Game Theory and Risk Analysis for Counterterrorism

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Game Theory and Risk Analysis for
Counterterrorism

• David Banks
• U.S. FDA

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1. Context

• Terrorists can invest in a portfolio of attacks.
• The U.S. can invest in various kinds of defense.
• If the U.S. fails to invest wisely, then we lose
  important battles.

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A Smallpox Exercise

• The U.S. government is concerned about the
  possibility of smallpox bioterrorism.
• Terrorists could make no smallpox attack, a small
  attack on a single city, or coordinated attacks
  on multiple cities (or do other things).

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• The U.S. has considered four defense strategies
• Stockpiling vaccine
• Stockpiling and increasing biosurveillance
• Stockpiling, surveilling, and inoculating first
  responders and/or key personnel
• Inoculating all consenting people with healthy
  immune systems.

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Deciding What To Do

• Currently, the U.S. government is
• Holding public meetings
• Soliciting scientific advice
• Seeking counterintelligence
• Calculating political support
• Balancing alternative threats
• Ensuring availability of options

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• But the U.S. government is not using
• Statistical risk analysis
• Game theory
• Cost-benefit analysis
• Portfolio theory
• These are methodologies that can and should
  inform decision-making to counter terrorist
  threats.

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2. Normal-Form Games

• Classical game theory uses a matrix of costs to
  determine optimal play.
• Optimal play is usually defined as a minimax
  strategy, but sometimes one can minimize expected
  loss instead.
• Both methods are unreliable guides to human
  behavior.

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Game Theory Matrix
No Attack Small Attack Big Attack
Stockpile C11 C12 C13
Surveillance C21 C22 C23
First Responders C31 C32 C33
Mass Inoculation C41 C42 C43
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Minimax Strategy

• The U.S. should choose the defense with smallest
  row-wise max cost.
• The terrorist should choose the attack with
  largest column-wise min cost.
• If these are not equal then a randomized strategy
  is better.

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Minimum Expected Loss

• Sometimes there is more information and different
  structure than classic game theory supposes.
• This occurs in serial games, where probabilities
  can be assigned to future actions. This
  extensive-form game theory generates decision
  trees.

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• Extensive-form game theory invites decision
  theory criteria based upon minimum expected loss.
• In our smallpox exercise, we shall implement this
  by assuming that the U.S. decisions are known to
  the terrorists, and that this affects their
  probabilities of using certain kinds of attacks.

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Game Theory Critique

• Game theory does not take account of resource
  limitations.
• It assumes that both players have the same cost
  matrix.
• It assumes both players act in synchrony (or in
  strict alternation).
• It assumes all costs are measured without error.

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3. Risk Analysis

• Statistical risk analysis makes probabilistic
  statements about specific kinds of threats.
• It also treats the costs associated with threats
  as random variables. The total random cost is
  developed by analysis of component costs.

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Cost Example

• To illustrate a key idea, consider the problem of
  estimating the cost C11 in the game theory
  matrix. This is the cost associated with
  stockpiling vaccine when no smallpox attack
  occurs.
• Some components of the cost are fixed, others are
  random.

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• C11 cost to test diluted Dryvax
• cost to test Aventis vaccine
• cost to make 209 x 106 doses
• cost to produce VIG
• logistic/storage/device costs.

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• Assume that an expert indicates that
• Dryvax and Aventis testing have costs that are
  independent and uniformly distributed on 2
  million, 5 million.
• New vaccine production is not random the
  contract specifies 512 million.
• VIG production is normally distributed with mean
  100 million, s.d. 20 million.
• Logistics costs are normally distributed with
  mean 940 million, s.d. 100 million.

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Other Cij s

• The other costs in the matrix are also random
  variables, and their distributions can be
  estimated in similar Delphic ways.
• Note that different matrix costs are not
  independent they often have components in common
  across rows and columns.

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• Other components include
• Number of attacks this is Poisson with mean 5,
  plus 2.
• Number of key personnel this is uniform between
  2 million and 12 million.
• Number of smallpox cases per attack this is
  Gamma with mean 10 and s.d. 100.

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• Cost to treat one smallpox case this is normal
  with mean 200,000 and s.d. 50,000.
• Cost to inoculate 25,000 people this is normal
  with mean 60,000 and s.d. 10,000.
• Economic costs of a single attack this is gamma
  with mean 5 billion and s.d. 10 billion.

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4. Games Risk

• Game theory and statistical risk analysis can be
  combined to give arguably useful guidance in
  threat management.
• We generate many random tables, according to the
  risk analysis, and find which defenses are best.

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• We run 100 game theory matrices and count how
  many times each defense is optimal in terms of
• Minimaxity
• Minimum Expected Loss
• We also calculate a score for each, since the
  second-best defense may have nearly the same cost
  as the best.

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Screenshot of Output
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Minimum Expected Loss

• The table in the lower right shows the elicited
  probabilities of each kind of attack given that
  the corresponding defense has been adopted.
• These probabilities are used to weight the costs
  in calculating the expected loss.

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Scores

• The scores beside the left-hand tables are found
  by
• Summing the maximum costs in each row
• Dividing each maximum by the sum
• Allocating weight to the decisions
  proportionally.

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5. Conclusions

• For our rough risk analysis, minimax favors
  universal inoculation, minimum expected loss
  favors stockpiling.
• This accords with the public and federal thinking
  on threat preparedness.
• And the approach can be generalized.