Co-opetition in Network TasksYoram Bachrach, Peter Key, Jeff Rosenschein, Morteza

Zadimoghaddam, Ely Porat

Agenda

Joint Network Tasks

Advertising in Networks

Network Security

3

Negotiation

4

Negotiation

“Collective Buying Power”

Quota: 100 BuyersReward: Discount of $10 (total saving 10*100=$1000)

25 Users 70 Users 50 Users 30 Users

Transferable Utility Games

• Agents: • Coalition: • Characteristic function: • Simple coalitional games:

– Win or Lose

• Agreements (imputations):– A payoff vector

• Efficiency:

– Coalition’s payoff:

Solution ConceptsC v(C)

…

GAME IMPUTATION

Solution ConceptsC v(C)

…

GAME IMPUTATION

Stability

Unblocked agreements

The Core: imputation such that:

Solution ConceptsC v(C)

…

GAME IMPUTATION

Fairness (Power)

Average contribution across all agent permutations

Shapley’s value:

[

Solution ConceptsC v(C)

…

GAME IMPUTATION

Fairness (Power)

Average contribution across all agent coalitions

Banzhaf’s index:

Solving the Groupon Game

• Average contribution across all permutations𝜙𝑖 (𝑣 )= 1

𝑛 ! ∑𝜋∈Π [𝑣 (𝑠𝜋 (𝑖 )∪ {𝑖 } )−𝑣 (𝑠𝜋 (𝑖)¿)]¿

Users 25 70 50 30

8.33% 41.67% 25% 25%

Required:100 Users

25 Users 70 Users 50 Users 30 Users

Solving the Groupon Game

• Average contribution across all permutations𝜙𝑖 (𝑣 )= 1

𝑛 ! ∑𝜋∈Π [𝑣 (𝑠𝜋 (𝑖 )∪ {𝑖 } )−𝑣 (𝑠𝜋 (𝑖)¿)]¿

Users 15 70 50 30

0% 66.67% 16.66% 16.66%

Required:100 Users

15 Users 70 Users 50 Users 30 Users

Solving the Groupon Game

• Core: no deviations – Cannot win without the 70 usersUsers 15 70 50 30

0% 100% 0% 0%

Required:100 Users

15 Users 70 Users 50 Users 30 Users

Display Advertising

Sponsored Search Advertising

Social Network Advertising

Social Advertising In Groupon

Connectivity Games

s

t

Connectivity Games

s

t

Coalition

Connectivity Games

s

t

Coalition

Connectivity Games

s

tCoalition

Connectivity Games

s

tCoalition

Richer Model

p

p

p

b

Network Reliability

p

p

p

b

Connectivity Games

• Agents are vertices in a graph – Vertices are either primary or backbone

• wins if it connects all primary vertices – Using the graph induced by

• Extension of single source-target vertices– Advertise to target audience– Allow reliable network communication

pp

p

b

Example Network (1)

Example Network (2)

Hotspots and Bargaining

• Fair payment for advertising?– Power indices reflect contribution– Probabilistic assumptions

• Target vertex survives, other vertices fail with probability

• Bargaining power– Core reflects stable agreements

• Alternative coalitions and agreements

– Empty unless veto vertices exist• Relaxation:

Computational LimitationsCG Solution Computation

Power indicesBanzhaf, Shapley

#P-Complete (even without backbones)Polynomial algorithm for treesGeneral approximations

Core Polynomial algorithmFinding veto agents

Maximal Excess (-core)

coNP-completePolynomial algorithm in trees

Network Security

• Physical networks– Placing checkpoints – Locations for routine checks

• Computer networks– Protecting servers and links from attacks

• Various costs for different nodes and links– How easy it is to deploy a check point– Performance degradation for protected servers

• What agreements would be reached regarding related budgets and rewards?

Security Crowdsourcing

• Texas Virtual Boarder Watch– Individuals observe US-Mexico border for suspicious behavior

Blocking an adversary

s

t

Blocking an adversary

s

t

Blocking an adversary

s

t

Blocking an adversary

s

t

Blocking an adversary

s

t

Blocking an adversary

s

t

Blocking an adversary

s

t

Incorporating costs

s

t8

2

5

3

3

2

1

2

2

7

Incorporating costs

s

t8

2

5

3

3

2

1

2

2

7

Multiple Adversaries

s1

t1

8

2

5

s2

3

2

t2

2

2

7

Coalitions in Network Security

• Agents must for coalitions to successfully block the adversary– How should they split costs and rewards?

• Security resources are limited– Which node should be allocated these resources first?

• Similar tools from Game Theory

s

t

8

2

5

3

3

2

1

2

2

7

Path Disruption Games

• Games played on a graph G=<V,E> (a network)– Simple version (PDGs): coalition wins if it can block the adversary and

loses otherwise

– Model with costs (PDGCs): a coalition is guaranteed a reward r for blocking the adversary, but incurs the cost of its checkpoints

Computational LimitationsPDG Solution Computation

Coalition utility (optimal strategy) NP-Hard for multiple adversaries and costsPolynomial algorithm for other cases

Power indicesBanzhaf, Shapley

#P-Complete even for single adversary and no costs

Core Polynomial algorithm

Maximal Excess (-core)

Polynomial algorithm for single adversaryNP-Complete for multiple adversaries

Related Models• Network Flow Games

– C’s value: the maximal flow it can send between s and t

• Collusion in network auctions– Procurer buys a path from s to t in an auction– C’s value: obtained price when rigging the auction

Conclusions

pp

p

b

st

8

2

5

3

3

2

1

2

2

7