Теория игр, весна 2015: Теорема об ожидаемой полезности и матричные игры

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  1. 1. .. ( ) 2015 1 / 24
  2. 2. , . - . 90 , 60 . , 1 3 . - 20. ? .. ( ) 2015 2 / 24
  3. 3. , . - . 90 , 60 . , 1 3 . - 20. ? 90 1 3 60 2 3 = 10 > 20. ? .. ( ) 2015 2 / 24
  4. 4. . : 1. $1 2. 0.5 $2 .. ( ) 2015 3 / 24
  5. 5. . : 1. $1 2. 0.5 $2 . 1/2 1/4 1/8 1/16 1/32 ... 2 4 8 16 32 ... .. ( ) 2015 3 / 24
  6. 6. .. ( ) 2015 4 / 24
  7. 7. - . .. ( ) 2015 4 / 24
  8. 8. - . - . . .. ( ) 2015 4 / 24
  9. 9. X, (X). > (X). : .. ( ) 2015 5 / 24
  10. 10. X, (X). > (X). : . () .. ...... x, y (X) : x > y, x < y, x = y. .. ( ) 2015 5 / 24
  11. 11. X, (X). > (X). : . () .. ...... x, y (X) : x > y, x < y, x = y. . () .. ...... x > y, y > z, x > z. .. ( ) 2015 5 / 24
  12. 12. X, (X). > (X). : . () .. ...... x, y (X) : x > y, x < y, x = y. . () .. ...... x > y, y > z, x > z. . () .. ...... x > y > z, (0, 1), x + (1 )z > y. .. ( ) 2015 5 / 24
  13. 13. X, (X). > (X). : . () .. ...... x, y (X) : x > y, x < y, x = y. . () .. ...... x > y, y > z, x > z. . () .. ...... x > y > z, (0, 1), x + (1 )z > y. . ( ) .. ...... x > y, z x + (1 )z > y + (1 )z. .. ( ) 2015 5 / 24
  14. 14. . .. ...... (1)-(4), U : X R, x, y (X) x > y EU(x) > EU(y). .. ( ) 2015 6 / 24
  15. 15. 1. u < v (0, 1) , - .. ( ) 2015 7 / 24
  16. 16. 1. u < v (0, 1) , - 2. ( 0 1) .. ( ) 2015 7 / 24
  17. 17. 1. u < v (0, 1) , - 2. ( 0 1) 3. .. ( ) 2015 7 / 24
  18. 18. 1. u < v (0, 1) , - 2. ( 0 1) 3. 4. , 0 1. , .. ( ) 2015 7 / 24
  19. 19. 1. u < v (0, 1) , - 2. ( 0 1) 3. 4. , 0 1. , 5. , .. ( ) 2015 7 / 24
  20. 20. 1. X - . > (X), (, 4 ). .. ( ) 2015 8 / 24
  21. 21. ( ) = {N, {Xi}iN, {Ki}iN}. N - , Xi, i N i N, Ki : iN Xi R i N, .. ( ) 2015 9 / 24
  22. 22. ( ) = {N, {Xi}iN, {Ki}iN}. N - , Xi, i N i N, Ki : iN Xi R i N, : |N| = 2, N = {1, 2}, X1, X2 , K1(x1, x2) + K2(x1, x2) = 0(const) x1 X1, x2 X2. .. ( ) 2015 9 / 24
  23. 23. X1 = {1, ..., m}, X2 = {1, ..., n} m n A = a11, ..., a1n a21, ..., a2n ... ... ... am1, ..., amn , aij = K1(i, j). . .. ( ) 2015 10 / 24
  24. 24. A = 1 2 0 3 2 1 1 1 2 0 0 1 4 1 0 2 .. ( ) 2015 11 / 24
  25. 25. . .. ...... , i, j max i min j aij min j max i aij .. ( ) 2015 12 / 24
  26. 26. . .. ...... , i, j max i min j aij min j max i aij (i, j), ( i,j ) max i min j aij = min j max i aij .. ( ) 2015 12 / 24
  27. 27. , . 1 x = x1 x2 ... xm , xi 0, m i=1 xi = 1, 2 y = y1 y2 ... yn , yj 0, n j=1 yj = 1. 1 x, 2 y, 1 A(x, y) = xT Ay = m i=1 n j=1 aijxiyj. .. ( ) 2015 13 / 24
  28. 28. . ( , (1928)) .. ...... max x min y xT Ay = min y max x xT Ay. .. ( ) 2015 14 / 24
  29. 29. : Ax b, x 0 cT x max : AT y c, y 0 bT y min .. ( ) 2015 15 / 24
  30. 30. A = 1 2 0 3 2 1 1 1 2 0 0 1 4 1 0 2 .. ( ) 2015 16 / 24
  31. 31. (x, y) , x i > 0, y j > 0 i, j. , , . . .. ...... , m = n, x, y . .. ( ) 2015 17 / 24
  32. 32. : A = a11 0 . . . 0 0 a22 . . . 0 . . . . . . . . . . . . 0 0 . . . ann , aii > 0, i = 1, . . . , n. .. ( ) 2015 18 / 24
  33. 33. : A = a11 0 . . . 0 0 a22 . . . 0 . . . . . . . . . . . . 0 0 . . . ann , aii > 0, i = 1, . . . , n. . .. ...... . .. ( ) 2015 18 / 24
  34. 34. 2. (x, y) , x i > 0, y j > 0 i, j. , , . , , , x, y . .. ( ) 2015 19 / 24
  35. 35. 3. a = aij , aij = aji i, j. , . , . . . . .. ( ) 2015 20 / 24
  36. 36. 4. 2 n . , 2 . k = 1, ..., n , k > 0, , . . .. ( ) 2015 21 / 24
  37. 37. 5. . , . , 3, 4 5 . . . , . 1, , 0, -1, . , .. 3,4 5 . .. ( ) 2015 22 / 24
  38. 38. 6. m n f(x). . .. ( ) 2015 23 / 24
  39. 39. 7. 33. a13 = a22 = a31 = 0. . 1 1,2,3. 2 -1,-2,-3. , 1, . , 1 , . , 1 , (.. ). .. ( ) 2015 24 / 24