Price of Stability Li Jian Fudan University May, 8 th,2007 Introduction to.
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Transcript of Price of Stability Li Jian Fudan University May, 8 th,2007 Introduction to.
Price of Stability
Li Jian
Fudan University
May, 8th ,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 (si,ti)
Si : some path that connects si to ti (Si is called the strategy of player i)
State S=(S1,S2,…,Sn)
Cost definition
• c(e) – cost of edge e
• xs(e) – number of users that use edge e in state S
• cost to the player:
• total cost:
( )( )
( )i
Se S s
c eC i
x e
( ) ( ) ( )i i
Si e S
C S C i c e
w
C(v) = 8
$2
$6
$5C(w)= 5
t
u
vC(v) = ?
C(w)= ?
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
'( ) ( )S SC i C 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
t1, t2, … tk
s1, s2, … sk
Price of Stability
Example:
t
s
1+ k
t1, t2, … tk
s1, s2, … sk
t
s
1+ k
Nash eq
Price of Stability
Example:
t
s
1+ k
t1, t2, … tk
s1, s2, … sk
t
s
1+ k
OPT(also Nash eq)
t
s
1+ k
Nash eq
POS=1 (not k)
Price of Stability
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 1n
12
13
1 2 3 n
t
0 0 0 0
1+ . . . n-1
0
1n-1
Example: High Price of Stability
1 1n
12
13
1 2 3 n
t
0 0 0 0
1+ . . . n-1
0
1n-1
C(OPT) = 1+ε
Example: High Price of Stability
1 1n
12
13
1 2 3 n
t
0 0 0 0
1+ . . . n-1
0
1n-1
C(OPT) = 1+ε
…but not a NE:
player n
pays (1+ε)/n,
could pay 1/n
Example: High Price of Stability
1 1n
12
13
1 2 3 n
t
0 0 0 0
1+ . . . n-1
0
1n-1
so player n
would deviate
Example: High Price of Stability
1 1n
12
13
1 2 3 n
t
0 0 0 0
1+ . . . n-1
0
1n-1
now player n-1
pays (1+ε)/(n-1),
could pay 1/(n-1)
Example: High Price of Stability
1 1n
12
13
1 2 3 n
t
0 0 0 0
1+ . . . n-1
0
1n-1
so player n-1
deviates too
Example: High Price of Stability
1 1n
12
13
1 2 3 n
t
0 0 0 0
1+ . . . n-1
0
1n-1
Continuing this process, all players defect.
This is a NE!
(the only Nash)
cost = 1 + + … +
Price of Stability is Hn = Θ(ln n) !
1 12 n
The Price of StabilityThus: 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: (fn from outcomes to reals) is a Փpotential function if for all outcomes S, player i, and deviations by i from S:
Δ = Δc(i)Փ
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:
Potential Function
Consider some solution S. Suppose player i is unhappy and decides to deviate.
What happens to Ф(S)?
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.
e
e’
C(e)[1+ 1/2 +… +1/xS(e)]
C(e’)[1+ 1/2 +… +1/xS(e’)]
i
SO, Δ = Δc(i)Փ
Let’s consider the state S with min (S) Փ:
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)
sj
Common terminal: t
LCA(i,j)
d(si,sj)si
Pij Pj
i
O(logn/loglogn) upper bound
• Add together:
Common terminal: t
si
Pij Pj
i
sj
• 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
O(logn/loglogn) upper bound
Suppose there is a dummy player at t.Relabel players according to the TSP, i A(i,i+1)· 2i 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
O(logn/loglogn) upper bound
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 fN(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 Ineffici
ency 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.