WPA(Win Probability Added) 1 WPA WPA ( ) WPA WPA WPA WPA WPA

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2 WPA(Win Probability Added) 1 WPA WPA ( ) WPA WPA WPA WPA WPA

3 1 1 2 WPA WPA(Win Probability Added) () WPA A m m A 29 A A

4 1 [7, 8, 9, 12, 14] 1959 Lindsey [13] 1 Bukiet, Harold and Palacios [4] 9 9 Bukiet [4] ( ) 2 Bukiet WPA(Win Probability Added) , , = WPA WPA WPA () Win Expectancy Finder web [18] [17] 1 1 ( ) p(1 ) x λ x λ x

5 () WPA 6 9 WPA 4 1 () ( ) WPA Bukiet, Harold and Palacios [4] [15] Bukiet WPA () WPA WPA WPA 2012 vs 6 ( ) 1 WPA WPA 1 WPA WPA WPA 0 [10] [16] WPA WPA 2 ( [15])

6 5 2 WPA 2.1 WPA(Win Probability Added) 1 WPA(Win Probability Added) 1 WPA 1 1 WPA WPA WPA WPA WPA WPA WPA 3 WPA WPA () 2.2 WPA 1 WPA [15] WPA t 1, t 2, t 3, i j i j 3 1 Leverage Index WPA Leverage Index 3

7 1: 1 i j (i, j) P P = P n i(i = 1, 2, 3) u n i ( u n 1 u n 2 u n 3 ) ( = u n 1 1 u2 n 1 u n 1 3 ) [4] (0 1 2 ) 8 ( ) : i j (i, j) P P A 0 B 0 C 0 D 0 P = O A 1 B 1 E 1 O O A 2 F 2 O O O 1 A i (i = 0, 1, 2) i i A i (i = 0, 1, 2) ( 2 ) 8 8 B i (i = 0, 1) 4

8 3 C 0 3 D 0, E 1, F 2 3 D E F P 0 P P (16,13) A 1 (8,5) () 9 [4] P (16, 9) [4, 15]4 2 D Esopo and Lefkowitz [6] 5 2: D Esopo and Lefkowitz D Esopo and Lefkowitz [1] [4] P A B O O P = O A B O O O A F O O O D Esopo and Lefkowitz

9 C 0, D 0, E 0 0 A, B A, B, F 1 p s, p d, p t, p h, p w, p out A, B, F p h p s + p w p d p t p h 0 0 p t p s + p w 0 p d 0 p h p s p d p t p w A = p h p s p d p t 0 p w 0 0 p h 0 0 p t p s 0 p d p w p h 0 0 p t p s 0 p d p w p h p s p d p t p w p h 0 0 p t p s 0 p d p w B = p out I p out F =. B p out n s s 25 u n j (j ) P j 7 n j P j u n 1 u n u 0 = ( ) u n = u n 1 P j (1) n r (r = 0, 1, 2, ) s (r, s) U n 8 r R max U n (R max + 1) 25 u n (1) U n U n 1 P P (0), P (1), P (2), P (3), P (4) P (1) (i, j) i 1 j () () 1() () () ( r) P (r) (i, j) ˆr r P (ˆr) (i, j) 0 P P (0) + P (1) + P (2) + P (3) + P (4) 7 8 U n

10 D Esopo and Lefkowitz P (r) (r = 0, 1, 2, 3, 4) A (0) B O O P (0) = O A (0) B O O O A (0) F O O O 1 A (r) O O O P (r) = O A (r) O O O O A (r) O (r = 1, 2, 3, 4) O O O 0 0 p s + p w p d p t p s + p w 0 p d p w A (0) = p w p w p w p w p h p t p s p d p t A (1) = 0 p s p d p t p s 0 p d p s 0 p d p w p h p h A (2) = p h p t p t p s p d p t p s 0 p d A (3) = p h p h p h p t

11 A (4) = p h U n r U n r U n r n j P j U n r = U n 1 r 0 P (0) j + U n 1 r 1 P (1) j + U n 1 r 2 P (2) j + U n 1 r 3 P (3) j + U n 1 r 4 P (4) j (2) 1 U 0 = R max (2) U n U n 25 n 3 U n (r, 25) n r 1 n U U n 1 9 U n 9 R max U n n (R max + 1) 25 U n r U n r u n 25 U n 25 ( n ) (R max + 1) u n 25 1 u n 25 (1-)( n ) q (ij) i j 8

12 R q(ij) i j (R max + 1) p (m,n) m n R p(m,n) m n m (R max + 1) R (m) m (R max + 1) m q (ij) R q(ij) u n 25 1 n n u n 25 1 u n i n(n = 1, 2, ) n j 1 ( j ) q (ij) i(i = 1, 2,, 9) q (ij) u n 25 u n 1 25 n i n(n = 1, 2, ) n j 1 ( j ) () R q(ij) i(i = 1, 2,, 9) R q(ij) n 1 n u n ε i j i, j R q(ij) R max q (ij) R q(ij) 2 Step1. i = 1 Step2. n = 0, j = i Step3. n = n + 1 Step4. r = 0, 1, 2,, R max U n r = U n 1 r 0 P (0) j + U n 1 r 1 P (1) j + U n 1 r 2 P (2) j + U n 1 r 3 P (3) j + U n 1 r 4 P (4) j Step5. j = (j mod 9) + 1 Step6. q (ij) = q (ij) + ( u n 25 1 u n ), R q(ij) = R q(ij) + (u n 25 u n 1 25 ) Step7. u n ε Step3. Step8. Step9. j = 1, 2,, 9 R q(ij) i = i + 1 i 9 Step2. i = 10 2: q (ij) R q(ij) q (ij) m n p (m,n) 1 1 p (1,1) = 1, p (1,n) = 0 (n = 2, 3,, 9) p (m,n) (m = 2, 3,, 10) q (ij) 9 p (m,n) = p (m 1,k) q (kn) (m = 2, 3,, 10, n = 1,, 9) (3) k=1 9

13 p (m,n) m n m R p(m,n) (R max + 1) c (m 1,k,n) i c (m 1,k,n)i = R p(m 1,k)r R q(kn)i r (i = 0, 1,, R max ) r=0 R p(m 1,k)r m 1 k m 2 r R q(kn)i r k n i r c (m 1,k,n)i m 1 k m n m 1 i m n m 1 k p (m 1,k), q (kn), p (m,n) p (m 1,k) q (kn) p (m,n) R p(m,n) (m = 2, 3,, 10) R p(m,n) = 9 k=1 p (m 1,k) q (kn) p (m,n) c (m 1,k,n) (m = 2, 3,, 10, n = 1,, 9) (4) 1 0 ( R p(1,n) = ) T (n = 1, 2,, 9) R p(m,n) m R (m) 10 ( 9 ) 10 n p (10,n) R p(10,n) R (10) = 9 p (10,n) R p(10,n) (5) n= A B A 9 R (10) B 9 R (10) A i B i 1 A 9 A P W (9) P W (9) = R max i=1 i 1 R (10)i j=0 R (10)j (6) A (B ) P L(9) P D(9) 10

14 A R (11) 10 B R (11) m 1 (m = 10, 11, ) m () n p (m,n) ( p (m,n) ) (5) R (10) = 9 p (10,n) R p(10,n) (7) n=1 10 A n p tie(10,n) (n = 1, 2,, 9) R (10) p tie(10,n) = R max r=0 R p(10,n)r R (10)r (8) 9 10 n p (10,n) p (10,n) = p (10,n)p tie(10,n) P D(9) (9) (7)-(9) 9 10 B n p (10,n) m = 10, 11, p (m,n), p (m,n) m m A B 3 P W (m), P L(m), P D(m) 2.2.3, P W (m), P L(m), P D(m) (m = 10, 11, ) 12 A P W (9) + P D(9) (P W (10) + P D(10) (P W (11) + P D(11) P W (12) )) (10) P D(9) P D(10) P D(11) P D(12) (11) 15 A P W (9) +P D(9) (P W (10) +P D(10) (P W (11) +P D(11) (P W (12) +P D(12) (P W (13) +P D(13) (P W (14) +P D(14) P W (15) ))))) (12) P D(9) P D(10) P D(11) P D(12) P D(13) P D(14) P D(15) (13) B 11

15 3 1 WPA () WPA m 0 d 0 s 0 (s 0 = 1, 2,, 24)i (6) 10 R (10) (5) 10 n p (10,n) 10 R p(10,n) R (10) m 0 s 0 (s 0 = 1, 2,, 24)i 0 m 0 +1 n p (m0+1,n) R p(m0 +1,n) q (ij) R q(ij) (3) (4) m p (m,n) R p(m,n) p (m0 +1,n) R p(m0 +1,n) p (m0 +1,n) R p(m0 +1,n) q (ij) R q(ij) q (ij) R q(ij) i 1 j 1 i s 0 (s 0 = 1, 2,, 24) j 1 q s0(ij) R qs0 (ij) q (ij) = q 1(ij), R q(ij) = R q1(ij) p (m0 +1,n) p (m0+1,n) = q s0(i 0n) m 0 R p(m0 +1,n) R p(m0 +1,n) = R q s0 (i 0 n) R p(m0+1,n) R p (m,n) R (m) P W (9), P L(9), P D(9) d 0 m 0 d 0 s 0 i 0 q s0(ij) i s 0 j R qs0 (ij) i s 0 j (R max + 1) p s0i 0(m,n)m 0 s 0 i 0 m n R ps0 i 0 (m,n) m n m 0 s 0 i 0 m (R max + 1) 12

16 R s0 i 0 (m)m 0 s 0 i 0 m (R max + 1) U n, R max, U n r, u n p s 0i 0(m,n), R ps0 i 0 (m,n), R s 0i 0(m) p (m,n), R p(m,n), R (m) i 1 j 1 q 1(ij) R q1(ij) q (ij) R q(ij) 2 q 1(ij) R q1(ij) i s 0 j q s0 (ij) R qs0(ij) q (ij) R q(ij) Step3. U 0 Step1. n = 0, j = i 0 Step2. n = n + 1 Step3. r = 0, 1, 2,, R max U n r = U n 1 r 0 P (0) j + U n 1 r 1 P (1) j + U n 1 r 2 P (2) j + U n 1 r 3 P (3) j + U n 1 r 4 P (4) j 1 s U 0 = Step4. j = (j mod 9) + 1 Step5. Step6. Step q s0 (i 0 j) = q s0 (i 0 j) + ( u n 25 1 u25 n 1 1),R qs0 (i 0 j) = R q s0 (i 0 j) + (un 25 u n 1 u n ε Step2. j = 1, 2,, 9 R qs0(i 0j) 3: q s0 (i 0 j) R qs0 (i 0 j) 25 ) p s0 i 0 (m,n) p s0 i 0 (m 0 +1,n) = q s0 (i 0 n) (n = 1, 2,, 9) 9 p s0 i 0 (m,n) = p s0 i 0 (m 1,k)q 1(kn) (m = m 0 + 2, m 0 + 3,, 10, n = 1,, 9) k=1 R ps0 i 0 (m,n) R ps0i0(m0+1,n) = R qs0(i0n) (n = 1, 2,, 9) R ps0 i 0 (m,n) = 9 k=1 p s0 i 0 (m 1,k)q 1(kn) c (m 1,k,n) (m = m 0 + 2, m 0 + 3,, 10, n = 1,, 9) p s0i 0(m,n) c (m 1,k,n)i := i R ps0i 0(m 1,k) R q1(kn)i r (i = 0, 1,, R max ) r r=0 R s0 i 0 (10) R s0 i 0 (10) = 9 n=1 p s0 i 0 (10,n)R ps0 i 0 (10,n) 13

17 R s0a i 0A (10) A m 0A () s 0A i 0A 9 R s0b i 0B (10) B m 0B () s 0B i 0B 9 A d 0 A m 0A s 0A i 0A B m 0B s 0B i 0B 9 A P W sit(9) P W sit(9) = R max i=0 R s0a i 0A (10) i d 0 +i 1 j=0 R s0b i 0B (10) j (m 0A, s 0A, i 0A ) (m 0B, s 0B, i 0B ) (m 0A, s 0A, i 0A ) = (2, 1, 4), (m 0B, s 0B, i 0B ) = (1, 24, 6) 9 A (B ) P Lsit(9) P Dsit(9) P Lsit(9) = R max i=0 P Dsit(9) = R s0b i 0B (10) i R max i=0 d 0+i 1 j=0 R s0a i 0A (10) j ( ) R s0a i 0A (10) i R s0b i 0B (10) d0 +i d 0 A (7)-(9) m 1 (m = 10, 11,, 15) m () n p (m,n) ( p (m,n) ) m m A B 3 P W sit(m), P Lsit(m), P Dsit(m) 2.2.3, P W sit(m), P Lsit(m), P Dsit(m) (m = 10, 11, ) (10)-(13) P W sit(9) = 0, P Lsit(9) = 0, P Dsit(9) = 1 4 () WPA CPU Intel Core i5, 2.27GHz 4.00GBOS Windows7 Home Premium Matlab ε = 10 3, R max = 20 [19] [5] [2, 3, 11] 14

18 4.1 WPA WPA 1 WPA 2012 vs 6 1 WPA WPA WPA WPA (144 ) () 19 () () 19 () () () p w p w = () + () () 9 + () + () + () DH 9 () = () () () () () () () 15

19 ( ) ( ) ( ) WPA WPA : () () () : WPA : WPA : WPA = WPA = WPA WPA 3 WPA WPA 16

20 4: p s p d p t p h p w p out : 2012 WPA () WPA WPA 2 1 () () () () () () () () () () () () () () WPA WPA 6 WPA WPA( WPA) 1. WPA( WPA) 2. WPA WPA 10 WPA WPA WPA WPA 1 () WPA WPA 17

21 3. 6,7 6 7 WPA 1 2 6: WPA () WPA WPA WPA WPA 1 (0.352) (0.407) (0.262) (0.472) 2 (0.300) (0.305) (0.200) (0.268) 3 (0.224) (0.304) (0.014) (0.004) 7: WPA () WPA WPA WPA WPA 1 (0.274) (0.357) (0.211) (0.184) 2 ()(0.193) ()(0.265) (0.205) (0.183) 3 (0.170) (0.154) (0.102) (0.106) 8: p s p d p t p h p w p out WPA WPA 2 3 WPA WPA 2012 MVP ( 8 ) WPA () WPA WPA ()

22 WPA WPA WPA WPA WPA WPA 0 WPA [16] 9 [10] 2005 [16] WPA (0 ) P 0 1 P s 1 11 P f p 0 WPA (P s P 0 )p + (P f P 0 )(1 p) (14) (14) 0 p p out = p out =

23 9: 5 p s p d p t p h p w p out () ( 167/ 264) (0.848) : (1 ),(-2+2 ) : (24 ),(-2+2 ) +1(4 ),+2-2(2 ), (4 ) : (5 ),(-2+2 ) : (6,7 ),(-2+2 ) 0+2-1(6 ), (6 ), (7 )

24 14: (8 ),(-2+2 ) : (912 ),(-20 ) 0(9,12 ) ( ) () [2, 3, 11] 13 (p s, p d, p t, p h, p w, p out ) = (0.1919, , , , , ) (2012 ) ( ())

25 : (1,2 ),( : (3,4 ),(-2+2 ) +1, (2 ) ) 0+2 0(4 ),+1,+2 0(4 ),+1,+2 +1, : (58 ),( : (915 ),(-2 ) 0(8 ) 0+2 0(8 ) 0+2 0(8 ) 0+2 0(8 ) )

26 (2 ) (2 ) : 2 (68 ),(-2 20: 2 (15 ),(-2 +2 ) +2 ) -2+1, +2(1,2 ) ,-1, 0(3 ) -2+1, +2(13 ) -2(5 ), -1(5 ) -20(6,8 ), 0(7 ) -20,+1(6 ) -2+1,+2(6 ) , -20, +2(6,7 ) +1(6,7 ) , -2+1, -20, +2(6,7 ) +2(6,7 ) +1(6,7 ) -20(7 ), 0(8 ) 22: 2 (911 ),(-2 23: 2 (12 ),(-2,-1 0 ) ,+1(11 ) 0 0,+1 0,+1(11 ) 0 0,+1 0,+1 0,+1 0 ) , ( 8 ) ()

27 25: 2 (58 ),(-2 24: 2 (14 ),(-2 +2 ) +2 ) (8 ) -2+1,+2(7,8 ) : 2 (911 ),(0 27: 2 (12 +2 ) +1(9,10 ) 0,+1 0,+1 0, ,+1 0, ,+1,+2(11 ) 0,+1 ),(+1,+2 ) +2 +1, ,+2 +1,+2 +1,+2 +1,+2 +1,+2 +1,+2 +1,+2 +1,+2 +1, ( ) 5 () ( ) () WPA 0 () 4,5,6 4:

28 5: 6: 1 ( 4) 2 ( 5) 0,1 2 ( 6) ( 48/ 94) 0.673( 981/ 1457) ()

29 (1 ) (2 1 ) 28,29 28: 0 (18 ) , : 0 (912 ) , (1 ) (2 ) 26

30 28, () WPA WPA WPA 0 WPA 0 [1] Albert, J. and Bennett, J., :,, (2004). [2] , 27

31 [3] , [4] Bukiet, B., Harold, E.R. and Palacious, J.L.: A Markov Chain Approach to Baseball, Operations Research, 45, 1, (1997). [5], [6] D Esopo, D.A. and Lefkowitz, B.: The Distribution of Runs in the Game of Baseball, Optimal Strategies in Sports (edited by Ladany, S.P. and Machol, R.E.), 55-62, North-Holland, New York (1977). [7] : OR, --, 24, 4, (1979). [8], : - -, --, 49, 6, (2004). [9] James, B: The New Bill James Historical Baseball Abstract, Free Press, New York (2001). [10], :,, (2008). [11], [12] Lewis, M, : --,, (2004). [13] Lindsey, G.R.: Statistical Data Useful for the Operation of a Baseball Team, Operations Research, 7, 2, (1959). [14],, Student,,,,, : 1,, (2012). [15], :,, Vol.18, No.3, (2008). [16] : 9,, (2011). [17] : Win Probability Added in Sabermetrics,, 1703, 1-9 (2010). [18] Win Expectancy Finder, gregstoll/baseball/stats.php [19] Yahoo!JAPAN, 28

32 A m m A A.1 [15] m = 10, 11, p (m,n), p (m,n) m m A B 3 P W (m), P L(m), P D(m) m = 10, 11, 1. (7)-(9) p (m,n) p (m,n) 2. p (m+1,n) ( (3) ) 9 p (m+1,n) = p (m,k) q (kn) k=1 3. R p(m+1,n) ( (4) ) ( R p(m,n) = 9 p (m,k) q (kn) R p(m+1,n) = c (m,k,n)i := k=1 p (m+1,n) c (m,k,n) i R p(m,k)r R q(kn)i r r=0 ) T (n = 1, 2,, 9) 4. R (m+1) ( (5) ) 9 R (m+1) = p (m+1,n) R p(m+1,n) n= B q (ij), R q(ij) R (m+1) 6. (6) P W (m), P L(m), P D(m) ( ) T 4. R p(m,n) = (4) R p(m,n) 1 0 A.2 A d 0 A m 0A s 0A i 0A m 1 (m = 10, 11, ) m n p s0a i 0A (m,n) A d 0 B m 0B s 0B i 0B m 1 (m = 10, 11, ) m n p s0b i 0B (m,n) 29

33 m 1 (m = 10, 11, ) m () n p s0a i 0A (m,n)( p ) m s0b i 0B (m,n) m A B 3 P W sit(m), P Lsit(m), P Dsit(m) (i)9 10 n p s0a i 0A (10,n) p (m,n) ( p (m,n) ) (7)-(9) A d 0 B d 0 10 A n R max p ties0a i 0A (10,n) = r=0 R ps0a i 0A (10,n) r R s0b i 0B (10) d0 +r (15) (9) 9 10 n p s0a i 0A (10,n) = p s 0A i 0A (10,n)p ties0a i 0A (10,n) P Dsit(9) (16) A d B n p s0b i 0B (10,n) A.1 P W sit(10), P Lsit(10), P Dsit(10) R ps0a i 0A (m,n) R p s0b i 0B (m,n) 1 m = 11 p s0a i 0A (m,n), p s0b i 0B (m,n) P W sit(m), P Lsit(m), P Dsit(m) (ii)9 m 0A = m 0B + 1 m 0B 9 p s0a i 0A (m 0A,n) 9 (m 0B = 9) m 0A = 10, s 0A = 1 10 i 0A 9 10 n p s0a i 0A (10,n) p s0a i 0A (10,n) = p 1i0A (10,n) = { 1 (n = i 0A ) 0 () p s0b i 0B (10,n) (15),(16) A d A.1 P W sit(10), P Lsit(10), P Dsit(10) R ps0a i 0A (m,n) R p s0b i 0B (m,n) 1 m = 11 p s0a i 0A (m,n), p s0b i 0B (m,n) P W sit(m), P Lsit(m), P Dsit(m) 9 P W sit(m), P Lsit(m), P Dsit(m) 30

34 (iii)10 m 0A = m 0B m 0B 10 9 m A n p ties0a i 0A (m 0+1,n) A d 0 R max p ties0a i 0A (m 0 +1,n) = r=0 R ps0a i 0A (m 0 +1,n) r R s0b i 0B (m 0+1) d0+r (9) m 0 m n p s0a i 0A (m 0 +1,n) = p s 0A i 0A (m 0 +1,n)p ties0a i 0A (m 0 +1,n) P Dsit(m0 ) A d 0 p s0b i 0B (m 0 +1,n) A.1 P W sit(10), P Lsit(10), P Dsit(10) R ps0a i 0A (m,n) R p s0b i 0B (m,n) 1 m = m p s0a i 0A (m,n), p s0b i 0B (m,n) P W sit(m), P Lsit(m), P Dsit(m) 31

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