The Cost of Stability in Network Flow Games

1
The Cost of Stability in Network Flow Games

• Ezra Resnick
• Yoram Bachrach
• Jeffrey S. Rosenschein

1
2
Overview

• Goal In cooperative games, distribute the grand
  coalitions gains among the agents in a stable
  manner
• This is not always possible (empty core)
• Stabilize the game using an external payment
• Cost of Stability minimal necessary external
  payment to stabilize the game
• Focus on Threshold Network Flow Games

2
3
Cooperative games

• A set of agents N
• A characteristic function v 2N ? R
• the utility achievable by each coalition of
  agents
• Example
• N 1,2,3
• v(F) v(1) v(2) v(3) 0
• v(1,2) v(1,3) v(2,3) 2
• v(1,2,3) 3

3
4
Threshold Network Flow Games (TNFGs)

• A TNFG is defined by a flow network and a
  threshold value
• Each agent controls an edge
• The utility of a coalition is 1 if the flow it
  allows from source to sink reaches the threshold,
  0 otherwise
• TNFGs are simple, increasing games

4
5
TNFG example
a
2
2
1
1
b
t
s
1
1
c

• Threshold 3

5
6
TNFG winning coalition
a
2
2
1
1
b
t
s
1
1
c

• Threshold 3

6
7
TNFG losing coalition
a
2
2
1
1
b
t
s
1
1
c

• Threshold 3

7
8
Distributing coalitional gains

• Imputation a distribution of the grand
  coalitions gains among the agents
• pa is the payoff of agent a
• is the payoff of a coalition C
• Solution concepts define criteria for imputations
• Individual rationality

8
9
The core

• Coalitional rationality
• A coalition C blocks an imputation p if
• An imputation p is stable if it is not blocked by
  any coalition
• The core is the set of all stable imputations

9
10
The core of a TNFG
a
2
2
0.5
0.5
1
1
b
t
s
0
0
1
1
c
0
0
Threshold 3
In a simple game, the core consists of
imputations which divide all gains among the veto
agents
10
11
A TNFG with an empty core
a
2
2
1
1
b
t
s
1
1
c
Threshold 2
If a simple game has no veto agents then the core
is empty
11
12
Supplemental payment

• An external party offers the grand coalition a
  supplemental payment ? if all agents cooperate
• This produces an adjusted game
• v(N) ? are the adjusted gains
• A distribution of the adjusted gains is a
  super-imputation

12
13
The Cost of Stability (CoS)

• The core of the adjusted game may be nonempty
  if ? is large enough
• The Cost of StabilityCoS min v(N) ? the
  core of the adjusted game is
  nonempty

13
14
CoS in TNFG example
a
2
2
1
0
1
1
b
t
s
1
0
1
1
c
0
0
Threshold 2
Q. What is the CoS?
A. 2
14
15
CoS in simple games

• Theorem If a simple game contains m
  pairwise-disjoint winning coalitions, then CoS
  m
• Theorem In a simple game, if there exists a
  subset of agents S such that every winning
  coalition contains at least one agent from S,
  then CoS S

15
16
Connectivity games

• A connectivity game is a TNFG where all
  capacities are 1 and the threshold is 1
• A coalition wins iff it contains a path from
  source to sink
• Theorem The CoS of a connectivity game equals
  the min-cut (and max-flow) of the network

16
17
CoS in connectivity games
a
d
b
t
s
e
c
17
18
CoS in connectivity games
a
d
b
t
s
e
c
CoS min-cut max-flow 2
18
19
CoS in TNFG upper bound

• Theorem If the threshold of a TNFG is k and the
  max-flow of the network is f, then CoS f/k
• Proof Find a min-cut, and pay each c-capacity
  edge in the cut c/k
• This gives a stable super-imputation with
  adjusted gains of f/k
• f/k can serve as an approximation of the CoS
  (useful if the ratio f/k is small)

19
20
CoS in equal capacity TNFGs

• Theorem If all edge capacities in a TNFG equal
  b, and the threshold is rb (r ? N), and f is
  the max-flow of the network, then CoS f/rb
• Connectivity games are a special case (r b
  1)
• Proof We already know that CoS f/rb, so it
  suffices to prove CoS f/rb

20
21
CoS in equal capacity TNFGs
a
1
1
1
1
b
t
s
1
1
c
Threshold 2

• b 1, r 2, f 3
• CoS 1.5

22
Serial TNFGs
1
1
1
1
2
s
t
s
t
1
2
3
3
3
1
23
Serial TNFGs
1
1
1
1
2
s
t
1
2
3
3
3
1
24
CoS in serial TNFGs

• Theorem The CoS of a serial TNFG equals the
  minimal CoS of any of the component TNFGs
• Proof Show that a super-imputation which is
  stable and optimal in the component with the
  minimal CoS is also a stable and optimal
  super-imputation for the entire series

25
CoS in bounded serial TNFGs

• Theorem If the number of edges in each component
  TNFG is bounded, then the CoS of a serial TNFG
  can be computed in polynomial time
• Runtime will be linear in the number of
  components, but exponential in the number of
  edges in each component

26
CoS in bounded serial TNFGs

• Proof Describe the CoS of each component TNFG as
  a linear programMinimizeConstraints

27
TNFG super-imputation stability

• TNFG-SIS Given a TNFG, a supplemental payment,
  and a super-imputation p in the adjusted game,
  determine whether p is stable
• Theorem TNFG-SIS is coNP-complete
• Proof Reduction from SUBSET-SUM

28
TNFG super-imputation stability
v1
a1
a1
a2
a2
v2
t
s
an
an

vn

• Threshold b
• Super-imputation p gives an edge with capacity ai
  a payoff of

29
Summary

• CoS defined for any cooperative game
• coNP-complete to determine whether a
  super-imputation in a TNFG is stable
• For any TNFG, CoS max-flow/threshold
• CoS in special TNFGs
• Connectivity games
• Equal capacity TNFGs
• Serial TNFGs