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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 jldarby@sandia.gov 505-284-7862. Acknowledgments.

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Adversary/Defender Model of Risk of Terrorist Attacks using Belief

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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

jldarby@sandia.gov

505-284-7862


Acknowledgments

  • 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

jldarby@sandia.gov 505-284-7862


Goal

  • 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

jldarby@sandia.gov 505-284-7862


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

jldarby@sandia.gov 505-284-7862


Terrorist Risk (Intentional)

Safety Risk (Random)

Likelihood

Likelihood

Maximum Risk

Maximum Risk

(Uncertainty)

(Uncertainty)

Consequence

Consequence

Safety Risk vs. Terrorist Risk

jldarby@sandia.gov 505-284-7862


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

jldarby@sandia.gov 505-284-7862


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

Plausibility

Probability is somewhere in [Belief, Plausibility] Interval

Belief

jldarby@sandia.gov 505-284-7862


Fuzzy Sets for Vagueness

  • Consequences (Deaths) are “Major”

    • “Major” is fuzzy: between about 500 and about 5000 deaths

jldarby@sandia.gov 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

jldarby@sandia.gov 505-284-7862


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

jldarby@sandia.gov 505-284-7862


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

jldarby@sandia.gov 505-284-7862


Example Result from Defender Model

Probability assumes evidence for an interval uniformly distributed over interval

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Worst

Scenarios: Ranked

By Decreasing Expected Value

Best

0

106

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

jldarby@sandia.gov 505-284-7862


Expected Value of

Likelihood: f

0

Expected Value of Deaths: P*C

106

Next Level Of Detail for Defender Ranking

jldarby@sandia.gov 505-284-7862


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

jldarby@sandia.gov 505-284-7862


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

jldarby@sandia.gov 505-284-7862


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

jldarby@sandia.gov 505-284-7862


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}

jldarby@sandia.gov 505-284-7862


Example of Adversary Model

  • Part of Example Rule Base

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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}

jldarby@sandia.gov 505-284-7862


Example of Adversary Model

jldarby@sandia.gov 505-284-7862


Example of Adversary Model

jldarby@sandia.gov 505-284-7862


Example of Adversary Model

jldarby@sandia.gov 505-284-7862


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

jldarby@sandia.gov 505-284-7862


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

jldarby@sandia.gov 505-284-7862


“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

jldarby@sandia.gov 505-284-7862


Evidence

Aggregated

Evidence

*

Bins (linear or log10)

Aggregation in BeliefConvolution Code

jldarby@sandia.gov 505-284-7862


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

jldarby@sandia.gov 505-284-7862


Assumptions

  • 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!

jldarby@sandia.gov 505-284-7862


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

jldarby@sandia.gov 505-284-7862


Summary

  • 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

jldarby@sandia.gov 505-284-7862


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