Hierarchical Choice Models for Adversarial Risk

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Hierarchical Choice Models for Adversarial Risk

• Michael Porter
• SAMSI AR Working Group
• 29 Nov 2007

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A1 A2 An
D1 UD(1,1) UD(1,2) UD(1,j) UD(1,n)
D2 UD(2,1) UD(2,2) UD(2,j) UD(2,n)
UD(i,1) UD(i,2) UD(i,j) UD(i,n)
Dm UD(m,1) UD(m,2) UD(m,j) UD(m,n)
A1 A2 An
D1 P(1,1) P(1,2) P(1,j) P(1,n)
D2 P(2,1) P(2,2) P(2,j) P(2,n)
P(i,1) P(i,2) P(i,j) P(i,n)
Dm P(m,1) P(m,2) P(m,j) P(m,n)
UD(i,j) Defenders Utility of Attackers choice
j given Defender chose i. P(i,j) Estimated
probability Attacker chooses j given Defender
chooses i.
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Defenders Action
EUD
D1 ¹D1 ? UD(1,j) P(1,j)
D2 ¹D2 ? UD(2,j) P(2,j)
¹Di ? UD(i,j) P(i,j)
Dm ¹Dm ? UD(m,j) P(m,j)
Which action should Defender take?
D argmaxi mDi
D maximizes expected utility
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Estimating Attacker Probabilities

• The estimation of P(i,j) seems very difficult as
  it must account for
• A1 The utility of outcome (i,j) to the attacker
• A2 The probability that the attacker will even
  evaluate action j
• Financial constraints
• Supply constraints
• Technological constraints
• Etc.

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?

• Estimate A1 and A2 separately
• Let UA(i,j) V(i,j) e(i,j)
• Random utility where e(i,j) describes the
  uncertainly in estimating the attackers utility
  for outcome (i,j)
• V(i,j) is the point estimate for the attackers
  utility (non-stochastic)
• Let P(j 2 Ci) be the estimated probability that
  action j is in the attackers choice set when
  defender chooses action i

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Multinomial Choice Models

• Pi(j) P(Uj gt maxk?j Uk)
• P(Vj ej gt maxk?j Vk ek)
• Assume ej are independent with Type I extreme
  value distribution (Gumbel Distribution)
• ej EV1(j,µ)

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Type I Extreme Value R.V.s

• P1 If X EV1(,µ)
• then Xb EV1( b,µ)
• P2 If X EV1(X,µ) and Y EV1(Y,µ)
• then FX-Y(x-y)
• (1expµ(X- Y - (x-y)))-1
• P3 If Xi EV1(i,µ)
• then
• maxi Xi EV1(µ-1 ?expiµ, µ)

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Multinomial Logit Model

• Ignore P(j 2 Ci) for the moment
• Pi(j) P(Vj ej gt maxk?j Vk ek)
• Assume ej EV1(0,µ)
• Vjej EV1(Vj,µ) (from P1)
• Vjej maxk?j Vk ek
• EV1(µ-1 ?k?j expVkµ,µ)
• (from P3)
• Pi(j) P(Vj ej gt Vj ej)
• P(Vj ej - Vj ej lt 0)
• (1expµ(Vj - Vj))-1 (from
  P2)

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Defenders Probability of Action j

• Further derivation leads to
• Hierarchical Multinomial Choice

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Defenders Action

• The expected utility for defender action i
  becomes
• ¹Di ? UD(i,j) Pi(j)
• And the optimal action for defender