On the Price of Stability for Designing Undirected Networks with Fair Cost Allocations

1
On the Price of Stability for Designing
Undirected Networks withFair Cost Allocations

• Svetlana Olonetsky
• Joint work with Amos Fiat, Haim Kaplan,
• Meital Levy, Ronen Shabo

2
Network design game
t1
t2

• Si strategy of player i is some path that
  connects si to ti
• State S(S1,S2,,Sn)

s1
s2
3
Network design game
2
C(1) 2 8/2 6 C(2) 1 8/2 3 1 9
3
t1
2
t2
1
8
5
2

• cost to the player
• total cost

2
v
2
2
s1
1
s2
4
Definitions

• Nash Equilibrium
• State S is a Nash equilibrium if for every
  state S'(S1,,Si-1, S'i, Si1,,Sn)
• Price of stability

C(best NE) C(OPT)
(Min cost Steiner forest)
5
Summary

• Known Results
• Price of stability on directed graphs ?(log
  n)
• The Price of Stability for Network Design
  with Fair Cost Allocation
• E.Anshelevich, A.Dasgupta, J.Kleinberg,
  E.Tardos, T. Roughgarden
• Open problem
• Price of stability on undirected graphs

6
Our results

• Undirected graphs
• Common target vertex r (Multicast)
• Player at every vertex
• Theorem
• The Price of Stability for this game is
  O(loglog n).

7
Proof overview

• Start with OPT tree (OPT is some MST)
• Describe algorithm that produces a particular
  sequence of improvement moves leading to Nash
  equilbirium
• Bound cost of resulting Nash equilbirium

8
Improvement moves
Edges in graph
r
Edges in OPT
9
Improvement moves
Edges in graph
r
Edges in OPT
10
EE move use Existing Edges
r
v
no new edges were added by v
11
OPT move use edges in MST
r
v
new OPT edge was added
12
move
Edges in OPT
- change of strategy
w
r
previous
new
w
new edge, not OPT, not EE, first on path from w
13
EE, OPT, and moves

• Lemma 1
• If no EE moves possible ? S is a tree
• Proof

r
v
u
14
EE, OPT, and moves

• Lemma 2
• If no OPT moves possible
• ?
• - calculated in similar way as
  CS(w), except that additional player counted
  on path from w to LCAS(v,w).
• Proof
• If S' differs from S by strategy of v, only edges
  on path from w to LCAS(v,w) can become cheaper
  for w.
• If Lemma doesnt hold, connect v to w and
  continue with w

15
EE, OPT, and moves

• Lemma 3 (without proof)
• If no EE, OPT, or moves possible
• ? state S is in Nash equilibrium

16
EE, OPT, and moves

• EE moves do not increase the total cost
• OPT moves increase the Price of Stability by a
  factor 2
• moves can increase the total cost
• Every move adds one new edge to S

17
Scheduling algorithm

• The scheduler works in phases
• In the beginning of a phase no OPT or EE moves
  are possible.

18
Scheduling phase
OPT edges
r
graph edges
dashed edges unused in S
19
Scheduling phase
OPT edges
r
graph edges
dashed edges unused in S
u
20
Scheduling phase
OPT edges
r
graph edges
dashed edges unused in S
x
u
u performs move
21
Scheduling phase
OPT edges
r
graph edges
dashed edges unused in S
x
u
1
loop on distOPT(u,w)
22
Scheduling phase
OPT edges
r
graph edges
dashed edges unused in S
x
u
1
2
loop on distOPT(u,w)
23
Scheduling phase
OPT edges
r
graph edges
unused edge
dashed edges unused in S
5
3
x
6
u
1
2
loop on distOPT(u,w)
unused edge
4
24
Scheduling phase
OPT edges
r
graph edges
dashed edges unused in S
5
3
x
6
u
x/8
1
2
4
25
Scheduling phase
OPT edges
r
graph edges
dashed edges unused in S
5
3
x
6
u
x/8
1
2
4
26
Scheduling phase

• Player u performs some move
• For all players w in order of increasing
  distOPT(u,w)
• If, PathOPT(w,u) followed by the path from u to r
  is better for w, then w chooses this strategy.
• While possible, schedule OPT and EE moves

27
Potential function
This game has an exact potential function
If user i changes its strategy from Si to S'i
?
?
28
Properties of Scheduling algorithm(1)
r
Sv
v

• Let e(u,v), e? OPT,
• added to S by an move
• Lemma During the remainder of the phase
• All users w within distOPT(u,w) c(e)/8 modify
  their strategy to include u?? r as the tail of
  their strategy.
• After each move potential drops by a constant
  fraction of c(e)

Sw
S'u
c(e)
Su
u
w
distOPT(u,w)ltx/8
29
Proof sketch
r
Sv
v

• S' strategy after move
• Step 1 In state S', strategy of w is an
  improving move

Sw
S'u
c(e)
Su
u
w
distOPT(u,w)ltx/8
30
Proof sketch of Step 1

• Cost of proposed strategy of w is at most

r
We show, that
Sv
v
Sw
S'u
c(e)
Su
u
w
distOPT(u,w) ltx/8
31
Proof sketch of Step 1

• Since no OPT move allowed,
• u made an improvement move, so
• ? result follows from (2) and (3).

r
Sv
v
Sw
S'u
c(e)
Su
u
w
distOPT(u,w) ltx/8
32
Proof sketch
r
Sv
v

• Step 2 It can be shown by induction, that all
  players will take proposed strategy

Sw
S'u
c(e)
Su
u
w
distOPT(u,w) ltx/8
33
Properties of Scheduling algorithm(2)

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

34
Proof
OPT edges
r
graph edges
dashed edges unused in S
e2
c(e2)/8
e1
u2
u1
c(e1)/8
distOPT(v,w)
c(e1)c(e2)distOPT(u1,u2)c(e1)/8.
35
Crowded edge amortization

• At least logn players inside the ball
• Moves of players inside the ball dropped the
  potential by ?(x logn)
• Initial potential value is at most C(OPT) logn

Lemma The total cost of crowded edges is C(OPT)
36
Light edge amortization

• At most logn players inside the ball of radius
  xv/8
• Lemma
• The total cost of light edges is C(OPT)
  loglogn

37
Proof

• Look at Nash Equilibrium
• Mark light vertices

10
3
1
3
38
Proof

• Choose vertex with maximum weight W and draw a
  ball with radius W/8
• Remove light vertices inside this ball with
  weight less then W / log n
• Total cost of removed vertices at most W

10
10
3
1
3
39
Proof

• Continue the process

10
3
3
40
Proof

• Draw a ball of radius W/24 around remained
  vertices
• Every point of tree can be covered by balls with
  radiuses
• max W / log n lt R lt max W
• Radius size decreases by at least factor 2
• ? every point of tree can be covered by loglogn
  balls

10
3
3
41
Summary

• Total cost of crowded edges
• C(OPT)
• Total cost of light edges
• C(OPT) loglogn
• Price of Stability loglogn

42
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?