Path Disruption Games (Cooperative Game Theory meets Network Security)

1
Path Disruption Games(Cooperative Game Theory
meets Network Security)

• Yoram Bachrach, Ely Porat
• Microsoft Research Cambridge

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Agenda
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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
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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

9M
p1
p2
p3
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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?

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Treasure Island Forming Coalitions
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Treasure Island Sharing Rewards

• Some agreements wont 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
• Smolletts ship really helps the gains

1000
p1
p2
p3
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UK Elections 2010 Budgets and Politics
Conservatives Labour Lib-Dems
306 258 57

• 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 57/6508.8 of seats
• But large influence on policy
• Other alternative for the conservatives
  government with labour
• Not very appealing to the conservatives

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An Alternate Universe
Conservatives Labour Liberals Democrats
306 258 28 29

• 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

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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?

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Coalitional Game Theory
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Transferable Utility Games

• Agents
• Coalition
• Characteristic function
• Two flavors cost and surplus sharing
• Simple coalitional games
• Coalitions either win or lose
• Monotone games gt
• More agents gt More money
• Super-additive games
• It is always worthwhile for coalitions to merge
• The Grand Coalition would form

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Transferable Utility Games
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Agent properties

• Veto agent
• Cant win without the agent (simple games)
• Cant generate any value without the agent
  (Non-simple games)
• Dummy agent
• Never contributes to any coalition
• Equivalent agents , gt
• 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

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Payoffs

• 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) lt v(C)

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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!

1000
p1 900
p2 50
p3 50
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The Core (Stability)

• 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)

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Treasure Island the Core

• Two coalitions can block
• Only need to make sure get at least
  200

1000
p1
p2
p3
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X-Ray Problem the Core

• c1 c2 c3 9M
• For any imputation c, some pair must pay at least
  6M
• So cicj gt 5
• However v( I,j ) 5
• Thus any imputation c is blocked by some pair
  i,j
• The core is empty

9M
c1
c2
c3
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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)

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WVGs and the UK Elections
Conservatives Labour Lib-Dems
306 258 57

• 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 Liberals Democrats
306 258 28 29
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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?

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Fairness

• Return of the Pirates

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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
• (vw)(C) v(C)w(C)
• E.g. playing both treasure island and treasure
  cave
• Then an agents payoff in vw is the sum of her
  payoffs in v and in w
• Is there a solution concept that fulfills all
  these fairness axioms?

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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
  1000-010000 to coalition

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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
  1000-200800 to coalition

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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 agents 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

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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
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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?

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Power in the UK Elections
Conservatives Labour Lib-Dems
306 258 57
66.66 16.66 16.66

• 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 Liberals Democrats
306 258 28 29
75 8.33 8.33 8.33
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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?

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Blocking an adversary
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Blocking an adversary
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Blocking an adversary
s
t
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Blocking an adversary
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Blocking an adversary
s
t
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Blocking an adversary
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Blocking an adversary
s
t
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Incorporating costs
3
1
2
s
2
t
8
5
2
3
7
2
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Incorporating costs
3
1
2
s
2
t
8
5
2
3
7
2
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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

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Path Disruption Games

• Games played on a graph GltV,Egt (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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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

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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

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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

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Results (cont.)

• Model with costs (PDGCs)
• Computing the value of a coalition is NP-hard
• Min cost vertex cut
• Can do better for trees

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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