Transit price negotiation: repeated game approach Sogea 23 Mai 2007 Nancy, France D.Barth, J.Cohen, L.Echabbi and C.Hamlaoui

Interdomain Routing : example 3b 1d 3a 1c 2a AS3 AS1 AS2 1a 2c 2b 1b 3c source 4b 4a 4c AS4 5b 5a 5c AS5 6a AS6 6c 6b 7a AS7 7c 7b Destination

Interdomain routing : BGP AS3 AS1 AS2 source AS4 AS5 AS7 AS6 Shortest Vs cheapest Price Routing informations Destination

Interdomain routing : economic model AS3 Provider1 AS1 AS2 Provider2 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 ∑ prices of AS on the route

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 AS3 p1p1 p3p3 p5p5 p4p4 p2p2

A particular case  1 source, 1 destination, N providers (Identical Quality)  Discret prices, price min = C i, price max = p max  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 Destination Provider 2 Provider N

Bertrand game with two players: equal costs p 1 =p 2 p1p1 p2p2 p max p* 1 = f (p 2 ) p* 2 = f (p 1 )  The only one Nash equilibrium is to propose a price= price min  When costs are different, the lowest cost provider should propose the cost of the other provider minus one in order to get the market p min

Two providers: equal costs (minimum price)  Share the market while maintaining higher prices  Alternate p max 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. ( Optimal strategy based on cooperation p max P max +1Player 2 p max +1p max Player 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 Destination Provider 2Provider N Cost of provider i = c i with c 1 < c 2 < …< c n Provider 1 has to make a choice :  Take all the market by announcing c 2 -1  Share the market with provider 2 by announcing c 3 -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 SourceProvider 1 Destination Provider 2Provider 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 SourceProvider 1 Destination Provider 2Provider 3 x 1 P max =8 Get all the market

General case : sketch idea SourceProvider 1 Destination Provider 2Provider 3 x P max =8  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 Share the market Alternate their announced price

General case : sketch idea SourceProvider 1 Destination Provider 2Provider 3 x 8 8 P max =8 Share the market 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 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 variables N N NNN N NN 1. At the beginning : routes are not established. N O NNN N NN State Update msg OOO O N O NNN N NN OOO O NNN O O O 2. Source chooses acceptable route->state=O Node'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 p i p i +1 else if (p i > p min ) then p i p i -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:United StatesCanadaEUantitrustprice leadership Price fixing and market division among manufacturers of heavy electrical equipment in the 1960s.Price fixing electrical1960s An attempt by Major League Baseball owners to restrict players' salaries in the mid-1980s.Major League Baseballrestrict players' salaries1980s Price fixing within food manufacturers providing cafeteria food to schools and the military in 1993.food schoolsmilitary1993 Market division and output determination of livestock feed additive by companies in the US, Japan and South Korea in 1996.livestockJapanSouth Korea1996