The Cost of Stability in Network Flow Games - PowerPoint PPT Presentation

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The Cost of Stability in Network Flow Games

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This gives a stable super-imputation with adjusted gains of f/k ... coNP-complete to determine whether a super-imputation in a TNFG is stable ... – PowerPoint PPT presentation

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Title: 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 Stability CoS 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 program Minimize Constraints

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
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