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


1 1n

12

13

1 2 3 n

t

0 0 0 0

1+ . . . n-1

0

1n-1

C(OPT) = 1+ε


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


1 1n

12

13

1 2 3 n

t

0 0 0 0

1+ . . . n-1

0

1n-1

so player n

would deviate


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)


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


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)


• Consider NASH (also a steiner tree)

sj

Common terminal: t

LCA(i,j)

d(si,sj)si

Pij Pj

i


• 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


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



t

Nash Tree

TSP tour


t

Nash Tree

TSP tour

i A(i,i+1) contains:


t

Nash Tree

TSP tour

i A(i,i+1) contain:


Let

And

It is easy to see

|NASH|=i fN(i)=g(1)


Since Every edge in Nash tree appears in i A(i,i+1) at least once.

So


Define:


• We can also get:

• So,


• So,

• The right hand side of the equality is maximized at

• So,


• 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.

price of stability li jian fudan university may, 8 th,2007 introduction to

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