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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) ,

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Price of Stability

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  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.

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