Defender Acts 1st

A1 A2

D1 C11 C12

D2 C21 C22

where Ci,j=Cost to defender from play (Aj|Di)

A1 A2

D1 1 12

D2 21 22

Random Cost Matrix Expected Cost Matrix

where i,j=E[Ci,j]

Minimax Strategy

A1 A2 Maxj Ci,j

D1 C11 C12 C*1

D2 C21 C22 C*2

A1 A2 Maxj i,j

D1 1 12 *1

D2 21 22 *2

Random Cost Matrix Expected Cost Matrix

Minimax Strategy

A1 A2 Maxj Ci,j

D1N(6,10) N(6,2) C*

1

D2N(6.5,3.5) N(7.5,3) C*2

A1 A2 Maxj i,j

D1 *1

D2 *2

Random Cost Matrix Expected Cost Matrix

Example: Ci,j=N(i,j, i,j)

Minimax Strategy

A1 A2 Maxj Ci,j

D1N(6,10) N(6,2) C*

1

D2N(6.5,3.5) N(7.5,3) C*2

A1 A2 Maxj i,j

D1 *1

D2 *2

Random Cost Matrix Expected Cost Matrix

Example: Ci,j=N(i,j, i,j)

Which action should Defender take?

D*=argmini maxj E[Ci,j]

=argmini *i

Banks and Anderson Strategy #1

Choose D1, but rather close to indifferent

A1 A2 Maxj Ci,j

D1N(6,10) N(6,2) C*

1

D2N(6.5,3.5) N(7.5,3) C*2

D*=argmaxi P(C*i < mink C*k)

Histogram of Dif

C*_1 - C*_2

De

nsi

ty

-20 -10 0 10 20 30 40

0.0

00

.02

0.0

40

.06

0.0

8

P(C*_1 < C*_2)= 0.5053

Histogram of Dif

C*_1 - C*_2

De

nsi

ty

-20 -10 0 10 20 30 40

0.0

00

.02

0.0

40

.06

0.0

8

P(C*_1 < C*_1)= 0.5053

Banks and Anderson Strategy #2

From this, choose D2

Score(i)=mink {C*k} / C*i

Score(i) 2 (0,1] – Larger is better

0.0 0.2 0.4 0.6 0.8 1.0

01

23

45

6

density(x = Score[, 1])

N = 10000 Bandwidth = 0.03499

De

nsi

ty

Score(1)Score(2)

E[Score(1)]=0.815

E[Score(2)]=0.822

A1 A2 Maxj Ci,j

D1N(6,10) N(6,2) C*

1

D2N(6.5,3.5) N(7.5,3) C*2

D*=argmaxi E[Score(i)]

An Alternative Approach

0 5 10 15 20 25

0.0

00

.05

0.1

00

.15

C*=max Cost

De

nsi

ty

f(C*_1)f(C*_2)

m*_1= 10.14

m*_2= 8.9

where m*i=E[C*i]=E[maxj Ci,j]

A1 A2 Maxj Ci,j

D1N(6,10) N(6,2) C*

1

D2N(6.5,3.5) N(7.5,3) C*2

Choose D2, since worst case has lower expected cost

D*=argmini E[maxj Ci,j]