On the Price of Stability for Designing Undirected Networks with Fair Cost Allocations M.Sc. Thesis...

On the Price of Stability for Designing Undirected Networks

withFair Cost Allocations

M.Sc. Thesis Defense

Svetlana Olonetsky

Definition of the problem

• Graph G=(V,E)• n players

• Player i wants to connect vertices 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

( ) ( )Si

C S C i

w

C(v) = 8

$2

$6

$5C(w)= 5

r

u

vC(v) = ?

C(w)= ?

Nash Equilibrium

• State S is a Nash equilibrium if for every state S’=(S1,…,Si-1, S’

i, Si+1,…,Sn)

'( ) ( )S SC i C i

Price of Stability

Price of Stability = C(best NE)

C(OPT)

(Min cost Steiner forest)

Price of Stability

For this game on directed graphs:

Price of stability 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(log n) !

1 12 n

Potential function

( )

1

( )( )

sx e

e E j

c eS

j

This game is a special case of congestion games, therefore has a potential function:

If user i changes its strategy from Si to S’i:

'( ) ( ') ( ) ( )S SS S C i C i

Upper bound on the Price of Stability

( )

1

( )( ) ( ) log * ( )

sx e

e E j

c eC S S n C S

j

(NASH) (NASH) ( ) log * ( )C n C

Summary

• Results of Anshelevich et. al:

Price of stability on directed graphs

(log n)

• Open problem:

Price of stability on undirected graphs

Our results

• We consider a restricted version of a game:

– undirected graph

– all players want to connect to the same vertex r

– every vertex v has at least one player associated with it

• Theorem: The Price of Stability for this game is O(loglog n).

Overview of the proof

• Start with OPT tree (OPT is MST)

• Schedule sequence of improvement moves

• When no moves are possible => NE

• Bound cost of NE

Improvement moves

r

Improvement moves

r Edges in OPT

Edges in graph

EE move

r

v

Edges in OPT

Edges in graph

v - change of strategy

EE move

r

v

Edges in OPT

Edges in graph

no new edges were added by v


OPT move

r

v

Edges in OPT

Edges in graph


OPT move

r

v

Edges in OPT

Edges in graph


new OPT edge was added

w

r

w

Edges in OPT

Edges in graph

- change of strategy

- move

r

w

Edges in OPT

Edges in graph

- change of strategy

- move

w

new edge, not in OPT, was added

Given a state S, and user u, improvement moves for u can be classified as follows:

• EE – All edges in the path u->r are in S. • OPT – All edges in the path u->r are in SOPT.

• – The first edge e=(u,v) of S’u is neither in S nor in OPT, all other edges are in S.

• Other – All other improvement moves

Improvement moves

We never schedule Other improvement moves

• EE moves do not increase the total cost• OPT moves increase the Price of Stability by

at most a factor of 2

• moves can increase the total cost • Every move adds one new edge to S

EE, OPT, and moves

EE, OPT, and moves

Lemma: If no EE moves possible S is a tree

Lemma: If no EE, OPT, or moves possible

state S is in Nash equilibrium

Scheduling algorithm

• The scheduler works in phases

• In the beginning of a phase no OPT or EE moves are possible.

Scheduling phaser OPT edges

graph edges

dashed edges unused in S

Scheduling phaser

u

OPT edges

graph edges

Scheduling phaser

u

OPT edges

graph edges

u performs move

x

Scheduling phase

1

r

u

OPT edges

graph edges

x

loop on distOPT(u,w)

Scheduling phase

12

r

u

OPT edges

graph edges

x


Scheduling phase

12

r

u

OPT edges

graph edges

x63

4

5

unused edge

unused edge


Scheduling phaser OPT edges

graph edges

x

x/8

12

u

63

4

5

Scheduling phase

12

r

u

OPT edges

graph edges

x63

4

5

x/8

Scheduling phase

1. Player u performs move

2. Loop over players w in increasing order distOPT(u,w):

– If strategy S’w that consist of PathOPT(w,u)

followed by current strategy of u is an improvement move perform it

3. While possible, schedule OPT and EE moves

Scheduling algorithm properties (1)

Let e=(u,v), e OPT, that was added to S. It must have been added by an move that started a phase.

Lemma: During the remainder of the phase

– All users w within distOPT(u,w) ≤ c(e)/8 use the strategy u … r as the tail of their strategy.

– When each of these users modify their strategy to include u … r, the potential drops by a constant fraction of c(e)

Proof:

r

u

v

w

Sw

Sv

Su

u performs move

Used OPT edges

Unused OPT edges

Proof:

r

u

v

w

Sw

Sv

Su

distOPT(u,w) <x/8

xS’u

u performs move

S’’w

Used OPT edges

Unused OPT edges

Proof:

r

u

v

w

Sw

Sv

SuxS’u

player w with distOPT(u,w)≤x/8 will decrease potential by at least x/4

u performs move

S’’w

CS(u) < CS(w) + distOPT(u,w)

CS(w) > CS(u) – x/8

CS’’(w) < x/8 + CS’(u) – x/2

CS’’(w) < CS’(w) - x/4

distOPT(u,w) <x/8

Scheduling algorithm properties (2)

Let e1=(u1,v1), e2=(u2,v2) be two edges that belong to Nash, e1 OPT and e1 OPT.

Lemma: 1 2

OPT 1 2

min{c( ),c( )}dist ( , )

8

e eu u

Proof:

w

r

v

OPT edges

graph edges

dashed edges unused in S

e1

e2

c(e1)≤c(e2)Suppose distOPT(v,w)≤c(e1)/8.

c(e1)/8distOPT(v,w)

c(e2)/8

Definition

Let u v … r be the strategy for u in the final state (Nash equilibrium).

Classify edge e = (u,v)OPT with c(e) = x, as either– a light edge – if there are ≤ log n vertices

within distOPT ≤x/8 of u,

or – a crowded edge - otherwise

Lemma:The total cost of all crowded edges is (OPT)

Proof: – In the phase such an edge was added to S, the

potential drops by at least (c(e)log n).– Thus, the total drop in potential during phases

with crowded first edges

( ) log ( ) C(OPT) loge crowded

c e n n

( ) C( )

e crowded

c e

Lemma:The total cost for all light edges is (OPT loglog n)

Proof:

Let u v … r be the strategy for u in the final state and let e=(u,v) be a light edge, define the cost of u to be the cost of e=(u,v)/16.

Call u light vertex.

Light edge amortization

r

OPT tree


r

OPT tree

light vertices

remove all vertices that are not light and don’t have light descendants


r

v

OPT tree

light vertices

take furthest vertex from r


r

v

OPT tree

light vertices

mark v's cost in r-direction


r

v

OPT tree

light vertices

mark subtree

mark v's cost in r-direction


r

v

OPT tree

light vertices

amortize the cost of subtree and remove it from tree


r

OPT tree

light vertices

continue with the rest of the tree

mark its cost in r-direction

take furthest vertex from r

amortize the cost of subtreeremove from tree


r

OPT tree

light vertices

continue with the rest of the tree

Subtree amortization

Lemma: The total cost of light edges starting from vertices in a subtree is at most loglog n times the cost of the subtree


v

Direct a path from u towards v, of length equal to the cost of vertex u

• Paths starting at vertices of the same cost don’t intersect• The total cost of vertices of equal cost with a subtree is no

more than the cost of the subtree.


Proof:– suppose all costs are powers of 2

– at most C(subtree)/ 2i vertices with cost 2i

– at most logn vertices

So the cost: loglog n C(subtree)

vTheorem finished!!!

Open problems

• We believe that the price of stability for this version is constant.

• Can our result be applied to a single source setting where there may not be an agent in every node?

• Generalization to the case where agents want to connect to different sources?

On the Price of Stability for Designing Undirected Networks with Fair Cost Allocations M.Sc. Thesis...

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