Co-opetition in Network Tasks
Yoram Bachrach, Peter Key, Jeff Rosenschein, Morteza
Zadimoghaddam, Ely Porat
Agenda
Joint Network Tasks
Advertising in Networks
Network Security
Negotiation
3
Negotiation
“Collective Buying Power”
Quota: 100 Buyers
Reward: Discount of $10 (total saving 10*100=$1000)
25 Users
70 Users
50 Users
30 Users
4
Transferable Utility Games
•
•
•
•
Agents: 𝐼 = {𝑎1 , … , 𝑎𝑛 }
Coalition: 𝐶 ⊆ 𝐼
Characteristic function: 𝑣: 2𝐼 → ℛ
Simple coalitional games: 𝑣: 2𝐼 → {0,1}
– Win or Lose
• Agreements (imputations):
– A payoff vector (𝑝1 , … , 𝑝𝑛 )
• Efficiency:
𝑖∈𝐼 𝑝𝑖
= 𝑣(𝐼)
– Coalition’s payoff: 𝑝 𝐶 =
𝑖∈𝐶 𝑝𝑖
Solution Concepts
C
…
v(C)
Solution Concepts
C
v(C)
…
Stability
The Core: imputation 𝑝 such that:
Unblocked agreements
∀𝐶 𝑝 𝐶 ≥ 𝑣(𝐶)
Solution Concepts
C
v(C)
…
Fairness (Power)
Shapley’s value:
Average contribution across all agent
permutations
𝑠ℎ𝑖 𝑣 =
1
𝑛!
[𝑣 𝑠𝜋 𝑖 ∪ 𝑖
𝜋∈Π
[w1 = 50, w2 = 26, w3 = 26; q = 51]
− 𝑣(𝑠𝜋 (𝑖))]
Solution Concepts
C
v(C)
…
Fairness (Power)
Average contribution across all agent
coalitions
Banzhaf’s index:
𝑠ℎ𝑖 𝑣 =
1
𝟐𝒏−𝟏
[𝑣 𝑪 ∪ 𝑖
𝐂 | 𝒊∉𝑪
− 𝑣(𝑪)]
Solving the Groupon Game
25 Users
70 Users
50 Users
30 Users
Required:
100 Users
• Average contribution across all permutations
1
𝜙𝑖 𝑣 =
𝑛!
Users
[𝑣 𝑠𝜋 𝑖 ∪ 𝑖
− 𝑣(𝑠𝜋 (𝑖))]
𝜋∈Π
25
70
50
30
8.33%
41.67%
25%
25%
Solving the Groupon Game
15 Users
70 Users
50 Users
30 Users
Required:
100 Users
• Average contribution across all permutations
1
𝜙𝑖 𝑣 =
𝑛!
Users
[𝑣 𝑠𝜋 𝑖 ∪ 𝑖
− 𝑣(𝑠𝜋 (𝑖))]
𝜋∈Π
15
70
50
30
0%
66.67%
16.66%
16.66%
Solving the Groupon Game
15 Users
70 Users
50 Users
30 Users
Required:
100 Users
• Core: no deviations
– Cannot win without the 70 users
Users
15
70
50
30
0%
100%
0%
0%
Display Advertising
Sponsored Search Advertising
Social Network Advertising
Social Advertising In Groupon
Connectivity Games
t
s
Connectivity Games
Coalition 𝐶 ⊆ 𝐼
t
s
Connectivity Games
Coalition 𝐶 ⊆ 𝐼
t
s
Connectivity Games
Coalition 𝐶 ⊆ 𝐼
t
s
Connectivity Games
Coalition 𝐶 ⊆ 𝐼
t
s
Richer Model
p
b
p
p
Network Reliability
p
b
p
p
Connectivity Games
• Agents 𝐼 ⊆ 𝑉are vertices in a graph 𝐺 = (𝑉, 𝐸)
– Vertices 𝑉 ∖ 𝐼 are either primary 𝑉𝑝 or backbone 𝑉𝑏
• 𝐶 ⊆ 𝐼 wins if it connects all primary vertices 𝑉𝑝
– Using the graph induced by 𝐶 ∪ 𝑉𝑝 ∪ 𝑉𝑏
• Extension of single source-target vertices
– Advertise to target audience
– Allow reliable network communication
p
p
b
p
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: 𝜖 − 𝑐𝑜𝑟𝑒
• ∀𝐶 𝑝 𝐶 ≥ 𝑣 𝐶 − 𝜖
1
2
Computational Limitations
CG Solution
Computation
Power indices
Banzhaf, Shapley
#P-Complete (even without backbones)
Polynomial algorithm for trees
General approximations
Core
Polynomial algorithm
Finding veto agents
Maximal Excess
(𝜖-core)
coNP-complete
Polynomial 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
3
1
2
s
2
t
8
5
2
3
7
2
Incorporating costs
3
1
2
s
2
t
8
5
2
3
7
2
Multiple Adversaries
3
t2
2
s1
2
t1
8
5
2
s2
7
2
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
3
1
2
s
2
t
8
5
2
3
7
2
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 Limitations
PDG Solution
Computation
Coalition utility (optimal strategy) NP-Hard for multiple adversaries and costs
Polynomial algorithm for other cases
Power indices
Banzhaf, Shapley
#P-Complete even for single adversary and
no costs
Core
Polynomial algorithm
Maximal Excess
(𝜖-core)
Polynomial algorithm for single adversary
NP-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
p
b
p
p
3
2
s
1
2
t
8
5
2
3
7
2