Defender Acts 1st

1
Defender Acts 1st
A1 A2
D1 C11 C12
D2 C21 C22
A1 A2
D1 m11 m12
D2 m21 m22
Random Cost Matrix Expected Cost Matrix
where Ci,jCost to defender from play (AjDi)
where mi,jECi,j
2
Minimax Strategy
A1 A2 Maxj Ci,j
D1 C11 C12 C1
D2 C21 C22 C2
A1 A2 Maxj mi,j
D1 m11 m12 m1
D2 m21 m22 m2
Random Cost Matrix
Expected Cost Matrix
3
Minimax Strategy
Example Ci,jN(mi,j, si,j)
A1 A2 Maxj Ci,j
D1 N(6,10) N(6,2) C1
D2 N(6.5,3.5) N(7.5,3) C2
A1 A2 Maxj mi,j
D1 6 6 m1 6
D2 6.5 7.5 m2 7.5
Random Cost Matrix
Expected Cost Matrix
4
Minimax Strategy
Example Ci,jN(mi,j, si,j)
A1 A2 Maxj Ci,j
D1 N(6,10) N(6,2) C1
D2 N(6.5,3.5) N(7.5,3) C2
A1 A2 Maxj mi,j
D1 6 6 m1 6
D2 6.5 7.5 m2 7.5
Random Cost Matrix
Expected Cost Matrix
Which action should Defender take?
Dargmini maxj ECi,j argmini mi
5
A1 A2 Maxj Ci,j
D1 N(6,10) N(6,2) C1
D2 N(6.5,3.5) N(7.5,3) C2
Banks and Anderson Strategy 1
Dargmaxi P(Ci lt mink Ck)
Choose D1, but rather close to indifferent
6
A1 A2 Maxj Ci,j
D1 N(6,10) N(6,2) C1
D2 N(6.5,3.5) N(7.5,3) C2
Banks and Anderson Strategy 2
Score(i)mink Ck / Ci Score(i) 2 (0,1
Larger is better
EScore(1)0.815 EScore(2)0.822
Dargmaxi EScore(i)
From this, choose D2
7
An Alternative Approach
A1 A2 Maxj Ci,j
D1 N(6,10) N(6,2) C1
D2 N(6.5,3.5) N(7.5,3) C2
Dargmini Emaxj Ci,j
Choose D2, since worst case has lower expected
cost
where miECiEmaxj Ci,j