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1 Bradley-Terry W 03D L

2 Bradley-Terry W Bradley-Terry FIFA Bradley-Terry 1998 W 2002 W 2006 W Bradley-Terry W

3 Bradley-Terry BT FIFA BT BT FIFA BT W W W

4 1 4 1 W W W W 2 W 2006 W 1998 W 2002 W FIFA FIFA Bradley-Terry Bradley-Terry 1998 W 2002 W 2006 W 1

5 2 Bradley-Terry 2 2 Bradley-Terry [1] 2.1 m ( ;, = 1,2, L, m) (2.1) = 1 ( ) (2.2) + m = < < < < < < (2.3)

6 m,, 2, 1 L m = ( ) + (2.4) = ( ) (2.5) Bradley and Terry1952Bradley-Terry BT 2.2 BT BT k k k k k k k k k : k k k k k k k k + k k (2.6) k k k k = (2.7) + k k k k 3

7 k k = (2.8) k k m = ( 1,2,, 1) m ( ) k = L m = 1 (2.9) m m m = = (2.10) m m (2.5) (2.5) = (2.11) k k 3 1 k k (2.11) k k 2.3 m n = n ) ( X ( ;, = 1,2, L, m) (2.12) X + X = n ( ) (2.13) X 2 B n, ) ( n! x x Pr { X = x } = ( x = 0,1, L, n ) (2.14) x! x! 4

8 n! x { } x Pr X = = = x ; ;, 1,2, L, m (2.15) < x! x! BT (2.15) Pr { X = x ; ;, = 1,2, L, m} = = m < x = < n! x! x! n! x! x! ( + ) ( + ) x 1 x n n m = 1 t (2.16) t = (2.17) (2.16) =,, L, ) ( 1 2 m const. L = const. n ( + ) < m = 1 T (2.18) X m T = (2.19) X = BT ( T, T2,, T m ) ( T, T2,, ) T m 1 L 1 L BT 2.4 =,, L, ) ( 1 2 m =,, L, ) ( 1 2 m (2.4) k m =1 k (2.20) = 5

9 (2.18) m l = log L = T log n log( + ) + const. (2.21) = 1 λ < l λ m = 1 k = 0 m l λ k = 0 λ = 1 (2.22) (2.23) T n ˆ ˆ + ˆ λ = 0( = 1,2, L, m) (2.24) (2.24) ˆ m =1 ˆ ˆ + ˆ ˆ k (2.25) = = ( ;, = 1,2, L, m) T = n (2.26) ˆ + λ ˆ ( = 1,2, L, m) (2.27) m = 1 T = = = < < < n n n ( ˆ + ˆ ) + λk + λ ˆ (2.28) λ = 0 ˆ n = T ( = 1,2, L, m) (2.29) ˆ + ˆ m =1 ˆ = k (2.30) 6

10 (0) (0) (0) (0) ˆ ( ˆ, ˆ, L, ˆ ) n n T = ( = 1,2, L, m) (2.31) ˆ + ˆ ˆ 1 2 = 1 m n r (0) = (0) (0 ˆ + ˆ ) ( = 1,2, L, m) (2.32) ˆ (1) T = r (0) (1) = 1 ˆ (1) (2.33) (1) (1) k ˆ ˆ = m ( = 1,2, L, m) (2.34) ˆ ( k ) (2.32)(2.34) ˆ ( = 1,2, L, m) 7

11 3 FIFA A FIFA [2] A FIFA 2 W A FIFA 207 A 9, PK BT FIFA FIFA 3.1 A FIFA FIFA FIFA 3.1 FIFA 8

12 4 BT 2 BT BT 4.1 BT FIFA 3BT BT FIFA FIFA 6 FIFA FIFA BT (2.28) W

13 FIFA FIFA y = x (4.1) R FIFA 2 BT 2 BT FIFA 2 FIFA 10

14 4.1 FIFA 11

15 FIFA 12

16 4.2 BT FIFA BT FIFA FIFA FIFA FIFA BT FIFA FIFA FIFA

17 W FIFA FIFA FIFA FIFA W W 4.3 FIFA 14

21 BT H A N BT 18

24 FIFA 4.1 FIFA

25 BT FIFA FIFA FIFA FIFA 22

27 5 BT W BT W 5.1 W BT 2 2 BT (2.4) BT 2 (5.1) + S = (5.1) 2 S = S ) 2 2 ( (2.2)(5.2) H = (log( )) (log( )) (5.2) H = H ) 2 = 0 = 1 ( = 1 = 0 H = 0 = 0. 5 H = 1 = 2 24

28 (5.1)(5.2) I = I ) ( I = S H (5.3) (5.3)

29 5.2 W W W 32 4 A H A 1 B W 2002 W 2006 W BT W W 26

30 W (5.3) I 1 W m 1 8 m = 1,2, L, m = 9, L, 12 2 m = 13, 14 1 m = 15 w 1 w = 1 2 w = 2 w = 4 w = 8 m I m ( = I ) = 16, 384 t I t 15 I = w ( t = 1,2, L,16384) (5.4) t I m m= 1 2 t 1 I = I t I t (5.5) 1998 W 2002 W 2006 W 1 I Real I max t= 1 27

31 I mn I I Real 1998 W W W 70.9 I I Real 1998 W W W W 2006 W 2002 W (5.1) IReal I max I mn I 5.2 W (5.5) I 16 I W 2002 W 2006 W I W 2002 W 2006 W I = 429,981,696 1 f I E ( f = 1,2, L, ) f 28

32 f BT f f E f E f E f E = f E f (5.6) 1998 W 2002 W 2006 W 1 E Real 1 E max 1 E mn E f E E 1998 W W W W E Real E W W f EReal Emax Emn E

33 5.3 W 1 W ,625,702,400 [3] W ,

35 6 6.1 BT W FIFA W 1998 W 2002 W 2002 W 6.2 BT W 8 W 16 W 32

36 33

37 [1] 1988 [2] <htt:// [3] 2000 [4] 2000 [5] W