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Adversary/Defender Model of Risk of Terrorist Attacks using Belief. SAND 2006-1470C Risk Symposium 2006 Risk Analysis for Homeland Security and Defense: Theory and Application March 20, 2006 John Darby Sandia National Laboratories 505-284-7862. Acknowledgments.

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adversary defender model of risk of terrorist attacks using belief

Adversary/Defender Modelof Risk of Terrorist Attacks using Belief

SAND 2006-1470C

Risk Symposium 2006

Risk Analysis for Homeland Security and Defense:

Theory and Application

March 20, 2006

John Darby

Sandia National Laboratories


  • Linguistic Model for Adversary
    • Logic Evolved Decision (LED) Methodology developed at Los Alamos National Laboratory by Terry Bott and Steve Eisenhawer
    • Belief/Plausibility measure of uncertainty added
  • Numerical Model for Defender
    • Uses work and suggestions from Jon Helton, Arizona State University
  • Belief/Plausibility with Fuzzy Sets
    • 1986 paper by Ronald Yager 505-284-7862

  • Apply Belief/Plausibility Measure of Uncertainty to Evaluating Risk from Acts of Terrorism
  • Why Belief/Plausibility?
    • We have considerable epistemic (state of knowledge) uncertainty
      • Terrorists acts are not random events, but we have considerable epistemic uncertainty for evaluating them 505-284-7862

random vs intentional
Random vs. Intentional
  • Random event: Earthquake
    • Magnitude of earthquake independent of structures exposed to earthquake
    • Event is a “dumb” failure
  • Intentional Event: Terrorist Blow Up Building
    • Are the Consequences worth the effort to gather the resources to be able to defeat any security systems in place and destroy the building
    • Event is a choice by a thinking, malevolent adversary with significant resources 505-284-7862

safety risk vs terrorist risk

Terrorist Risk (Intentional)

Safety Risk (Random)



Maximum Risk

Maximum Risk





Safety Risk vs. Terrorist Risk 505-284-7862

belief plausibility for epistemic uncertainty
Belief/Plausibility for Epistemic Uncertainty
  • Toss a fair coin
    • Uncertainty is aleatory (random)
    • Probability Heads is ½
    • Probability Tails is ½
  • But if we do not know coin is fair
    • May be two-headed or two-tailed
    • Epistemic (state of knowledge) uncertainty
    • Insufficient information to assign Probability to Heads and Tails
    • Belief/Plausibility for Heads is 0/1
    • Belief/Plausibility for Tails is 0/1
  • With more information (actually tossing the coin) we can reduce Epistemic Uncertainty
  • For Fair Coin we cannot reduce aleatory uncertainty 505-284-7862

belief and plausibility
Belief and Plausibility
  • Belief / Plausibility form a Lower / Upper Bound for Probability
  • Belief is what probability will be
  • Plausibility is what probability could be
  • Similar to a Confidence Interval for a Parameter of a probability distribution; a confidence measure that parameter is in interval, but exactly where in interval is not known
  • Belief/Plausibility both reduce to Probability if Evidence is Specific


Probability is somewhere in [Belief, Plausibility] Interval

Belief 505-284-7862

fuzzy sets for vagueness
Fuzzy Sets for Vagueness
  • Consequences (Deaths) are “Major”
    • “Major” is fuzzy: between about 500 and about 5000 deaths 505-284-7862

adversary defender
  • Adversary (them)
  • Defender (us)
  • Adversary and Defender each have different goals and different states of knowledge
  • Risk = Threat x Vulnerability x Consequence
    • Defender goal: Minimize Risk with available resources
    • Adversary goal: Maximize Consequence with available resources (working assumption)
  • Adversary is the Threat
    • Epistemic Uncertainty for Vulnerability and Consequence
  • Defender knows Vulnerability and Consequence
    • Epistemic Uncertainty for Threat 505-284-7862

scenario and dependence
Scenario and Dependence
  • Scenario defined as: Specific Target, Adversary Resources and Attack Plan
    • Resources are: attributes (numbers weapons, etc.) and knowledge
  • Risk for a Scenario
    • Risk = f x P x C
    • f is frequency of scenario (number per year)
    • P is probability scenario is successful
    • C is consequence
  • P is conditional on scenario (adversary resources and security in place, both physical security and intelligence gathering)
  • C is conditional on scenario (target)
  • Risk is scenario dependent
    • Adversary has choice (not random event)
    • Risk must consider millions of scenarios 505-284-7862

defender model for a scenario
Defender Model for a Scenario
  • Risk = f x P x C
  • f, P, and C are random variables with uncertainty
  • Degrees of Evidence to f, P, C based on state of knowledge
  • Convolution using Belief/Plausibility Measure of Uncertainty 505-284-7862

example result from defender model
Example Result from Defender Model

Probability assumes evidence for an interval uniformly distributed over interval 505-284-7862

defender ranking of scenarios


Scenarios: Ranked

By Decreasing Expected Value




Expected Value of Deaths per Year: f*P*C

Defender Ranking of Scenarios
  • For Belief/Plausibility Expected Value is an Interval [Elow, Ehigh]. Reduces to point (Mean) for Probability
  • Rank by Ehigh, Subrank by Elow

Scenario 505-284-7862

next level of detail for defender ranking

Expected Value of

Likelihood: f


Expected Value of Deaths: P*C


Next Level Of Detail for Defender Ranking 505-284-7862

adversary model
Adversary Model
  • Use surrogate Adversary (Special Forces)
  • Adversary has Choice
    • All Variables of concern must be “OK” or

we will pick another scenario

    • Recruit Insider? Not unless already placed
    • Large Team? Concern about being detected by Intelligence
    • Uncertainty?
      • Door was green yesterday, is red today…What else changed?
  • Variables for Adversary Decision are Not all Numeric
    • Consequence = Deaths x Economic Damage x Fear in Populace x Damage to National Security x Religious Significance x …..
    • Deaths and Economic Damage are numeric
    • Fear in Populace, Damage to National Security, and Religious Significance are not numeric 505-284-7862

adversary model16
Adversary Model
  • Linguistic Model
  • Develop Fuzzy Sets for Each Variable
  • Develop Approximate Reasoning Rule Base for Linguistic Convolution of Variables to Reflect Scenario Selection Decision Process (LANL LED process)
  • We are not the Adversary, we try to think like the Adversary
    • Considerable Epistemic Uncertainty
    • Use Belief/Plausibility Measure of Uncertainty Propagated up the Rule Base 505-284-7862

adversary model17
Adversary Model
  • Assume Adversary Goal is Maximize Expected Consequence
    • Expected Consequence ≡ P x C
    • Expected Consequence is Adversary estimate of Consequence, C, weighted by Adversary estimate of Probability of Success, P 505-284-7862

example of adversary model
Example of Adversary Model
  • Rule Base and Variables
    • Expected Consequence = Probability Of (Adversary) Success x Consequence
    • Probability Of Success = Probability Resources Required Gathered Without Detection x Probability Information Required Can Be Obtained x

Probability Physical Security System can be Defeated

    • Consequence = Deaths x Damage To National Security
  • Fuzzy Sets
    • Expected Consequence = {No, Maybe, Yes}
    • Probability Of Success = {Low, Medium, High}
    • Consequence = {Small, Medium, Large}
    • Probability Resources Required Gathered Without Detection =
    • {Low, Medium, High}
    • Probability Information Required Can Be Obtained = {Low, Medium, High}
    • Probability Physical Security System can be Defeated = {Low, Medium, High}
    • Deaths = {Minor, Moderate, Major, Catastrophic}
    • Damage To National Security = {Insignificant, Significant, Very Significant} 505-284-7862

example of adversary model19
Example of Adversary Model
  • Part of Example Rule Base 505-284-7862

example of adversary model20
Example of Adversary Model
  • Focal Elements (Evidence) for Particular Scenario
    • Deaths: 0.8 for {Major, Catastrophic}
    • 0.2 for {Moderate, Major}
    • Damage To National Security: 0.1 to {Insignificant, Significant}

0.9 to {Significant, Very Significant}

    • Probability Resources Required Obtained Without Detection:

0.7 to {Medium}

0.3 to {Medium, High}

    • Probability Information Required can Be Obtained:

0.15 to {Medium}

0.85 to {Medium, High}

    • Probability Physical Security System can be Defeated:

1.0 to {Medium, High} 505-284-7862

example of adversary model21
Example of Adversary Model 505-284-7862

example of adversary model22
Example of Adversary Model 505-284-7862

example of adversary model23
Example of Adversary Model 505-284-7862

adversary ranking of scenarios
Adversary Ranking of Scenarios
  • Defender thinking like Adversary Ranks by Plausibility
    • Rank scenarios based on the plausibility for the worst fuzzy set for expected consequence, “Yes” in the prior example, sub-ranked by plausibility of the next-worst fuzzy sets, “Maybe” and “No” in the prior example
  • Note: Actual Adversary using the Model would Rank by Belief
    • “We will not attempt a scenario unless we believe it will succeed”… Osama 505-284-7862

software tools
Software Tools
  • Numerical Evaluation of Risk for Defender
    • BeliefConvolution Java code (written by author)
    • RAMAS RiskCalc
  • Linguistic Evaluation for Adversary
    • LinguisticBelief Java code (written by author)
    • LANL LEDTools 505-284-7862

combinatorics issue for convolution with belief plausibility
“Combinatorics” Issue for Convolution with Belief/Plausibility
  • Convolution using Belief/Plausibility must be done at the focal element level
    • Convolution of 20 variables each with 3 focal elements results in a variable with 320 focal elements
  • Need to Condense or Aggregate Degrees of Evidence for Large Scale Problems
    • Must “condense” or “aggregate” focal elements with repeated convolution of variables to reduce number of focal elements to manageable size 505-284-7862

aggregation in beliefconvolution code





Bins (linear or log10)

Aggregation in BeliefConvolution Code 505-284-7862

aggregation in linguisticbelief code
Aggregation in LinguisticBelief Code
  • Aggregation is “Automatic” per Rule Base
    • Focal Elements for Inputs are on Fuzzy Sets For Input variables
    • Focal Elements for Output are on Fuzzy Sets of Output Variable
  • Happiness = Health x Wealth x Outlook on Life
    • Assume Health, Wealth, and Outlook on Life

each have 3 focal elements: implies Health x Wealth x Outlook on Life has 27 focal elements

    • Happiness = {Not So Good, OK, Good}
    • Evidence on Happiness is aggregated on subsets of

{Not So Good, OK, Good} so Happiness has at most

23 = 8 Focal Elements 505-284-7862

  • Focal Elements for Different Variables are Noninteracting for Convolution with Belief/Plausibility (independent for Probability)
    • SAND2004-3072 “Dependence in Probabilistic Modeling, Dempster-Shafer Theory, and Probability Boxes”, Ferson et al, Oct. 2004
  • Adversary Goal is to Maximize Expected Consequence
    • Don’t like this goal? Change the rule base! 505-284-7862

future work
Future Work
  • How Consider Millions of Scenarios?
    • Binning and Screening Criteria
    • Can reason at any level if honest about uncertainty
      • What are # deaths from a terrorist attack? [0, 6x109]
    • Use binning/filtering process at successively finer levels of detail, but capture uncertainty at each step
  • Dependence? Risk will always be dependent on scenarios
  • Implement Monte-Carlo Sampling with Belief/Plausibility (work by Helton et al)
    • Eliminates problem of dependence from a repeated variable that cannot be factored
    • Automate evaluation of large scale problems 505-284-7862

  • Adversary and Defender have Different States of Knowledge
  • Defender wants to Minimize Risk
  • Adversary wants to Maximize Expected Consequence (working assumption)
  • Adversary selection of Scenarios(s) is not random
  • Large Epistemic Uncertainty for Defender
    • Use Belief/Plausibility Measure instead of Probabililty Measure
  • Adversary selects Scenario(s) with “high” ranking
    • Defender thinks like adversary using linguistics
    • Linguistic variables modeled with Fuzzy Sets
  • Adversary/Defender Models Developed
    • Implemented in Java software 505-284-7862