1. Xujin Chen, Xiaodong Hu Institute of Applied Mathematics Chinese Academy of Sciences Network Routing Games on Ring Infrastructure G CN-GER Conf on Math & Industry, Mar 15-18, 2010, Beijing, China

2. Outline ◙ Why SRR (Selfish Ring Routing)? ◙ How SRR? ◘ when min max delay ◘ when min max load ◙ Beyond SRR …

3. Why SRR? Xiamen, CN Traffic! New York, USA Tokyo, JP HUGE networks with LITTLE control Congestion! 1

4. Why SRR? Network Be connected? Wait & wait  HUGE networks with LITTLE control Congestion! Pls. try again  2

5. Why SRR? HUGE networks with LITTLE control Cost?! Life GDP loss 5% - 8% Congestion! ? 3

6. Why SRR? HUGE networks with LITTLE control Cost?! Life GDP loss 5% - 8% Congestion GAME non-coparative players 3

7. Why SRR? HUGE networks with LITTLE control Cost?! Life GDP loss 5% - 8% Congestion GAME Selfish Players 3

8. Exist ? How Good? Why SRR? HUGE networks with LITTLE control Congestion Selfish Players GAME Opt Nash Equilibrium efficiency  ratio btw So, Selfish Routing 4

9. MIT Press 2005 So, Selfish Routing Cambridge 2007 Algorithmic Why SRR? Game Theory von Neumann Father of Computer 5

10. MIT Press 2005 So, Selfish Routing Cambridge 2007 A Why SRR? G T ◘ Kleinberg : network formation, PoS… ◘ Goemans: mechanism design, equilibria ◘ Papadimitriou : complexity, covergence… ◘ Roughgarden, Tardos: network routing games, PoA, … selfish routing …… 6

11. How Good ? Why SRR? HUGE networks with LITTLE control Congestion Selfish Players GAME Opt Nash Equilibrium efficiency  ratio btw So, Selfish Routing NE 7

12. How Good ? Why SRR? HUGE networks with LITTLE control Congestion Selfish Players GAME Opt NAS’09 ， Knuth’02 C. Papadimitriou ◙ worst = max NE Opt NE PoA = Price of Anarchy NE 7

13. How Good ? Why SRR? HUGE networks with LITTLE control Congestion Selfish Players GAME ◙ worst = max NE Opt NE PoA = Price of Anarchy Nevanlinna’06, ICM’06 ， MacArthur’05 J. Kleinberg ◙ best = Price of Stability min NE Opt NE PoS = Opt NE 7

14. How Good ? Why SRR? HUGE networks with LITTLE control Congestion Selfish Players GAME Selfish Routing Very Bad in General !!! Delay Opt = Min MaxDelay Load Opt = Min MaxLoad Opt NE 7

15. 0 nx x 0 x x 00 0 8 ◙ general network has Unbounded PoA & PoS s1s1s1s1 t1t1t1t1 s2s2s2s2 t2t2t2t2 …… snsnsnsn tntntntnst st ( s, t )  player 1 ， ( s i, t i )  other n 2 players, Opt = Min MaxDelay s1s1s1s1 t1t1t1t1 s2s2s2s2 t2t2t2t2 …… snsnsnsn tntntntnst Opt Opt = everybody’s delay = n 2 n n2n2 n n2n2 n n2n2 NE = delay of player 1 = n ( n 2  n ) s1s1s1s1 t1t1t1t1 s2s2s2s2 t2t2t2t2 …… snsnsnsn tntntntnst NE n2nn2n n2nn2n n2nn2n PoA  PoS  NE/Opt  n  1st Why SRR? [ Chen, Chen, Hu, Hu, JOC’09, … ]…

16. How Good ? Why SRR? HUGE networks with LITTLE control Congestion Selfish Players GAME Selfish Routing Very Bad in General !!! Really Hopeless Opt NE 9

17. How SRR In the real world … Network congestion games happen on concrete topologies, such as Rings Bangalore Beijing EDmoton 10

18. How SRR In the real world … Network congestion games happen on concrete topologies, such as Rings Res. & Edu. over Optical Ring Triple-play services over Resilient Packet Ring Business over Optical Ring 11

19. PoA  PoS  NE/Opt  n  1 When Min MaxDelay 12 ◙ Th ： ring has Small Constant PoA & PoS 0 nx x s1s1s1s1 t1t1t1t1 s2s2s2s2 t2t2t2t2 …… snsnsnsn tntntntn s t nx x x 0000 [ Chen, Chen, Hu, Hu, JOCO’09,… ] PoS < PoA 2  PoA  16 1.2564  PoS  3.9 ◙ general network has Unbounded PoA & PoS

20. PoA  NE/Opt  n  1 When Min MaxLoad 13 However … ◙ ring has Unbounded PoA Opt = 1 NE = n −1 = 3 [ Busch, Magdon-Ismail, ’09 ]

21. PoA  NE/Opt  n  1 When Min MaxLoad 14 [ Chen, Hu, Ma, ’10 ] allow small coalitions ◙ ring has Unbounded PoA [ Busch, Magdon-Ismail, ’09 ] ◙ (k- strong equilibrium ) is resilient to deviation by any coalition of at most k players k -SE max k -SE Opt } { k -SE = k -SPoA

22. ◙ (k- strong equilibrium ) is resilient to deviation by any coalition of at most k players k -SE ◙ dynamic significant ring performance PoA  NE/Opt  n  1 When Min MaxLoad 14 [ Chen, Hu, Ma, ’10 ] ◙ ring has Unbounded PoA [ Busch, Magdon-Ismail, ’09 ] small coalition significantly improves max k -SE Opt } { k -SE = k -SPoA PoA  2-SPoA = n  1 >  m -SPoA > 1 2 = 3-SPoA

23. Beyond SRR … ◙ In SRR ◘ Poly-time convergence to NE in delay game? ◘ Weighted (un)splittable version? ◘ …… ◙ Other topologies ◘ Try: directed ring … ◘ Characterize topology with constant PoS ◘ Extend small coalition method to …… 15

24. More challenges … ◙ In Algorithmic game theory ◘ Design games with efficient equilibria when involving selfish, altruistic, malicious players? cascading behaviors? (information) market pricing? network formation? … ◘ Prove PoS type bounds in more realistic setting under increasingly weak assumption on the rationality? when failing to converge to an eq.? general analytical technique?... “ an emerging new area of designing systems and algorithms for selfish users …” - E. Tardos “ a deep theory of algorithms & games of powerful tech as well as many challenges …” – C. Papadimitriou 16

25. More challenges … ◙ In Algorithmic game theory ◘ Rethink about equilibria idea credible/natural, guaranteed to exist & efficiently computable eq. concept related to Nash dynamic? Complexity of apx Nash? … ◘ Design (apx) truthful mechanism for auction, resource allocation, facility location…? Deterministic vs randomness (in expectation or universal sense) ?... ◘ …… “If your laptop cannot find it, neither the market.” --- Christoms H. Papadimitriou 17

26. Xujin Chen, xchen@amss.ac.cn Thank You ！

27. References ◘ Andelman, Feldman, Mansour, Strong price of anarchy, Games Econom. Behav., 65 (2009), 289-317. ◘ Anshelevich, Dasgupta, Kleinberg, Tardos, Wexler, Roughgarden, The price of stability for network design with fair cost allocation, SIAM J. Comput. 38 (2008), 1602-1623. ◘ Awerbucj, Azar, Epstein, The price of routing unsplittable flow, Proc. STOC, 2005, 57-66. ◘ Chen, Chen, Hu, The price of atomic selfish ring routing, J. Comb. Optim., 2008. ◘ Chen, Chen, Hu, Hu, Stability vs. optimality in selfish ring routing, submit to SIAM J. Discrete Math., 2009. ◘ Chen, Hu, Ma, Balancing ring load via small coalitions, submit to Games Econom. Behav., 2010.

28. References ◘ Fotakis, Kontogiannis, Spiraklis, Atomic congestion games among coalitions, ACM Trans. Algorithms, 4 (2008), Article 52. ◘ Koutsoupias, Papadimitriou, Worst-case equilibria, Lecture Notes in Comput. Sci. 1563 (1999) 404-413. ◘ Nisan, Roughgarden, Tardos, Vazirani, Algorithmic Game Theory, Cambridge Univ. Press, 2007. ◘ Roughgarden, Selfish Routing and the Price of Anarchy, MIT press, 2005. ◘ Roughgarden, Tardos, How bad is selfish routing? J. ACM 49 (2002) 236-259.

29. SRR model Selfish Ring Routing ( SRR ): Nash Opt

30. How efficient are equilibria? Nash Opt ◙ Best ( Price of Stability ): PoS = ◙ Worst ( Price of Anarchy ): PoA =

31. ◙ SRR admits a (9,1)- apx NR Better quantify instability Key. Otherwise,  ( f ) can be decreased by changing one or two players routes without destroying optimality ◙ Measure instability & inefficiency is an ( ,  )- apx Nash routing (NR) if & ◙ SRR admits a (54,1)- apx NR [CCH08] an opt routing f with min =

32. Faster search good routings ◙ QP rounding → routing f ’ with M ( f ’ )  2 Opt ◙ f ’ → f  NR in O ( n 2 k 3 W ) time [CCH08], where & M ( f )  13.7 Opt ◙ O ( nk 3 ) Combinatorial Alg → routing f o with M ( f o )  3 Opt ◙ f o → f  NR in O ( nk 3 W ) time s.t. M ( f )  11.7 Opt ◙ f’ → f  ( 9,3)-apx NR in O ( nk 3 W ) time