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.