Transit price negotiation: repeated game approach

1. Transit price negotiation: repeated game approach • Sogea 23 Mai 2007 Nancy, France D.Barth, J.Cohen, L.Echabbi andC.Hamlaoui chah@prism.uvsq.fr

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

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

4. P3>P2 The rest of the internet AS3 Provider1 AS2 Provider2 AS1 source Interdomain routing : economic model • Pay the first provider on the selected route • Bilateral nature of economic contracts • Problem: How AS should set their transit prices ? • Game : AS = Players ∑ prices of AS on the route

5. 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)

6. p2 AS2 p1 AS3 p3 AS4 AS5 p4 p5 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

7. Provider 1 Source Destination Provider 2 Provider N 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.

8. p1=p2 pmax p*1= f (p2) p*2= f (p1) pmax pmin pmin • The only one Nash equilibrium is to propose a price= pricemin Bertrand game with two players: equal costs p2 p1 • When costs are different, the lowest cost provider should propose the cost of the other provider minus one in order to get the market

9. Two providers: equal costs (minimum price) Optimal strategy based on cooperation • Share the market while maintaining higher prices • Alternate pmax as in the following table odd stages Player 1 pmax pmax+1 Player 2 Pmax+1 pmax even stages • 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)

10. N providers: different costs Provider 1 Destination Source Provider 2 Provider N 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 Cost of provider i = ci with c1< c2 < …< cn Provider 1 has to make a choice : Provider 1 chooses the best strategy. • 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) • …

11. Provider 1 More powerful to decide the strategy Destination Source Provider 2 Provider N Different disjoint routes: equal costs Price announced by AS i = price paid by AS i to its provider+ transit price of AS i • 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)

12. General case : sketch idea 1 Provider 1 Destination x Pmax=8 Get all the market Source Provider 2 Provider 3

13. General case : sketch idea 6 Provider 1 5 1 Destination x Alternate their announced price Pmax=8 Share the market Source Provider 2 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

14. General case : sketch idea 8 Provider 1 8 Destination x Pmax=8 Share the market Source Provider 2 Provider 3 8 5 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

15. 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 ?

16. Local information at node i 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 Node is informed of all the variables of his neighbors.

17. N N N N N N N N Protocol for communicating state variables 1. At the beginning : routes are not established . State Update msg State Update msg N N N State Update msg 2. Source chooses acceptable route->state=O Node's state is updated when it receives « state update message » O O O O N O N N N State Update msg State Update msg N N N State Update msg O O O 3. Source switch on a new received route -> State of node on new route (better price) is updated iteratively into O O N N N N O State Update msg N N N O O O State Update msg State Update msg

18. 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 To attract trafic Provider that have transit trafic increase price To reach the maximum possible benefit

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

20. 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).

21. 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

22. 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?

23. 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.