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Defender Acts 1st A1A1 A2A2 D1D1 C 11 C 12 D2D2 C 21 C 22 where C i,j =Cost to defender from play (A j |D i ) A1A1 A2A2 D1D1 1 12 D2D2 21 22 Random Cost Matrix Expected Cost Matrix where i,j =E[C i,j ]

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Minimax Strategy A1A1 A2A2 Max j C i,j D1D1 C 11 C 12 C*1C*1 D2D2 C 21 C 22 C* 2 A1A1 A2A2 Max j i,j D1D1 1 12 * 1 D2D2 21 22 * 2 Random Cost Matrix Expected Cost Matrix

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Minimax Strategy A1A1 A2A2 Max j C i,j D1D1 N(6,10)N(6,2) C*1C*1 D2D2 N(6.5,3.5)N(7.5,3) C* 2 A1A1 A2A2 Max j i,j D1D1 * 1 D2D2 * 2 Random Cost Matrix Expected Cost Matrix Example: C i,j =N( i,j, i,j )

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Minimax Strategy A1A1 A2A2 Max j C i,j D1D1 N(6,10)N(6,2) C*1C*1 D2D2 N(6.5,3.5)N(7.5,3) C* 2 A1A1 A2A2 Max j i,j D1D1 * 1 D2D2 * 2 Random Cost Matrix Expected Cost Matrix Example: C i,j =N( i,j, i,j ) Which action should Defender take? D*=argmin i max j E[C i,j ] =argmin i * i

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Banks and Anderson Strategy #1 Choose D 1, but rather close to indifferent A1A1 A2A2 Max j C i,j D1D1 N(6,10)N(6,2) C*1C*1 D2D2 N(6.5,3.5)N(7.5,3) C* 2 D*=argmax i P(C* i < min k C* k )

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Banks and Anderson Strategy #2 From this, choose D 2 Score(i)=min k {C* k } / C* i Score(i) 2 (0,1] – Larger is better E[Score(1)]=0.815 E[Score(2)]=0.822 A1A1 A2A2 Max j C i,j D1D1 N(6,10)N(6,2) C*1C*1 D2D2 N(6.5,3.5)N(7.5,3) C* 2 D*=argmax i E[Score(i)]

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An Alternative Approach where m* i =E[C* i ]=E[max j C i,j ] A1A1 A2A2 Max j C i,j D1D1 N(6,10)N(6,2) C*1C*1 D2D2 N(6.5,3.5)N(7.5,3) C* 2 Choose D 2, since worst case has lower expected cost D*=argmin i E[max j C i,j ]

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