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Path Disruption Games(Cooperative Game Theory meets Network Security)
Yoram Bachrach, Ely PoratMicrosoft Research Cambridge
Agenda
Hospitals and Cost Sharing
• Three private hospitals need an X-Ray machine
• Optimal solution – Two cheap machines cost £10M– Buy the £9M machine share it
• Private sector problem– Private hospitals negotiate
• What to buy• How to share the costs
Machine Cost Serving
Cheap £5M 2 hospitals
Expensive £9M 3 hospitals
X-Ray Problem
• Some hospital pair must pay at least £6M• These hospitals can simply buy the cheap
machine and pay only £5M• Any cost sharing agreement is unstable
p1 p2 p3
£9M
Treasure Island
• Jim, Billy and Smollett are looking for a buried treasure, worth a £1000– Billy and Jim each have half of the map
• Each half is useless on its own
– Smollett has a ship that can sail to treasure island• Renting a ship from anyone else costs £800
– v(J)=v(B)=v(S)=v(J,S)=v(B,S)=£0– v(J,B)=£200– V(J,B,S)=£1000
• How should they split the gains?
Treasure Island – Forming Coalitions
£200 £1000
Treasure Island – Sharing Rewards
–Some agreements won’t last long, and others are stable• E.g. giving Smollett £900 and Jim and Billy £50 each
–What is a fair way to divide the money?• Cannot win without Jim and Billy• Smollett’s ship really helps the gains
p1 p2 p3
£1000
UK Elections 2010: Budgets and Politics
• No party had the required majority (326 seats)– Hung parliament
• Second time since World War II – Previous time was 1974
• First coalition government to eventuate from elections– The Lib-Dems only had 57/650=8.8% of seats
• But large influence on policy• Other alternative for the conservatives – government with labour
– Not very appealing to the conservatives…
Conservatives Labour Lib-Dems
306 258 57
An Alternate Universe
• Would the Conservatives be more powerful or less powerful in this alternate universe?– Intuition: much more alternatives to choose from!
• What determines the balance of power?– Suppose parties have to allocate a budget…
Conservatives Labour Liberals Democrats
306 258 28 29
Cooperative Games
• Agents must cooperate to achieve their goals…• … but are still selfish– Maximize their share of the rewards– Obtain the outcome maximizing their utility– Minimize their own cost– Maximize their influence
• What teams and agreements would form?
Coalitional Game Theory
Transferable Utility Games
• Agents: • Coalition:• Characteristic function:
– Two flavors: cost and surplus sharing
• Simple coalitional games:– Coalitions either win or lose
• Monotone games => – More agents => More money
• Super-additive games – It is always worthwhile for coalitions to merge– The Grand Coalition would form
Transferable Utility Games
Agent properties
• Veto agent– Can’t win without the agent (simple games)– Can’t generate any value without the agent (Non-simple games)
• Dummy agent– Never contributes to any coalition
• Equivalent agents , => – Contribute equally to any coalition that contains neither of them
• Critical agent for a coalition– The coalition wins with the agent, but loses without the agent
• Imputations define how the total utility is distributed
• A payoff vector such that
• Individual rationality
– Otherwise, an agent can do better alone• The payoff of a coalition C is
• A coalition C is blocking if p(C) < v(C)
Payoffs
Treasure Island – Imputations
–Is the vector p=(900,50,50) blocked? By what coalition?–What about p=(100,500,400)?–And p=(100,899,1)?–Or p=(0,1,999)?
• Stability does not mean fairness!
p1900£
p250£
p350£
1000£
• All imputations that are not blocked by any coalition• For any coalition C, p(C) ≥ v(C)
– For cost sharing games, the inequality is reversed
• No coalition is incentived to defect from the grand coalition• Gillies (1953) and von Neumann & Morgenstein (1947)
The Core (Stability)
Treasure Island – the Core
• Two coalitions can block:
• Only need to make sure get at least 200£
p1 p2 p3
1000£
£200 £1000
X-Ray Problem – the Core
• c1 + c2 + c3 = £9M– For any imputation c, some pair must pay at least £6M
• So ci+cj > 5– However v( {I,j} ) = 5– Thus any imputation c is blocked by some pair {i,j}
• The core is empty
c1 c2 c3
£9M
Weighted Voting Games (WVG)
• Set of agents• Each agent has a weight • A game has a quota• A coalition C wins if • A simple game (coalitions either win or lose)
ia A iw R
i
ia C
w q
q
WVGs and the UK Elections
• Game 1: [306, 258, 57; 326]
• Game 2: [306, 258, 28, 29; 326]
• What is a fair way of allocating the budget?• How does this “weight splitting” affect power?– Is power proportional to the weight?
Conservatives Labour Lib-Dems
306 258 57
Conservatives Labour Liberals Democrats
306 258 28 29
Power in WVGs• Consider
– No single agent wins– Any coalition of two agents wins– The grand coalition wins– No agent has more power than any other
• Voting power is not proportional to voting weight– Ability to change the outcome of the game with your vote– How do we measure voting power?
1 2 351, 50, 26, 26q w w w
Fairness
• Return of the Pirates
Treasure Island (1000£) Treasure Cave (2000£)
Fairness Requirements
• A solution concept maps a game (characteristic function) to an imputation for that game
• Efficiency Axiom: • Dummy Axiom: dummy agents get nothing• Symmetry Axiom: Equivalent agents get the same• Additivity axiom:
– If a game is composed of two sub-games• (v+w)(C) = v(C)+w(C)• E.g. playing both treasure island and treasure cave
– Then an agent’s payoff in v+w is the sum of her payoffs in v and in w
• Is there a solution concept that fulfills all these fairness axioms?
Marginal Contribution
• Treasure island
• The coalition has a value of 0£– No full map
• The coalition has a value of 1000£
• Agent has a marginal contribution of 1000£-0£=10000£ to coalition
Marginal Contribution
• Treasure island
• The coalition has a value of 200£– Full map, no ship
• The coalition has a value of 1000£
• Agent has a marginal contribution of 1000£-200£=800£ to coalition
The Shapley Value: Fairness
• Given an ordering of the agents in I, denotes the set of agents that appear before i in
• The Shapley value is an agent’s marginal contribution to its predecessors, averaged across all permutations
• The only solution concepts that fulfills all of the previously defined fairness axioms
• Can also be used to measure power
Treasure Island – the Shapley Value
0 0 1000
0 1000 0
0 0 1000
800 0 200
800 200 0
0 1000 0
Average 266.66 366.66 366.66
Power Indices
• Power in weighted voting games can be computed using the Shapley value– WVGs are simple games
• The Shapely value measures the proportion of coalitions where an agent is critical
• Each permutation has exactly one critical agent• Simple generative model
• Are there alternative models or power indices?
Power in the UK Elections
• Game 1: [306, 258, 57; 326]
• Game 2: [306, 258, 28, 29; 326]
• Split makes the labour less powerful– But the power goes to the conservatives…– … not the Lib-Dems
Conservatives Labour Lib-Dems
306 258 57
66.66% 16.66% 16.66%
Conservatives Labour Liberals Democrats
306 258 28 29
75% 8.33% 8.33% 8.33%
Security in Networks
• Physical network security– Placing checkpoints – Locations for routine checks
• Network security– 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
• How should the budget be spent on security resources?
Blocking an adversary
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Blocking an adversary
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Blocking an adversary
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Blocking an adversary
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Blocking an adversary
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Blocking an adversary
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Blocking an adversary
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Incorporating costs
s
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2
5
3
3
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1
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2
7
Incorporating costs
s
t8
2
5
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3
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1
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7
Network Security Hotspots
• Agents must for coalitions to successfully block the adversary– Obtain a certain reward or budget for achieving the task– How should this reward be shared between the agents
• Stability– No subset of the coalition should have an incentive to form an alternative coalition
• Fairness– Reflect the contribution of the each agent
• Security resources are limited– Which node / link should be allocated these resources first?– Power indices allow finding such reliability hotspots
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
Power and Security
• Suppose all check points have equal probability,50%, of blocking the adversary or not blocking– We have limited security resources– Which nodes should be protected first?
• “Powerful” nodes are more critical– Suppose we can only choose one node where the adversary is blocked
with 100% probability – The Banzhaf index of a node is the probability of stopping the
adversary when:• This node blocks with probability 100%• All other nodes block with probability of 50%
Stability in PDGs: the Core
• Given a reward for blocking the adversary what check point coalitions would form?– We want the agents to work under enforceable contracts:
• Which check points are used and • How to share the reward
• The core constitutes a stable allocation– A distribution not in the core would break down the
coalition structure– Unable to agree on a contract and infinite negotiation
Results
• PDGs (several adversaries, no cost)– Can test for veto agents and compute the core in
polynomial time– Computing the maximal excess for an imputation
(payoff vector) is NP-complete• NP-hard to compute the least core
– Testing for dummy agents is coNP-Complete– Computing the Banzhaf index is #P-complete• But for trees it is computing in polynomial time
Results (cont.)
• Model with costs (PDGCs):– Computing the value of a coalition is NP-hard• Min cost vertex cut
– Can do better for trees
Conclusion & Future Directions
• Suggested a game theoretic model for network security based on blocking adversaries
• Future work– Other solution concepts: power indices, nucleolus, kernel– More complex network security domains