11

The Cost of Stability in Network Flow Games

Ezra ResnickYoram Bachrach

Jeffrey S. Rosenschein

22

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

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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

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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

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TNFG example

Threshold: 3

a

b ts

c

2

1

1

2

1

1

66

TNFG winning coalition

Threshold: 3

a

b ts

c

2

1

1

2

1

1

77

TNFG losing coalition

Threshold: 3

a

b ts

c

2

1

1

2

1

1

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Distributing coalitional gains

Imputation: a distribution of the grand coalition’s 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: })({: avpNa a

Na

a Nvp )(

Ca

apCp )(

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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

)()(: CvCpNC

)()( CvCp

1010

The core of a TNFG

Threshold: 3

In a simple game, the core consists of imputations which divide all gains among the veto agents

a

b ts

c

2

1

1

2

1

1

0.5 0.5

0 0

00

1111

A TNFG with an empty core

Threshold: 2

a

b ts

c

2

1

1

2

1

1

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

Threshold: 2

Q. What is the CoS?

a

b ts

c

2

1

1

2

1

1

1 0

1 0

00

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 max-flow) of the network

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CoS in connectivity games

a

b ts

c

d

e

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CoS in connectivity games

a

b ts

c

d

e

CoS = min-cut = max-flow = 2

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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

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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

b = 1, r = 2, f = 3CoS = 1.5

Threshold: 2

a

b ts

c

1

1

1

1

1

1

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Serial TNFGs

s t

1

2

2

31 s t

1

3

1

31

1

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Serial TNFGs

s

1

2

2

31 t

1

3

1

31

1

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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

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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:

Constraints:

Ee

ep

)(:

0:

CvpEC

pEe

Cee

e

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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 baaa n ,,,, 21

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TNFG super-imputation stability

Threshold: b Super-imputation p gives an edge

with capacity ai a payoff of

v1

v2 ts

vn

…

a1

a2

an

a1

a2

an

)1(2 bai

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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