Achieving Network Optima Using Stackelberg Routing Strategies

1. Achieving Network Optima Using Stackelberg Routing Strategies Yannis A. Korilis, Member, IEEE Aurel A. Lazar, Fellow, IEEE & Ariel Orda, Member IEEE IEEE/ACM transactions on networking, vol. 5, No. 1, February 1997 Sanjeev Kohli EE 228A

2. Presentation Outline Introduction to non cooperative networks Overview of approach Model and Problem Formulation Non cooperative User & Manager Single Follower Stackelberg Routing game Multi Follower Stackelberg Routing game Issues

3. Non-cooperative Networks

4. Non-cooperative Networks Users take control decisions individually to max own performance

5. Non-cooperative Networks Users take control decisions individually to max own performance Similar to non cooperative games

6. Non-cooperative Networks Users take control decisions individually to max own performance Similar to non cooperative games Operating points of such networks are determined by Nash equilibria

7. Non-cooperative Networks Users take control decisions individually to max own performance Similar to non cooperative games Operating points of such networks are determined by Nash equilibria Nash Equilibria – Unilateral deviation doesn’t help any user

8. Non-cooperative Networks Users take control decisions individually to max own performance Similar to non cooperative games Operating points of such networks are determined by Nash equilibria Nash Equilibria – Unilateral deviation doesn’t help any user Inefficient, leads to sub optimal performance

9. Non-cooperative Networks Users take control decisions individually to max own performance Similar to non cooperative games Operating points of such networks are determined by Nash equilibria Nash Equilibria – Unilateral deviation doesn’t help any user Inefficient, leads to sub optimal performance Better solution needed !

10. Network Manager

11. Network Manager Architects the n/w to achieve efficient equilibria

12. Network Manager Architects the n/w to achieve efficient equilibria Run time phase

13. Network Manager Architects the n/w to achieve efficient equilibria Run time phase Awareness of users behavior

14. Network Manager Architects the n/w to achieve efficient equilibria Run time phase Awareness of users behavior Aims to improve overall system performance through maximally efficient strategies

15. Network Manager Architects the n/w to achieve efficient equilibria Run time phase Awareness of users behavior Aims to improve overall system performance through maximally efficient strategies Maximally efficient strategy Optimizes overall performance

16. Network Manager Architects the n/w to achieve efficient equilibria Run time phase Awareness of users behavior Aims to improve overall system performance through maximally efficient strategies Maximally efficient strategy Optimizes overall performance Individual users are well off at this operating point [Pareto Efficient]

17. Presentation Outline Introduction to non cooperative networks Overview of approach Model and Problem Formulation Non cooperative User & Manager Single Follower Stackelberg Routing game Multi Follower Stackelberg Routing game Issues

18. Overview of this approach

19. Overview of this approach Total flow: Flow of users + Flow of manager

20. Overview of this approach Total flow: Flow of users + Flow of manager Example of manager’s flow Traffic generated by signaling/control mechanism

21. Overview of this approach Total flow: Flow of users + Flow of manager Example of manager’s flow Traffic generated by signaling/control mechanism Users traffic that belongs to virtual networks

22. Overview of this approach Total flow: Flow of users + Flow of manager Example of manager’s flow Traffic generated by signaling/control mechanism Users traffic that belongs to virtual networks Manager optimizes system performance by controlling its portion of flow

23. Overview of this approach Total flow: Flow of users + Flow of manager Example of manager’s flow Traffic generated by signaling/control mechanism User traffic that belongs to virtual networks Manager optimizes system performance by controlling its portion of flow Investigates manager’s role using routing as a control paradigm

24. Non Cooperative Routing Scenario IPv4/IPv6 allow source routing User determines the path its flow follows from source- destination

25. Goal of Manager Capability of Manager Optimize overall network performance according to some system wide efficiency criterion • It is aware of non cooperative behavior of users and performs its routing based on this information

26. Central Idea

27. Central Idea Manager can predict user responses to its routing strategies

28. Central Idea Manager can predict user responses to its routing strategies Allows manager to choose a strategy that leads of optimal operating point

29. Central Idea Manager can predict user responses to its routing strategies Allows manager to choose a strategy that leads of optimal operating point Example of Leader-Follower Game [Stackelberg]

30. MAN VP’s k VP’s k VP’s k Org1 Org n Org2 User 1 User p User 2 User 3

31. Need to derive A necessary and sufficient condition that guarantees that the manager can enforce an equilibrium that coincides with the network optimum Above condition requires – Manager’s flow Control > Threshold

32. Need to derive A necessary and sufficient condition that guarantees that the manager can enforce an equilibrium that coincides with the network optimum Above condition requires – Manager’s flow Control > Threshold If the above criterion is met, we can show that the maximally efficient strategy of manager is unique and we will specify its structure explicitly

33. Presentation Outline Introduction to non cooperative networks Overview of approach Model and Problem Formulation Non cooperative User & Manager Single Follower Stackelberg Routing game Multi Follower Stackelberg Routing game Issues

34. 1 2 Source Destination L Model and Problem Formulation User set I = {1,…..,I} Communication Links L= {1,.....,L}

35. Model and Problem Formulation (contd) Manager is referred at user 0 I0= I U {0} cl = capacity of link l c = (c1,….cL) : capacity configuration C = lL cl : total capacity of the system of parallel links c1 >= c2 >= …. >= cL Each i  I0 has a throughput demand ri > 0 r1 >= r2 >= …. >= rI r = iI ri R = r + r0 Demand is less than capacity of links  R < C

36. Model and Problem Formulation (contd) User i  I0 splits its demand ri over the set of parallel links to send its flow Expected flow of user i on link l is fli Routing strategy of user i fi = (f1i,….fLi) Strategy space of user i  Fi = {fi IRL : 0 <= fli <= cl, l  L; lL fli = ri} Routing strategy profile f = {f0, f1,….,fI) System strategy space  F = iIoFi

37. Model and Problem Formulation (contd) Cost function quantifying GoS of user i’s flow is Ji : F  IR i  I0 Cost of user i under strategy profile f is Ji(f) Ji(f) = lL fliTl(fl); Tl(fl) is the average delay on link l, depends only on the total flow fl = iIofli on that link Tl(fl) = (cl - fl)-1, fl < cl = , fl >= cl Total cost J(f) = iIoJi(f) = lL fl/ (cl - fl) Higher cost  lower GoS provided to the user’s flow, higher average delay

38. Model and Problem Formulation (contd) is a convex function of (f1, …, fL)  a unique link flow configuration exists – min cost (f1*,….fL*) ; Above is solution to classical routing opt problem, routing of all flow (users+manager) is centrally controlled; referred to as network optimum.

39. Kuhn – Tucker Optimality conditions (f1*,….fL*) is the network optimum if and only if there exists a Lagrange Multiplier , such that for every link lL

40. Presentation Outline Introduction to non cooperative networks Overview of approach Model and Problem Formulation Non cooperative User & Manager Single Follower Stackelberg Routing game Multi Follower Stackelberg Routing game Issues

41. Non cooperative users

42. Non cooperative users Each user tries to find a routing strategy fi Fi that minimizes its cost Ji (average time delay)

43. Non cooperative users Each user tries to find a routing strategy fi Fi that minimizes its cost Ji (average time delay) This minimization depends on strategies of the manager and other users, described by strategy profile f-i = (f0 , f1 ,… fi-1, fi+1 ,… fI )

44. Non cooperative users Each user tries to find a routing strategy fi Fi that minimizes its cost Ji (average time delay) This minimization depends on strategies of the manager and other users, described by strategy profile f-i = (f0 , f1 ,… fi-1, fi+1 ,… fI ) Routing strategy of manger is FIXED  f0

45. Non cooperative users Each user tries to find a routing strategy fi Fi that minimizes its cost Ji (average time delay) This minimization depends on strategies of the manager and other users, described by strategy profile f-i = (f0 , f1 ,… fi-1, fi+1 ,… fI ) Routing strategy of manger is FIXED  f0 Each user adjusts its strategy to other users actions

46. Non cooperative users Each user tries to find a routing strategy fi Fi that minimizes its cost Ji (average time delay) This minimization depends on strategies of the manager and other users, described by strategy profile f-i = (f0 , f1 ,… fi-1, fi+1 ,… fI ) Routing strategy of manger is FIXED  f0 Each user adjusts its strategy to other users actions Can be modeled as a non cooperative game, any operating point is Nash Equilibrium; dependent on f0 !

47. Non cooperative users Each user tries to find a routing strategy fi Fi that minimizes its cost Ji (average time delay) This minimization depends on strategies of the manager and other users, described by strategy profile f-i = (f0 , f1 ,… fi-1, fi+1 ,… fI ) Routing strategy of manger is FIXED  f0 Each user adjusts its strategy to other users actions Can be modeled as a non cooperative game, any operating point is Nash Equilibrium; dependent on f0 ! From users view point, manager reduces capacity on each link l by fl0 , the system reduces to a set of parallel links with capacity configuration c – f0 has a unique Nash Equilibrium f0  f-0 ……. N0(f0)

48. Non cooperative users For a given strategy profile f-i of other users in I0, the cost of i, Ji(f) = lL fliTl(fl), is a convex fn of its strategy fi , hence the following min problem has a unique solution

49. Kuhn – Tucker Optimality conditions fi is the optimal response of user i if and only if there exists a (Lagrange Multiplier) , such that for every link lL, we have

50. Non cooperative users f-0  F-0 is a Nash Equilibrium of the self optimizing users induced by strategy f0 of the manger. The function N0 : F0 F-0 that assigns the induced equilibrium of the user routing game (to each strategy of the manger) is called the Nash Mapping. It is continuous.