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Co-opetition in Network TasksYoram Bachrach, Peter Key, Jeff Rosenschein, Morteza
Zadimoghaddam, Ely Porat
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Agenda
Joint Network Tasks
Advertising in Networks
Network Security
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3
Negotiation
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4
Negotiation
“Collective Buying Power”
Quota: 100 BuyersReward: Discount of $10 (total saving 10*100=$1000)
25 Users 70 Users 50 Users 30 Users
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Transferable Utility Games
• Agents: • Coalition: • Characteristic function: • Simple coalitional games:
– Win or Lose
• Agreements (imputations):– A payoff vector
• Efficiency:
– Coalition’s payoff:
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Solution ConceptsC v(C)
…
GAME IMPUTATION
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Solution ConceptsC v(C)
…
GAME IMPUTATION
Stability
Unblocked agreements
The Core: imputation such that:
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Solution ConceptsC v(C)
…
GAME IMPUTATION
Fairness (Power)
Average contribution across all agent permutations
Shapley’s value:
[
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Solution ConceptsC v(C)
…
GAME IMPUTATION
Fairness (Power)
Average contribution across all agent coalitions
Banzhaf’s index:
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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
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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
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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
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Display Advertising
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Sponsored Search Advertising
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Social Network Advertising
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Social Advertising In Groupon
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Connectivity Games
s
t
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Connectivity Games
s
t
Coalition
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Connectivity Games
s
t
Coalition
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Connectivity Games
s
tCoalition
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Connectivity Games
s
tCoalition
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Richer Model
p
p
p
b
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Network Reliability
p
p
p
b
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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
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Example Network (1)
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Example Network (2)
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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:
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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
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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?
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Security Crowdsourcing
• Texas Virtual Boarder Watch– Individuals observe US-Mexico border for suspicious behavior
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Blocking an adversary
s
t
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Blocking an adversary
s
t
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Blocking an adversary
s
t
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Blocking an adversary
s
t
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Blocking an adversary
s
t
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Blocking an adversary
s
t
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Blocking an adversary
s
t
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Incorporating costs
s
t8
2
5
3
3
2
1
2
2
7
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Incorporating costs
s
t8
2
5
3
3
2
1
2
2
7
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Multiple Adversaries
s1
t1
8
2
5
s2
3
2
t2
2
2
7
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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
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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
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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
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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
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Conclusions
pp
p
b
st
8
2
5
3
3
2
1
2
2
7