AS7
4c
4a
4b
AS4
6c
6a
AS6
5c
5a
5b
AS5
6b
3c
3a
3b
AS3
1a
2a
1c
1d
1b
source
2c
AS2
AS1
2b
Interdomain routing : BGP
Shortest Vs cheapest
Price
AS7
Destination
Routing informations
AS4
AS5
AS6
AS3
AS2
AS1
source
Interdomain routing : economic model
Pay the first provider
on the selected route

The rest of the internet
Bilateral nature of
economic contracts

Problem:
How AS should set
their transit prices ?


Game : AS = Players
AS3
Provider1
P3>P2
AS1
source
AS2
Provider2
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
p2
AS2
p1
AS3
p3
AS 1
AS5
p5
AS4
p4
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.
Provider 1
Source
Provider 2
Provider N
Destination
Bertrand game with two players: equal costs
p2
p1=p2
pmax
p*1= f (p2)
p*2= f (p1)
pmin
pmin

pmax
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

Two providers: equal costs (minimum price)
Optimal strategy based on cooperation

Share the market while maintaining higher prices
 Alternate p
max as in the following table
odd stages
even stages
Player 1
pmax
pmax+1
Player 2
Pmax+1
pmax
If one player deviates then the other one punishes him by
indefinitely playing the NE i.e announcing c

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)
N providers: different costs
Provider 1
Source
Provider 2
Destination
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 c -1
3
each 2 stages (we talk about coalition with provider 2)
 …

Provider 1
chooses the
best strategy.
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
Different disjoint routes: equal costs
Price announced by AS i = price paid by AS i to its provider+ transit
price of AS i
Provider 1
Source
Provider 2
More powerful to
decide the strategy
Destination
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)

General case : sketch idea
Get all the market
1
Provider 1
Pmax=8
Source
x
Provider 2
Provider 3
Destination
General case : sketch idea
Share the market
Alternate their
announced price
6
Provider 1
Pmax=8
Source
x
1
5
Provider 2
Destination
Provider 3
Why 6?
3rd route can not be proposed at this price
Provider 1 will gets 6 each 2 steps -> more interesting then to get all
the market with benefit = 1

General case : sketch idea
Share the market
8
8
Provider 1
8
Pmax=8
x
Source
Provider 2
Destination
5
3
Provider 3
Compute successive coalitions as long as that does not call into
question the preceding coalitions
The average benefit of each node is maximum considering the
strategy chosen by each node more powerful then him

Dynamic distributed
game
 Nodes have local view of game
 Price announcing follows an asynchronous model
Objectives :
 Stabilizing behaviour of the distributed system ?
 Whether theoretical results match results in distributed
framework ?
Distributed algorithmic model
Local information at node i
 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
Node is informed of all the variables of his neighbors.
Protocol for communicating state variables
N
N
N
N
1. At the beginning : routes are
not established .
N
N
N
N
State Update msg
2. Source chooses acceptable
route->state=O
Node's state is updated when
it receives « state update
message »
State Update msg
N
O
O
N
N
O
N
State Update msg
3. Source switch on a new
received route ->
State of node on new route
(better price) is updated
iteratively into O
State Update msg
N
O
O
N
State Update msg
N
O
State Update msg
N
O
N
N
O
State Update msg
N
O
N
N
O
State Update msg
N
O
N
N
O
State Update msg
N
O
Price adjustment strategy
Can some specific local strategies lead to a similar state that the one
expected by theoretical analysis ?
if state (i) = O then pi
pi+1
else if (pi > pmin) then
pi
pi-1
Intuition:
Provider with no transit
trafic decrease price
Provider that have
transit trafic increase
price
To attract trafic
To reach the maximum
possible benefit
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.