The Cost of Stability in
Network Flow Games
Ezra Resnick
Yoram Bachrach
Jeffrey S. Rosenschein
1
Overview

Goal: In cooperative games, distribute the
grand coalition’s 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
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(Φ) = v(1) = v(2) = v(3) = 0
v(1,2) = v(1,3) = v(2,3) = 2
v(1,2,3) = 3
3
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
TNFG example
2
s
1
a
b
1
2
1
t
1
c
Threshold: 3
5
TNFG winning coalition
2
s
1
a
b
1
2
1
t
1
c
Threshold: 3
6
TNFG losing coalition
2
s
1
a
b
1
2
1
t
1
c
Threshold: 3
7
Distributing coalitional gains

Imputation: a distribution of the grand
coalition’s gains among the agents


pa is the payoff of agent a:
p(C )   pa
p
aN
a
 v( N )
is the payoff of a coalition C
aC

Solution concepts define criteria for
imputations

Individual rationality: a  N : pa  v({a})
8
The core

Coalitional rationality


A coalition C blocks an imputation p if p(C )  v(C )
An imputation p is stable if it is not blocked by
any coalition:
C  N : p(C )  v(C )

The core is the set of all stable imputations
9
The core of a TNFG
2
s
2
0.5
0.5
1
1
b
0
0
1
1
0
0
c
a
t
Threshold: 3
In a simple game, the core consists of imputations
which divide all gains among the veto agents
10
A TNFG with an empty core
2
s
1
a
b
1
2
1
t
1
c
Threshold: 2
If a simple game has no veto
agents then the core is empty
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Supplemental payment


An external party offers the grand
coalition a supplemental payment Δ if
all agents cooperate
This produces an adjusted game


v(N) + Δ are the adjusted gains
A distribution of the adjusted gains is a
super-imputation
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The Cost of Stability (CoS)


The core of the adjusted game may be
nonempty – if Δ is large enough
The Cost of Stability:
CoS = min {v(N) + Δ : the core of the
adjusted game is nonempty}
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CoS in TNFG example
2
1
s
2
a
1
1
1
0
0
b
c
1
0
1
0
t
Threshold: 2
Q. What is the CoS?
A. 2
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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|
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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 maxflow) of the network
16
CoS in connectivity games
a
d
s
b
t
e
c
17
CoS in connectivity games
a
d
s
b
t
e
c
CoS = min-cut = max-flow = 2
18
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)
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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…
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CoS in equal capacity TNFGs
1
s
1
a
b
1
1
1
t
1
c
Threshold: 2
b = 1, r = 2, f = 3
CoS = 1.5
21
Serial TNFGs
1
1
s
2
1
2
1
3
1
t
s
t
3
1
3
22
Serial TNFGs
1
1
s
2
1
2
1
3
1
t
3
1
3
23
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
24
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
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CoS in bounded serial TNFGs

Proof: Describe the CoS of each component
TNFG as a linear program
Minimize:
p
eE
e
Constraints:
e  E : pe  0
C  E :  pe  v(C )
eC
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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
a1, a2 ,, an , b
27
TNFG super-imputation
stability
a1
a2
s
v2
…
an
v1
a1
a2
t
an
vn


Threshold: b
Super-imputation p gives an edge with
ai
capacity ai a payoff of
2(b  1)
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Summary
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
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
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