1 I. , ., , , ,Bayes, Markov. . () ( ). , (,, .. , .). , (Game Theory). Gardan,Pascal,Galileo,WaldergraveXVIIXVIII . John von-Neumann (1944TheoryandPracticeofGamesandEconomic Behavior),OskarMorgenstern1947 ,(Theoryof GamesandEconomicBehavior)., , ( ), (). ,, ( ). () , ,, ,, . 2 II. , : ) . )( ) . , . , :, ' () . , , . : ) . ( ) ( ) , . . (zero-sum games) . ) (non zero-sum games). ., ,, 0. .., , , . . III.(Two-person,zero-sum games) , . : 3 74: ( ). : i=() .Yj= .A(aij) = () i-j.i=i( ).qj = j ( ). () ( ). . , (purestrategy)., ,, , (mixed-strategy), , , . ( ), (value of the game). ( ),,V. 4 ,V, ( ). , V < 0, ,V>0, , , V =0, . ', minimax-maximin(Wald). minimax, ( ) ().,, ' , , . , . 1.(Strictlydeterminedgames)- : . , :1: . 2: . 3: . : 1: . 2: . , ( ) ( ): (. ) . , 5 (. ) . : , (.) 1 2 3. 1. 2 1. 2 3. 3 1 11. , . :' , , . minimax, . minimax . ( 3) 1 ( ), ( 1) 1 ( , ' ). (maximin strategy), 6 ()(maximinvalue)., (minimaxstrategy), ()(minimaxvalue). , V, : () V ( ) : ( ) = = 1 (saddle point), , (3,1) .,V,1(1 ) . , , ,,(31 ). , ( X3 = 1.0) ( 1 = 1.0). , V, (a3l=1), ( ). ( . ). 7 : () 3 3 2 ( )(ii) 1 1 3 1 ( )(iii) 1 1 0 ( ) 2.(Non-strictlydeterminedgames)- , , . 8 : maxinin, 2, 7. , 1, 9. ,, 1 1 ( 2). , , 2, 6 ( 9) . , , . . ,,, ,, .p1, p2, ... , pm q1, q2, ,qn , (. ), : minimax, , . pi, ,qj, 9 . , .. , , . , . , ..,maximin : minimax : , : () ( ). i j} ,, () , V, : = = V. ( ) . . , : 2x2 ( ). m x 2 2 x n ( ). 2x2 m x 2 2 x . m x n ( ). ) 2x2 . 10

2x2 11 ,1p1=p22. 1, (.) : ./1 =(p1 x 9) +[(1 p1)x7] =9p1 +7 7p1 =2p1 +7 , 2 : ./2 =(p1x6) +[(1 1) x11] =-5p1 +11 , ,./1./2 : 2p1 +7 =-5p1 +117p1=4 1 =4/7 1 p1 =3/7=p2. , 1 2 4:3, , V, : 91 +7(1 p1) =(9 x 4/7) +(7 x 3/7) =57/7 =8.14 :q1 1. , 1 q1 =q2 2. 1, (.): ./2 =(q1x 9) +[(1 q1) x 6] =9q1 +6 - 6q1 =3q1 +6 , 2, : ./2 =(q1 x 7) +[(1 q1) x 11] =7q1+11 - 11q1 =-4q1 +11 : ./1 = ./2 3q1+6 =-4q1+11 7q1=5 q1 =5/7, 1 q1 =2/7 =q2 ,1 2, 5:2. : V =(9x5/7) +(6x2/7) =57/7=8.14 8.14,8.14( 11 ). ( 2 x 2), ,,, .2x2 : 8.14,8.14( ).(2x2), ,,, .2x2 : :

q1 =q,q2 =1 - q p1 =p, p2 =1 - p 1 2, q1 q2 V : ,(1),, : 1, 1 - 2. B 1, : ./1 =a p +(1 - p) c =a p +c-c p : ./2 =b p +(1 - p) d =b p +d-d p : ./1 = ./2 :a p +c - c p =b p +d- d p a p - c p - b p +d p =d - cp (a - c - b +d) =d - cp =d-c/a+b-d-c... 2x212 : : ./1=2p1 +7(1) p1 = 1 (. 1), : ./1 =9 p1 = 0 (. 2), : ./1 =7 , : ./2 =-5p1 +11 (2) : p1 = 1, ./2 =6p1 = 0, ./2 =11 (1)(2) , . ( 1) , (2) . p1, 1 (1) (2). V , p, 0 1. 13 minimax, . V.V, : = V = 8.14 p1 =4/7, p2 =3/7 , ( 4/7 , 3/7 ). : , : : ./1 =3q1 +6 q, = 1, : ./1 =9 q, = 0, : ./1 =6 ./2 =-4q1 +11 q1 = 1, : ./2 =7 q1 = 0, : ./2 =1114 . 'V , () q1 0 1. minimax, . V. , V = 8.14 () ( 5/7 , 2/7 ). ) , . ,, ,, 2x2 ( ) mx2 2xn ( ). : 15 ) :( ) , , ' ( ). ): ,' . : 2x2, : 16 ) m x 2 2 x n ,. . ,, ( )., 2x2 : (1); : 2x2, , : ./1=p1 +(1-p1) 4=-3p1 +4 (1) p1 =1, ./1 =1 p1 =0, ./1 =4 ./2=4p1 +(1-p1) 3=p1 +3 (2) p1 =1, ./2 =4 p1 =0, ./2 =3 ./3=5p1 +(1-p1)=4p1 +1 (3) p1 =1, ./3 =5 p1 =0, ./3 =1 17 , ,, , . , V" V,( , ). V V =2.70 A (3/7, 4/7). , V (1, 3). maximin , (1) (3), (2), 0. 2 x 2, : , , , : 18 , (4/7,3/7). (2): : : ./1 =(-6p1) +[(1 p1) x (-5)] =-p1 - 5(1) p1=1,./1 =-6 p1=0,./1 =-5 ./2 =(-3p1) +[(1 p1) x (-6)] =3p1 - 6 (2) p1=1,./2 =-3 p1=0,./2 =-6 ./3 =(-8p1) +[(1 p1) x (-4)] =-4p1 - 4(3) p1=1,./3 =-8 p1=0,./3 =-4 ./2 =(-7p1) +(1 p1)=-8p1 +1 (4) p1=1,./4 =-7 p1=0,./4 = 1 19 .,, , V. V(1,2), , V = -5,25 ( ) ,p1=0.25=1/4p2=0.75=3/4. 2x2, . : q1 =3/4 q2 =1/4 V=-5.25 , (3/4,1/4). ) m n ( Simplex) : 20 2, , *: , ., , , V* (V**,V )p1, p2, p3 1, 2, 3, : 4p1+p2+p3V* 3p1+4p2+p3V* 4p1+p2+2p3V* p1+p2+p3=1 V* : p1/V*=x1, p2/V*=x2, p3/V*=x3, : 4x1+x2+x31 3x1+4x2+x31 x1+x2+2x31 x1+x2+x3=1/V* 21 V* 1/V*., , : x1, x2, x3 : 4x1+x2+x3- x4 +x7 =1 3x1+4x2+x3 - x5+x8=1 x1+x2+2x3 -x6+x9=1 : : max z=V

V . . , , , V* q1 q2,q3,1,2,3, : V* 22 q1/V*=y1, q2/V*=y2, q3/V*=y3, : V* 1/V* . y4, y5, y6 : y1, y2,y3 : : simplex , , , . I ( simplex) 23 II III

24 IV ( ) (IV) : : (1/3, 3/13, 9/13). , IV , ( 4, 5, 6), : X1=3/22, X2=1/22, X3=9/22 25 ,(3/13, 1/13, 9/13). ,V,V*=-2, V=(22/13)-2 =4/13( : 4/13 o A: -4/13).