1 / 43

Price of Stability

Price of Stability. Introduction to. Li Jian Fudan University May, 8 th ,2007. Part of my slides is drawn from Tim Roughgarden’s lecture on game theory and part from Svetlana Olonetsky’s Msc defense slides and part by myself…. Selfish Network Design. Given : G = (V,E) ,

Download Presentation

Price of Stability

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Price of Stability Introduction to Li Jian Fudan University May, 8th ,2007

  2. Part of my slides is drawn from Tim Roughgarden’s lecture on game theory • and part from Svetlana Olonetsky’s Msc defense slides • and part by myself…

  3. Selfish Network Design Given: G= (V,E), fixed costs c(e) for all e 2 E, k vertex pairs (si,ti) Si : some path that connects si to ti (Si is called the strategy of player i) State S=(S1,S2,…,Sn)

  4. Cost definition t • c(e) – cost of edge e • xs(e) – number of users that use edge e in state S • cost to the player: • total cost: $6 u $2 C(w)= 5 C(w)= ? w $5 v C(v) = 8 C(v) = ?

  5. Nash Equilibrium • In this case, the state S=(S1,…,Si-1, Si, Si+1,…,Sn) is a Nash equilibrium if for every state S’=(S1,…,Si-1, S’i, Si+1,…,Sn), Si’<>Si No player wants to change its path!

  6. Price of Stability C(best NE) C(OPT) Price of Stability(POS) = (Min cost Steiner forest)

  7. Price of Stability Example: t1, t2, … tk t 1+ k s s1, s2, … sk

  8. Price of Stability Example: t1, t2, … tk t t 1+ k 1+ k s s Nash eq s1, s2, … sk

  9. Price of Stability Example: t1, t2, … tk t t t 1+ k 1+ k 1+ k s s s Nash eq s1, s2, … sk OPT (also Nash eq) POS=1 (not k)

  10. Price of Stability For this game on directed graphs: POS:Θ(log n) “The Price of Stability for Network Design with Fair Cost Allocation “ [E. Anshelevich, A. Dasgupta, J. Kleinberg,E. Tardos, T. Roughgarden ]

  11. Example: High Price of Stability t 1 1 1 1 1 2 3 n n-1 . . . 1+ 1 2 3 n n-1 0 0 0 0 0

  12. Example: High Price of Stability C(OPT) = 1+ε t 1 1 1 1 1 2 3 n n-1 . . . 1+ 1 2 3 n n-1 0 0 0 0 0

  13. Example: High Price of Stability C(OPT) = 1+ε …but not a NE: player n pays (1+ε)/n, could pay 1/n t 1 1 1 1 1 2 3 n n-1 . . . 1+ 1 2 3 n n-1 0 0 0 0 0

  14. Example: High Price of Stability so player n would deviate t 1 1 1 1 1 2 3 n n-1 . . . 1+ 1 2 3 n n-1 0 0 0 0 0

  15. Example: High Price of Stability now player n-1 pays (1+ε)/(n-1), could pay 1/(n-1) t 1 1 1 1 1 2 3 n n-1 . . . 1+ 1 2 3 n n-1 0 0 0 0 0

  16. Example: High Price of Stability so player n-1 deviates too t 1 1 1 1 1 2 3 n n-1 . . . 1+ 1 2 3 n n-1 0 0 0 0 0

  17. Example: High Price of Stability Continuing this process, all players defect. This is a NE! (the only Nash) cost = 1 + + … + t 1 1 1 1 1 2 3 n n-1 . . . 1+ 1 2 3 n n-1 0 0 0 0 0 1 1 2 n Price of Stability is Hn = Θ(ln n) !

  18. The Price of Stability Thus: the price of stability of selfish network design can be as high as ln k. [k = # players] Our goals: in all such games, • there is at least one pure-strategy Nash eq • one of them has cost ≤ ln k • OPT • i.e. price of stability always ≤ ln k • [Anshelevich et al 04] Technique: potential function method.

  19. Potential Functions Defn:Փ (fn from outcomes to reals) is a potential functionif for all outcomes S, player i, and deviations by i from S: ΔՓ = Δc(i)

  20. Potential Function • State: S={S1,S2,…,Sn} • c(e) : cost of edge e • xs(e) : number of users that use edge e in state S • We define Potential Function:

  21. Potential Function Consider some solution S. Suppose player i is unhappy and decides to deviate. What happens to Ф(S)?

  22. Proof of Potential Function Фe(S) = ce[1+ 1/2 + 1/3 + … 1/xS(e)] So Ф(S)=e Фe(S) Suppose player i’s new path includes e. i pays c(e)/(xS(e)+1) to use e. Фe(S) increases by the same amount. If player i leaves an edge e’, Фe’(S) exactly reflects the change in i’s payment. C(e)[1+ 1/2 +… +1/xS(e)] e i e’ C(e’)[1+ 1/2 +… +1/xS(e’)]

  23. Proof of Potential Function SO, ΔՓ = Δc(i) Let’s consider the state S with min Փ(S) :

  24. Summary • Results of Anshelevich et. al: Price of stability on directed graphs (log n) • Open problem: Price of stability on undirected graphs. o(logn)? Conjecture: constant. only known results:O(loglogn), single source, every node has a player. [Fiat etc, ICALP06]

  25. My progress • Undirected, single source, O(logn/loglogn) • I am not clear how to get similar bound for general case (multi-source).

  26. better-response dynamics • If the current outcome is not a Nash equilibrium, there exists a player whose can decrease his cost by switching its strategy. • Update its strategy to an arbitrary superior one, and repeat until a Nash equilibrium is reached.

  27. better-response dynamics In this game, a NE must be reached by better response dynamics in finite step since: (1)Finite game -> finite number of states (2)Potential function strictly decrease -> no state appear more than once.

  28. O(logn/loglogn) upper bound • Consider a NE NASH reached by better response dynamics from OPT (OPT is a steiner minimum tree). • So (NASH)·(OPT)

  29. O(logn/loglogn) upper bound • Consider NASH (also a steiner tree) d(si,sj) si sj Pij Pji LCA(i,j) Common terminal: t

  30. O(logn/loglogn) upper bound • Add together: si sj Pij Pji Common terminal: t

  31. O(logn/loglogn) upper bound • Consider OPT(a steiner tree) • Double it and obtain a Eular tour T. • In the metric shortest path closure of G, Traverse T and do short cut to get a TSP= v1,v2,…,vn,vn+1 (w.l.o.g). So,  dis(vi,vi+1)·2OPT

  32. O(logn/loglogn) upper bound Suppose there is a dummy player at t. Relabel players according to the TSP, i A(i,i+1)·2i dis(vi,vi+1)·4OPT But what is i A(i,i+1) ? Now we show i A(i,i+1) contain term for every edge e2 NASH

  33. O(logn/loglogn) upper bound Nash Tree TSP tour t

  34. O(logn/loglogn) upper bound i A(i,i+1) contains: Nash Tree TSP tour t

  35. O(logn/loglogn) upper bound i A(i,i+1) contain: Nash Tree TSP tour t

  36. O(logn/loglogn) upper bound Let And It is easy to see |NASH|=i fN(i)=g(1)

  37. O(logn/loglogn) upper bound Since Every edge in Nash tree appears in i A(i,i+1) at least once. So

  38. O(logn/loglogn) upper bound Define:

  39. O(logn/loglogn) upper bound • We can also get: • So,

  40. O(logn/loglogn) upper bound • So, • The right hand side of the equality is maximized at • So,

  41. O(logn/loglogn) upper bound • It is not clear how to get similar bound for multi-source case, since the charging argument doesn’t work any more. • If you are interested, we can talk about it more.

  42. THANKS

  43. Reference • Roughgardan, “Selfish Routing”, Ph.d-Thesis. • Roughgarden, "Potential Functions and the Inefficiency of Equilibria", to appear in Proceedings of the ICM, 2006. • E. Anshelevich, A.Dasgupta, J.Kleinberg, E.Tardos, T.Wexler and T.Roughgarden. The price of stability for network design with fair cost allocation. FOCS,2004 • Amos Fiat, Haim Kaplan, Meital Levy, Svetlana Olonetsky and Ronen Shabo. On the prize of stability for designing undirected networks with fair cost allocations. ICALP06.

More Related