1. 1. : Brouwer Nash : . , . ... , 2014-2015 , katia_karavasili@yahoo.gr , evangelos@mitsokapas.gr , gprinos161168@yahoo.gr , seretifot@gmail.com , - siourisg@yahoo.com 25 2015
2. 2. 1 2 1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3.2 . . . . . . . . . . . . . . . . . 4 2 Simplexes 6 2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.1.1 n-Simplex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.1.2 Standard n-Simplex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.1.3 k-Face n-Simplex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.1.4 Symplotopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.2 Simplicial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.2.1 n-Simplex . . . . . . . . . . . . . . . . . . . . . . . . 8 2.3 Sperner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.3.1 Labeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.3.2 Labeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3 Sperner 12 3.1 Sperner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.2 . . . . . . . . . . . . . . . . . . . . . . . . 12 3.2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.2.2 . . . . . . . . . . . . . . . . . . . . . . . 14 3.2.3 n- . . . . . . . . . . . . . . . . . 16 4 Brouwer 17 4.1 n-simplex . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 4.2 - . . . . . . . . . . . . . . . 17 4.3 Brouwer Symplotopes . . . . . . . . . . . . . . . . . . . . . . 18 5 21 5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 6 (Game Theory) 23 6.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 6.2 Nash . . . . . . . . . . . . . . . . . . . . . 23 6.2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 6.2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 6.2.3 (Support) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2
3. 3. 6.2.4 . . . . . . . . . . . . . . . . 24 6.2.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 6.3 Nash . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 7 Nash-Nash Equilibrium 26 8 28 8.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 8.2 Debate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 8.2.1 Nash Minimax 30 8.3 2 2 . . . . . . . . . . . . . . . . . 31 8.4 2 5 . . . . . . . . . . . . . . . . . 32 3
4. 4. 2.1 n-simplexes faces . . . . . . 7 2.2 . . . . . . . . . . . . . . . . 9 2.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.4 simplex L. - simplexes . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.1 0, 1 . . . . . . . . . . . . . . 12 3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.3 k- . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.4 2-simplex. . . . . . . . . . . . . . . . . . . . 15 4.1 symplotope simplex. . . . . . . . . . . . . . . . . . . . . 19 4.2 . . . . . . . . . . . . . . . . . . . . 20 5.1 g(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 8.1 - . . . . . . . . . . . . . . . . . . . . 28 8.2 - . . . . . . . . . . . . . . . . . . . . 29 8.3 . . . . . . . . . . . . . . . . . . . . . . . . . 30 1
5. 5. Brouwer Nash, , , . , . , Poincare Picard Brower , 1910, Jacques Hadamard Luitzen Egbertus Jan Brouwer . Brouwer: , , . , Brouwer. 5 P. Bohl n = 3. , , Nash, John Nash , , , 4 , : The Nash Equilibrium. John Nash , . 1 - . Simplexes, - Sperner, Sperner. 3 Sperner , n = 1 n = 2 n . Brouwer Nash - .
6. 6. 1 1.1 1.1. ( ) {x0 , x1 , ..., xn} Rn : n i=0 ixi = 0 n i=0 i = 0 0 = 1 = ... = n = 0 1.2 1.2. E R u, v, E. u v, [u, v] V, [u, v] = {u + (1 )v/ [0, 1] R}. A E u, v A [u, v] A. . , A E , . 1.3. P0, P1, ..., Pn Rm m > n v Rm v = n i=0 iPi n i=0 i = 1, i [0, 1]. v = n i=0 iPi n i=0 i = 1, i 0. 1.3 1.3.1 1.1. - : x2 2 + y2 2 1 , A = (x, y) R2 / x2 2 + y2 2 1 . 2
7. 7. [u, v] = {u + (1 )v/ [0, 1] R}. u = (u1, u2), v = (v1, v2) u, v A. : (u1, u2) A/ u2 1 2 + u2 2 2 1 (v1, v2) A/ v2 1 2 + v2 2 2 1 , : (u1 + (1 )v1)2 2 + (u2 + (1 )v2)2 2 1 : (u1 + (1 )v1)2 2 + (u2 + (1 )v2)2 2 = 2 u2 1 + 2(1 )u1v1 + (1 )2 v2 1 2 + 2 u2 2 + 2(1 )u2v2 + (1 )2 v2 2 2 = 2 u2 1 + 2(1 )u1v1 + v2 1 2v2 1 + 2 v2 1 2 + 2 u2 2 + 2(1 )u2v2 + v2 2 2v2 2 + 2 v2 2 2 2 & & & & &&b 1 u2 1 2 + u2 2 2 + 2 & & & & &&b 1 v2 1 2 + v2 2 2 + & & & & &&b 1 v2 1 2 + v2 2 2 2 & & & & &&b 1 v2 1 2 + v2 2 2 + 2(1 )u1v1 2 + 2(1 )u2v2 2 (22 2) + 1 + (2 22 ) u1v1 2 + u2v2 2 u1 = v1 u2 = v2 . u1 v1 u2 v2: Cauchy-Schwarz: x, y 2 x, x y, y x = u1 , u2 y = v1 , v2 : x, y 2 = u1 , u2 , v1 , v2 2 = u1v1 2 + u2v2 2 2 u1 , u2 , u1 , u2 v1 , v2 , v1 , v2 = & & & & &&b 1 u2 1 2 + u2 2 2 & & & & &&b 1 v2 1 2 + v2 2 2 1 : u1 , u2 , v1 , v2 2 1 u1 , u2 , v1 , v2 1 3
8. 8. : (u1 + (1 )v1)2 2 + (u2 + (1 )v2)2 2 (22 2) + 1 + (2 22 ) B1 u1v1 2 + u2v2 2 (u1 + (1 )v1)2 2 + (u2 + (1 )v2)2 2 1 A = (x, y) R2 / x2 2 + y2 2 1 , . 1.3.2 Hahn-Banach, . . 1.4. A E f : A R. f u, v A : f(u + (1 )v) f(u) + (1 )f(v), [0, 1] 1.2. f(x) = x2 . f - . 1.4. x1, x2 R : f(x1) + (1 )f(x2) f(x1 + (1 )x2) : x2 1 + (1 )x2 2 (x1 + (1 )x2)2 x2 1 + (1 )x2 2 x2 1 + 2(1 x1x2 + (1 )2 x2 2 x2 1 + (1 )x2 2 x2 1 + 2(1 )x1x2 + x2 2 2x2 2 + x2 2 0 (2 )x2 1 + 2(1 )x1x2 + (2 )x2 2 0 (2 )x2 1 2(2 )x1x2 + (2 )x2 2 0 (2 )(x1 x2)2 0 (2 )(x1 x2)2 R x1 = x2 [0, 1] x1 x2. f(x) = x2 . f(x) = ex, f(x) =|x| , x R, f(x) = 1 x , x (0, +) . 1.5. f : E R . : epif := {(u, ) E R/ f(u) } f. 4
9. 9. 1.1. f : E R . () f (u, ), (v, ) epif. [0, 1] : f(u + (1 )v) f(u) + (1 )f(v) + (1 ) , (u, ) + (1 )(u, ) epif, f . () epif u, v E. (u, f(u)), (v, f(v)) f. epif : (u, f(u)) + (1 )(v, f(v)) epif, [0, 1] : f(u + (1 )v) f(u) + (1 )f(v), [0, 1]. f . 5
10. 10. 2 Simplexes 2.1 n-simplexes. faces, labeling n-simplex n-Sperner . 2.1.1 n-Simplex 2.1. {P0, P1, ., , , Pn} Rm m > n. n-simplex Pi, 0 i n n P : n P = n i=0 iPi n i=0 i = 1, i [0, 1] n-simplex n P {P0, P1, ..., Pn}. , simplex. simplex , n P = [P0, ..., Pn]. {P0, P1, ..., Pn} . 2.1. n-simplex n- . , 0-simplex , 1-simplex , 2-simplex , 3-simplex ... 2.1.2 Standard n-Simplex 2.2. Standard n-simplex n-simplex {e1, e2, ..., en+1} Rn+1 . n i=0 iPi, n i=0 i = 1 n+1 i=1 iei n+1 i=1 i = 1. - standard n-simplex : n S = y Rn+1 n+1 i=1 yi = 1, yi 0 n-simplex standard n-simplex. 6
11. 11. Simplexes 2.1.3 k-Face n-Simplex 2.3. k n. {V0, ...Vk} {P0, ..., Pn} k-simplex: S k = [V0, ..., Vk] k-face n P. 2.2. 2-simplex, . 3 3 . 1-faces. 2.3. 3-simplex, . 4 2-faces ( 4 ) 1-faces. : 2.1: n-simplexes faces . M = n+1 k+1 k-faces n-simplex. 2.4. n-simplex. (n 1)-face n P facet - : S n1 (i) = [P0, P1, ..., Pi1, Pi+1, ..., Pn] face n-simplex Pi. 7
12. 12. Simplexes n-simplexes . - , , n-simplexes. n-simplexes n P n Q . n-simplexes n P = [P0, ..., Pn] n Q = [Q0, ..., Qn] , P0 = Q0 = 0n . P1, ..., Pn Q1, ..., Qn n Rn, . T : Rn Rn : T(Pi) = Qi, i {1, ..., n} T n-simplexes, . , n-simplexes, n P n. 2.1.4 Symplotopes 2.5. Symplotope n-simplexes. n n ... n. 2.2 Simplicial 2.2.1 n-Simplex 2.6. Simplicial n-simplex n, - n-simplexes n i , i < : i n i = n , n i , n j n, n i n j , k-face k n. 2.4. n = 2. 2-simplex. . 2-simplex . 0, 1, 2 ( 1-faces), 4 2-simplexes. . 2.3, , 1-faces. . 0-faces . , labeling, Sperner, . 8
13. 13. Simplexes 2.2: . 2.3: . , n-simplex 0-simplexes , 0-simplexes . n-simplex n-simplexes, n- simplex. 2.7. y n. y = n i=0 iPi n i=0 i = 1, [0, 1]. : (y) = {i/i > 0} (y) y. 9
14. 14. Simplexes 2.3 Sperner 2.3.1 Labeling 2.8. K n-simplexes n i n - simplicial . Labeling K : L : K {1, 2, ..., n + 1} 2.4: simplex L. - simplexes . 2.3.2 Labeling 2.9. Labeling (proper labeling) Sperner labeling, labeling : (i) L(Pi) = L(Pj) i = j, Pi, Pj {P0, ..., Pn}. label. (ii) facet S n1 (i) n, Vj S n1 (i), L(Vj) 1, 2, ..., i 1, i + 1, ..., n + 1}. facet i, label i. L(y) (y). labels {1, 2, ..., n + 1} , Sperner. 2.5. n = 2. 2-simplex. - Sperner. 10
15. 15. Simplexes 2-simplex, , 0, 1, 2 . ( 1-faces) , 4 2-simplexes. 1-faces i i. , 2-simplex , 2-simplex 0, 1, 2 . 11
16. 16. 3 Sperner 3.1. n-simplex n-simplex . 3.1 Sperner n-simplex Sperner . n-simplexes . 3.2 3.2.1 Sperner N - , . 2 2N , N . . . . . 1-simplex, . 3.1: 0, 1 . ;; 1-simplex 0 1, 12
17. 17. Sperner 0 1 1 0 . : n . n = 1 . : 3.2: . 0 1 . n = k. k , 0 1 . n = k +1, : 3.3: k- . k + 1 . 1. : 2. : 3. : 4. : 13
18. 18. Sperner k m- , m 2Z + 1, (k + 1) m ( 1,2,3), (m+2) ( 4), . n = 2, 3 . 0 k 1 [i, i + 1] := i. : F(i) = {i, i = 0, 1, ..., k 1} R 0 . F(i): F(i) = 0, i = [1, 1] 1, i = [0, 1] [1, 0] 2, i = [0, 0] F(i) : i F(i) = 1 [(0, 1) ] + 2 [(0, 0) ] (3.1) 1-simplex - . , 0 2 0, . : i F(i) = 1 [( 0)] + 2 [( 0)] (3.2) i F(i) = 1 + 2 [( 0)] i F(i) = 2k + 1, k Z. i F(i) . (3.1) (3.2) : [(0, 1) ] = i F(i) 2 [(0, 0) ] - . Sperner. 3.2.2 , 2-simplex, , 14
19. 19. Sperner 0, 1, 2. 2-simplex. , . . . 3.4: 2-simplex. i 2-simplexes . F(i) (0, 1),1-simplexes i. : F(i) = 1, (0, 1, 2) simplex 2, (0, 1, 0) simplex (0, 1, 1) simplex 0, F(i) : i F(i) = 1 [(0, 1, 2) simplex] + 2 [(0, 1, 0) simplex] + 2 [(0, 1, 1) simplex] (3.3) , (1-simplex) 2 (2-simplexes). : i F(i) = 1 [(0, 1) simplexes ] + 2 [(0, 1) simplexes ] (3.4) i F(i) = 2Z + 1 . (3.3) (3.4) (0, 1, 2)-simplexes . 15
20. 20. Sperner 3.2.3 n- n-, n-simplex - . 1-simplex, n1-simplexes n-simplexes. - (n 1)-simplex n-simplex n-simplexes. K n-simplex, Sperner. i n-simplexes . F(i), (0, 1, ..., n 1)-simplexes (n + 1) i. F(i) : F(i) = 1, (0, 1, ..., n 1, n) simplex 2, (0, 1, ..., n 1, s) simplex s = {0, 1, ..., n 1} 0, (0, 1, ..., n 2, n 1)-simplex 0 n-simplex . F(i) : i F(i) = 1 [(0, 1, ..., n 1, n) simplexes] + 2 [(0, 1, ..., n 1, 0) simplexes] (3.5) + 2 [(0, 1, ..., n 1, 1) simplexes] + 2 [(0, 1, ..., n 1, 1) simplexes] + ... + 2 [(0, 1, ..., n 1, n 1) simplexes] : i F(i) = 1 [(0, 1, ..., n 1) simplexes ] + 2 [(0, 1, ..., n 1) simplexes ] (3.6) (n1)-simplex, [(0, 1, ..., n 1) simplexes ] i F(i) = 2Z + 1. (3.5) (3.6) (0, 1, ..., n 1, n)-simplexes . n . 16
21. 21. 4 Brouwer 4.1 n-simplex 4.1. n-simplex [x0 , x1 , ..., xn] : Centroid = 1 n + 1 n i=0 xi 4.2 - f : n n. f , z n f(z) = z. labeling n - simplicial , simplexes f. L labeling. > 0. simplicial n, n- simplex . labeling L : K {1, ..., n + 1} : v label : L(v) (v) {i : fi(v) vi} vi i- v fi(v) i- f(v). L(v) label i vi > 0 f i- v. L (v) {i : fi(v) vi} . . 17
22. 22. Brouwer (v) {i : fi(v) vi} = . , L(v) fi(v) > vi, i (v). standard simplex n i=0 vi = 1. , vj > 0 j (v), : j(v) vj = n i=0 vi = 1 fj(v) > vj, j (v), : j(v) fj(v) > n i=0 vi = 1 (4.1) f(v) simplex n. : j(v) fj(v) n i=0 fi(v) = 1 (4.2) (4.1) (4.2) j(v) fj(v) > 1 j(v) fj(v) 1 . L labeling . 0, simplexes f. L labeling Sperner, Sperner simplex [0 , 1 , ..., n], fi(j) j i i, j. 0 simplexes. n , . z . j = 0, 1, ..., n, j z 0. f fi(z) zi, i. f(z) = z, n i=0 fi(z) = 1, n i=0 zi = 1 n i= fi(z) < n i=0 zi, . Brower. 4.3 Brouwer Symplotopes K = k j=1 nj symplotope f : K K . f . n = k j=1 nj. K n, f : K K . h : n K . h1 f h : n n , h , h h1 . f , . Brower z n (h1 f h)(z ) = z . 18
23. 23. Brouwer z = h(z ), z K. : (h1 f)(z) = z = h1 (z) h1 (f(z)) = h1 (z) h1 , f(z) = z. , f. K n. , n j K . - . , n-simplex n ( n + 1 - n ). nj nj K n. K Rn+k n Rn+1 K, n n, Rn, (K) (n) . , K Rn+k n Rn+1 ( n + k n + 1 n), (K) (n) (K) , (n) . , , n n. standard 2-simplex R3 , - R2 . , standard 1-simplexes R2 . , . 4.1: symplotope simplex. , (K) (K) (K) (n) . , (K) (n) . . K Rn . K Bn, Sn1 K ( K). q K. K x x q ( Rn) 19
24. 24. Brouwer 0 K. > 0 B (0) K. x x/ Bn = B1(0) K. 0 K . K , x0 0 . K. 0 x0 K x0. B1(0) K K , x0 y B1(0) K. y B1(0) C . x0 , 0 < 1 1 C K. x0 K . f : K Sn1 : f(x) = x |x| f(x) 0 x . f , . K f . 4.2: . F : Bn K : F(x) = |x|f1 x |x| F f1 . F 0 Sn 0 f1 () K. , F K. F , . , y K 0. F . 20
25. 25. 5 5.1 Brouwer - . : B, B B[0, 1], B(0, 1). S , S := B = {x Rn : x, x = 1}. C1 , f : B S , f(x) = x, x S . . f : B B C1 Rn . f . f(x) x, x B. g : B S , : g(x) , f(x) x, S . g : B S g(x) = x, x S , . 5.1: g(x) . 21
26. 26. f : B B, . f . - ( Brouwer Symplotopes), - norm Rn. maximum norm : x := max{|xi|/1 i n} f = (f1, f2, ..., fn). > 0 - Weierstrass, gi fi(x) gi(x) < x 2. g := (g1, ..., gn) . : f g := sup x B f(x) g(x) < g B . : g(x) = g(x) f(x) + f(x) g(x) f(x) + f(x) < 1 + g : B(0, 1) B(0, 1 + ). h(x) := g(x) 1+ : f(x) h(x) = f(x) g(x) 1 + = f(x)(1 + ) g(x) 1 + f(x) g(x) + f(x) (1 + f ) 2 f(x) x, x B, , > 0, f(x) x > , x B. g f g < , h , x0 h(x0) = x0 : > f(x0) g(x0) = f(x0) x0 + x0 g(x0) = (f(x0) x0) + (x0 g(x0)) = f(x0) x0 > . f : B B x0, f(x0) = x0. 22
27. 27. 6 (Game Theory) 6.1 6.1. - , . 6.2. . , - . , - . 6.3. . 6.4. - , . . . . n 2 . 6.5. , , . 6.6. . 6.2 Nash 6.2.1 6.7. n- (N, A, O, , u), : 23
28. 28. (Game Theory) N n . A = (A1, ..., An), Ai / , - i. a = (a1, ..., an) A . O . : A O u = (u1, ...un), ui : O R ( ) i. 6.2.2 6.8. (N, (A1, ..., An), 0, , u) X, (X) X. i S i = (Ai). 6.9. Prole prole , : S = S1 S2 ... S n. si(ai) ai - si. 6.2.3 (Support) 6.10. si i ( ) {ai/si(ai) > 0}. . . 6.2.4 6.11. (N, A, u), ui i prole s = (s1, ..., sn) : ui(s) = aA ui(a) n j=1 sj(aj) 6.2.5 6.12. si = (s1, ..., si1, si+1, ..., sn) prole si i. s = (si, si) 24
29. 29. (Game Theory) prole si - s i S i : ui(s i , si)) ui(si, si1) si Si. . , . i Si = (Ai). Ai , i , {1, 2, ..., mi} f : f(i) 0, i mi i=1 f(i) = 1 S i simplex mi , : mi = {x Rm i / mi i=1 xi = 1} f. , prole S m1 ... mn symplotope 6.3 Nash 6.13. s = (s1, ..., sn) Nash , i, si si. Nash , prole , . 25
30. 30. 7 Nash-Nash Equilibrium (Nash 1951) prole , - Nash . prole s S , i N ai Ai : i,ai (s) = max{0, ui(ai, si) ui(s)} i , si si ai. f : S S f(s) = s , : si(ai) = si(ai) + i,ai (s) biAi si(bi) + i,bi (s) (7.1) si(ai) = si(ai) + i,ai (s) 1 + biAi i,bi (s) i ai , . prole s prole s prole s, . f , , si(ai) . si(ai) i,ai (s) , si(ai) - . i,ai (s) ui(ai, si) ui(s). ui - , . S , symplotope . f f : S S Brouwer Symplotopes, f . f Nash . 26
31. 31. Nash-Nash Equilibrium () s Nash i,ai = 0, i Nash . : f(si(ai)) = si(ai) = si(ai) + $$$$X0 i,ai (s) 1 + $$$$$$$X0 biAi i,bi (s) = si(ai) f(si(ai)) = si(ai) s f. () f, s. f si, ai ui(ai, s) ui(s). , i,ai (s) = 0 s f, si(ai) = si(ai). (7.1) : si(ai) = si(ai) + B0 i,ai (s) 1 + biAi i,bi (s) = si(ai) 1 + biAi i,bi (s) : si(ai) ai si. . i, bi Ai i,bi (s) = 0. , . s Nash . . 27
32. 32. 8 8.1 . 1950 Melvin Dresher Merrill Flood . - Stanford Albert W. Tucker . , . Tucker : , . : . . , , . : 8.1: - 8.2, : . : (confess), (not confess). . 28
33. 33. 8.2: - , . . , ( ), . . . , , . . . , . , . , , (1,1) (1,1), . - - . . , . - , , . . , . . : , . . 8.2 Debate (debate). , 29
34. 34. . ( ) , . A, - Nash. 8.3: . 8.2.1 Nash Mini- max - (minimax). (maximin). , Nash. 30
35. 35. 8.3 2 2 . - . A: A, V(A, B1) V(A, B2), B . A B, V(A, B1) V(A, B2) x (1 x). V(A) V(A, B1) V(A, B2). A A1 x, A2 (1 x). B B1, A : V(A, B1) = 2x + 5(1 x) = 7x + 5. B B2, A : V(A, B2) = 6x+1(1 x) = 5x+1. V(A, B1) = V(A, B2), A :7x + 5 = 5x + 1, x = 1/3 (1 x) = 2/3. A A1 x1 = 1/3 A2 x2 = 2/3. 31
36. 36. A : V(A) = 2(1/3) + 5(2/3) = 6(1/3) + 1(2/3) = 8/3. B. B: B B1 y, B2 (1 y). A A1, B :V(B, A1) = 2y + 6(1 y). A A2, B : V(B, A2) = 5y + 1(1 y). B , :V(B, A1) = V(B, A2), : 9 2y + 6(1 y) = 5y + 1(1 y), y = 5/12 1 y = 7/12. B1 y1 = 5/12 B2 y2 = 7/12. B : V(B) = 2(5/12) + 6(7/12) = 5(5/12) + 1(7/12) = 8/3 (- (, 1) V(B, A2)) V(A). A. 8.4 2 5 V(A, B1) = x1 + 4x2 = x1 + 4(1 x1) = 3x1 + 4 V(A, B2) = 4x1 + 3x2 = 4x1 + 3(1 x1) = x1 + 3 V(A, B3) = 2x1 + 5x2 = 2x1 + 5(1 x1) = 7x1 + 5 V(A, B4) = 3x1 + 2x2 = 3x1 + 2(1 x1) = 5x1 + 2 V(A, B5) = 5x1 x2 = 5x1 (1 x1) = 6x1 1 32
37. 37. 33
38. 38. [1] John H. Palmiery, The Brouwer xed point theorem, Department of Mathematics, University of Washington ACMS seminar [2] Tara Stuckless, M.Sc. Thesis: Brouwers Fixed Point Theorem: Methods of Proofs and Generalizations. Simon Fraser University, March 2003 [3] John M. Lee, Introduction to Topological Manifolds. Print ISBN: 978-1-4419-7939-1 Publisher: Springer Graduate Texts in Mathematics Volume 202 2011 [4] Albert Xin Jiang, Kevin Leyton-Brown A Tutorial on The Proof of the Existence of Nash Equilibria, Department of Computer Science, University of British Columbia [5] Warwick Math Organization, Sperners Lemma and Other Topics In Combinatorial Topology. [6] S. Kumaresan Vector Fields on Spheres and Brouwers Fixed Point Theorem Dept. of Math. and Stat. University of Hyderabad Hyderabad 500 046 [7] Vlachopoulou Athanasia, : Nash . - 2010 34