The Price of Anarchy in a Network Pricing Game (II)

1. The Price of Anarchy in a Network Pricing Game (II) SHI Xingang & JIA Lu 14-05-2008

2. Outline • Can We Find a Bound? • How Can We Find the Bound? • Let's Prove the Bound • Let's Prove It Again • How About Convex Latency • Conclusion and extension

3. Can We Find a Bound? Optimal price p=0, d=0, f=2, W=0+3/2=3/2 W / W*=1.5 Equilibrium price p*=1, d*=1, f*=1, W*=1+0=1 [3] has proved that 1.5 is the tight upper bound, using mathematical programming

4. How Can We Find the Bound? Linearization and Truncation [2] brings the idea for truncation

5. Linearized Disutility Function • Lemma : The Nash equilibrium flow and the price vectors of are the same as the Nash flow and the price vectors of • Remember the sufficient and necessary condition

6. Linearized Disutility Function • Lemma : The Optimal Welfare of is no more than that of And for a linearized game, d* < d • Remember this is an optimization problem This paper missed this point

7. Linearized and Truncated Disutility Function • Lemma : The Nash equilibrium flow and the price vectors of are the same as the Nash flow and the price vectors of • Remember the sufficient and necessary condition

8. Sufficient and Necessary Condition For It's also easy to see that the optimal flow and price vectors are the same as

9. Linearized and Truncated Disutility Function • Lemma : • Proof : introduce a truncated utility function , , so the optimization result is larger can decrease no more than from Now we only need to deal with linearized and truncated disutility function!

10. Deal with Linearized Truncated Disutility Function • There cannot exist links used in social optimum that are not used in Nash Equilibrium

11. Let's Find and • Linear truncated disutility function • Linear (not truncated) disutility function same (d,f) and (d*,f*)

12. and

13. and

14. Let's Prove

15. Let's Prove

16. Let's Prove • This paper proves by the following way: • restricting , and we can prove • since is decreasing in [0,1/2] • we only need to prove the diagonal elements are positive, where • But and • (we have and ) • there do exist chances that the sum is negative

17. Let's Prove Again • The reason it fails – bound is too low • When there is no unused flow in optimal, is actually 0 (restricting it by is too loose). We have proved successfully. • When there is unused flow in optimal, using to replace makes the value too small. We are walking on this way. Anyway, linearization is a very important step

18. Convex Latency Function linearization again! When equilibrium exists, we have

19. Conclusion and Extension • Analogy of circuit may give us some interesting ideas • Linearization is sometimes more simple and more powerful • Multi-commodity • Multiple source and destination pairs • Different type of sensitivity to latency

20. References [1] John Musacchio, The Price of Anarchy in a Network Pricing Game, Presentation at Allerton07. [2] A.Hayrapetyan, E. Tardos and T. Wexler, A Network Pricing Game for Selfish Traffic, Twenty-Fourth Annual ACM SIGACT-SIGOPS Symposium on Principles of Distributed Computing (PODC 2005) [3] D. Acemoglu and A.ozdaglar, Competition and Efficiency in Congested markets, Mathematics of Operations Research, 2007 [4] John Musacchio and Shuang Wu, The Price of Anarchy in a Network Pricing Game, The Forty-Sixth Annual Allerton Conference on Communication, Control, and Computing (Allerton07) [5] S. Boyd and L. Vandenberghe, Convex optimzation, Camebridge University Press, 2004 [6] T. Roughgarden, The Price of Anarchy is Independent of the Network Topology, 34th ACM Symposium on Theory of Computing (STOC 2002)