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

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), fixed costs c(e) for all e 2 E, k vertex pairs (s i,t i ) S i : some path that connects s i to t i (S i is called the strategy of player i) State S=(S 1,S 2,…,S n )

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

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

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

Example: t s 1+  k t 1, t 2, … t k s 1, s 2, … s k Price of Stability

Example: t s 1+  k t 1, t 2, … t k s 1, s 2, … s k t s 1+  k Nash eq Price of Stability

Example: t s 1+  k t 1, t 2, … t k s 1, s 2, … s k t s 1+  k OPT (also Nash eq) t s 1+  k Nash eq POS=1 (not k) 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 ]

Example: High Price of Stability 1 1 n n t ... n-1 0 1

Example: High Price of Stability 1 1 n n t ... n C(OPT) = 1+ε

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

Example: High Price of Stability 1 1 n n t ... n so player n would deviate

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

Example: High Price of Stability 1 1 n n t ... n so player n-1 deviates too

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

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.

Potential Functions Defn: Փ (fn from outcomes to reals) is a potential function if for all outcomes S, player i, and deviations by i from S: Δ Փ = Δc(i)

Potential Function State: S={S 1,S 2,…,S n } c(e) : cost of edge e x s (e) : number of users that use edge e in state S We define Potential Function:

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

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

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

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 ]

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

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.

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.

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)

O(logn/loglogn) upper bound Consider NASH (also a steiner tree) sjsj Common terminal: t LCA(i,j) d(s i,s j ) sisi PijPij PjiPji

O(logn/loglogn) upper bound Add together: Common terminal: t sisi PijPij PjiPji sjsj

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= v 1,v 2,…,v n,v n+1 (w.l.o.g). So,  dis(v i,v i+1 ) · 2OPT 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(v i,v i+1 ) · 4OPT But what is  i A(i,i+1) ? Now we show  i A(i,i+1) contain term for every edge e 2 NASH O(logn/loglogn) upper bound

t Nash Tree TSP tour

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

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

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

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

O(logn/loglogn) upper bound Define:

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

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

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.

THANKS

Reference –Roughgardan, “Selfish Routing”, Ph.d-Thesis. –Roughgarden, "Potential Functions and the Inefficiency of Equilibria", to appear in Proceedings of the ICM, –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.