transit price negotiation: repeated game approach sogea 23 mai 2007 nancy, france d.barth, j.cohen,...
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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.
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