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PPT – The Cost of Stability in Network Flow Games PowerPoint presentation | free to download - id: 598cd-ZDc1Z

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The Cost of Stability in Network Flow Games

- Ezra Resnick
- Yoram Bachrach
- Jeffrey S. Rosenschein

1

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

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

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

a

2

2

1

1

b

t

s

1

1

c

- Threshold 3

5

TNFG winning coalition

a

2

2

1

1

b

t

s

1

1

c

- Threshold 3

6

TNFG losing coalition

a

2

2

1

1

b

t

s

1

1

c

- Threshold 3

7

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

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

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

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

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

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

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

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

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

CoS in connectivity games

a

d

b

t

s

e

c

17

CoS in connectivity games

a

d

b

t

s

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)

19

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

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

Serial TNFGs

1

1

1

1

2

s

t

s

t

1

2

3

3

3

1

Serial TNFGs

1

1

1

1

2

s

t

1

2

3

3

3

1

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

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

CoS in bounded serial TNFGs

- Proof Describe the CoS of each component TNFG as

a linear program Minimize Constraints

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

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

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