22 [21] 2 (NPB) 2007 ( ) 3 ( ) J J ( ) [5] WBC J J J1 + [9] World Baseball Classic

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1 Transactions of the Operations Research Society of Japan Vol. 59, 2016, pp c J1 2 + ( ; ) 2015 J %, 35%, 3%( 3.4) : 1. J [8]( ) 1993 J ( ) ( ) ( ) ( ) ( [10]) ( [19]) & J ( ) [23]. J [17] 4 NFL( ) MLB( ) NBA( ) NHL( ) 4 MLS( ) 21

2 22 [21] 2 (NPB) 2007 ( ) 3 ( ) J J ( ) [5] WBC J J J1 + [9] World Baseball Classic

3 J st 1 2nd 1 3 1st 1 2nd Y1 Y2 Y3 W1 W2 1 5 (J [9] ) 1 8 1: 2015 J & 1 ( ) FIFA 50 ( ) 2

1 1 3 Y1 Y2 Y3 W1 W2 1 5 (J [9] ) 1 8 1:

4 24 case # overlap(s) teams 1 null 5 2 (Y3, W1) 4 3 (Y2, W1) 4 4 (Y2, W1), (Y3, W2) 3 1: case # overlap(s) teams 5 (Y1, W1) 4 6 (Y1, W1), (Y3, W2) 3 7 (Y1, W1), (Y2, W2) 3 8 (Y1, W1), (Y1, W2) 3 2: FIFA 50 ( ) Confederation Postseason YES NO All UEFA (Europa) 6 25 CONMEBOL (South America) 3 3 CONCACAF (North & Central America) 4 0 CAF (Africa) 2 7 AFC, OFC (Asia, Oceania) [6] V [22] 2 ( ) 2 (League Conference ) 4 8 (Division ) [14, 16] MLS(Major League Soccer) 2 + [13] ( MLS Cup Supporters Shield ) bj [1] J 1

(Europa) 6 25 CONMEBOL (South America) 3 3 CONCACAF (North & Central America) 4 0 CAF (Africa) 2 7 AFC, OFC (Asia, Oceania) 0 0 1

5 J (W1 W2) 3.1. i j λ ALL 1 λ i,gf i 1 ( ) (Goals For). λ i,ga i 1 ( ) (Goals Against). λ i,gf,h λ i,ga,h i 1 λ i,gf,a λ i,ga,a X i i 1 P o(λ) λ [2, 12] J Observed Poisson 0.3 Frequency Goals 2: 1 J1 ( 3 1 0) 1

λ i,gf,h λ i,ga,h i 1 λ i,gf,a λ i,ga,a X i i 1 P o(λ) λ. 3.2. 1 [2, 12].

6 26 [18] [4] Maher [12] ( ) M1: X i P o(λ ALL ) p(x) = e λ λx ALL ALL, x = 0, 1,. (3.1) x! P (X i = x, X j = y) = p(x)p(y). (3.2) M2: X i P o(λ i,gf ) p i (x) = e λ i,gf λx i,gf x!, x = 0, 1,. (3.3) P (X i = x, X j = y) = p i (x)p j (y). (3.4) ( ) λi,gf + λ j,ga M3: i, j X i P o 2 p i,j (x) = e µ µx i,j i,j x!, x = 0, 1,, µ i,j = λ i,gf + λ j,ga. (3.5) 2 P (X i = x, X j = y) = p i,j (x)p j,i (y). (3.6) ( ) λi,gf,h + λ j,ga,a M4: i j X i P o 2 p i,j (x) = e µ µx i,j i,j x!, x = 0, 1,, µ i,j = λ i,gf,h + λ j,ga,a. (3.7) 2 P (X i = x, X j = y) = p i,j (x)p j,i (y). (3.8)

, x = 0, 1,, µ i,j = λ i,gf + λ j,ga. (3.5) 2 P (X i = x, X j = y) = p i,j (x)p j,i (y). (3.6) ( ) λi,gf,h + λ j,ga,a M4: i j X i P o 2 p i,j (x) = e µ µx i,j i,j x!

7 J M5: i j i g (λ i,gf, λ j,ga, g) (λ i,gf, λ j,ga ) ( ) µ i,j µ(λ i,gf, λ j,ga ) = a 1 λ i,gf + a 2 λ j,ga + a 3 (3.9) i, j X i P o (µ i,j ) p i,j (x) = e µ µx i,j i,j, x = 0, 1,. (3.10) x! P (X i = x, X j = y) = p i,j (x)p j,i (y). (3.11) λ i,gf, λ i,ga J M5 5 ( ) 3 M5 (λ j,ga ) (λ i,gf ) ( 542) ( 3) 3: ( ) Year Average goals µ i,j = λ i,gf λ j,ga (4.1) r 2 =

10) x! 4. 4.1. P (X i = x, X j = y) = p i,j (x)p j,i (y). (3.11) λ i,gf, λ i,ga J M5 5 (2010 2014) 3 M5 (λ j,ga ) (λ i,gf ) 0.

8 28 3: : Teams M1 M2 M3 M4 M5 case # , 6, 7, , 3, mean M1 (case 1) 8% J 5

5196 0.5676 0.6173 4, 6, 7, 8 4 0.4876 0.3740 0.4282 0.3916 0.

9 J : Teams M1 M2 M3 M4 M5 case # M1 M (63) M2 M x M2 M3 M4 Frequency Points 4: M2 M4 ( , , ) M (Pts) (Mean) (Err) (Std) (Err/Std) µ ± 1.2σ 2 M1 M (18 ) ± 5 M5

2766 0.2456 0.2639 0.2864 7 3 0.0559 0.1198 0.0873 0.1172 0.1311 8 M1 M4 4 2013 (63) M2 M4 3.5 4 x 104 3 M2 M3 M4 Frequency 2.5 2 1.5 1 0.

10 30 6: M5 ( ) Standing Pts Mean Err Std Err/Std M χ 2 3 χ 2 = p = > J (1996, ) 1st 2nd : Teams Frequency 16/22 (73%) 5/22 (23%) 1/22 (4%) χ 2 2 χ 2 = p = > 0.05

549 13 45 48.65 3.65 7.41 0.492 14 45 44.90 0.10 7.47 0.013 15 37 39.43 2.43 7.10 0.342 16 25 30.72 5.72 6.90 0.828 17 23 37.37 14.37 7.22 1.990 18 14 26.62 12.62 6.55 1.926 M5 5 0.0326 5.

11 J Cumulative distribution function Points difference Approx. cdf Points difference between real and simulation 5: ( ) [7, 15] 2015 J ( ) (Y1 Y3) Y1 ( 6) 4 3 (W1) 4 (W2) W2 1

cdf 0 20 10 0 10 20 Points difference between real and

12 32 6: W1 Y2 Y3 W1 1 W2 1 ( 7(a)) W1 Y1 W2 1 Y2 Y3 1 ( 7(b)) 7: 4 2 (W1, W2) 1 1 Y2 W2 Y3 W1 4 Y1 ( 8) 8: J1 1

13 J (4 ) 1 [9] J ( ) [23] J [7, 15] [20] [3] ( ) 1 [17, 67 ] J1 [11]

14 ( ) [17, 49 ] J J 5 3 ( )2 J J (1 ) 1 +

15 J [1] bj : , php ( ). [2] J.S. Croucher: Using Statistics to Predict Scores in English Premier League Soccer. In S. Butenko, J. Gil-Lafuente, and P.M. Pardalos (eds.): Economics, Management and Optimization in Sports (Springer Berlin Heidelberg, 2004), [3] R. Gilmour: Badminton match-fixing scandal: how and why the four pairs were disqualified from the london 2012 olympics, badminton/ /badminton-match-fixing-scandal-how-and-why-the-four-pairswere-disqualified-from-the-london-2012-olympics.html ( ). [4] J. Greenhough, P.C. Birch, S.C. Chapman, and G. Rowlands: Football goal distributions and extremal statistics. Physica A: Statistical Mechanics and its Applications, 316 (2002), [5],, : (WBC)., 57 (2012), [6] : 39, /a/03.html#39. ( ). [7] J : 10 30, ( ). [8] J : 2015 ( ), ( ). [9] J : 2015 J1, aboutj/construction/ ( ). [10] Jupiler Pro League: Formule de championnat. league/competitieformule/ (in French) ( ). [11] : J1 ( ), frontale.co.jp/info/2013/0922_1.html ( ). [12] M.J. Maher: Modelling association football scores. Statistica Neerlandica, 36 (1982), [13] Major League Soccer: Competition rules and regulations, com/content/competition-rules-and-regulations. ( ). [14] Major League Baseball: World series history, ( ). [15] MSN : 2 1, jp.msn.com/sports/news/131030/scr n1.htm. ( ). [16] National Football League: History , / ( ). [17], ( ), ( ): J (, 2014). [18] C. Reep, R. Pollard, and B. Benjamin: Skill and chance in ball games. Journal of the Royal Statistical Society. Series A (General), 134 (1971),

Gilmour: Badminton match-fixing scandal: how and why the four pairs were disqualified from the london 2012 olympics, http://www.telegraph.co.

16 36 [19] Scottish Professional Football League: The rules of the Scottish Professional Football League, therulesofthescottishprofessionalfootball leagueasat11september2014_ pdf ( ). [20] R. Smyth: World cup: 25 stunning moments... no3: West germany 1-0 austria in 1982, ( ). [21] : 2 ( issue65 ), ( ). [22] V : 2014/15 V, vleague.or.jp/news_topics2/article/id=12463 ( ). [23] ( ): J 2, ( ) konaka@meijo-u.ac.jp

com/football/blog/2014/feb/25/world-cup-25-stunningmoments-no3-germany-austria-1982-rob-smyth (2015 3 3 ). [21] : 2 ( issue65 ), http:// www.footballchannel.

17 J ABSTRACT STATISTICAL ANALYSIS OF TWO-STAGE AND POSTSEASON FORMAT OF J1 FOOTBALL LEAGUE Takeshi IZUMI Eiji KONAKA Meijo University In this paper, the new postseason format from 2015 season of J1 League, the top division of professional football in Japan, is simulated and analyzed statistically. Official regulation defines that the new postseason from the 2015 season consists of five teams selected by different principles the top three teams by total season points, and the winning teams of the 1st and 2nd half of the season. The simulation results concludes that there are overlaps within these five teams very frequently, therefore the postseason tournament will be held with 3, 4, and 5 teams (i.e., without any overlaps) with probability 62%, 35%, and 3%. The result is obtained by using numerical simulation of 10 5 seasons. The goal scoring model is based on the results in 5 years (from 2010 to 2014 seasons), and is constructed by regression analysis technique. This result clarifies that the new postseason format is inherently one-stage, NOT two-stage as officially defined, format because the selection condition for the postseason tournament is not designed correctly.

Official regulation defines that the new postseason from the 2015 season consists of five teams selected by different principles the top three teams by total season points, and the winning teams of

arXiv201412_JP.dvi

arXiv201412_JP.dvi 1 2015 J ( 2 ) a a 2014 12 12 2015 J 2 3 4 5 62%, 35%, 3%( 3.4) µ±1.2σ 16 (18 ) 2 1 1 J1 2015 ( ) [1]. 1993 J 2004 2 + 2005 [2] [3] a 1-501 konaka@meijo-u.ac.jp J [4]. 2 1 2 1 5 *1 20 5 J 08 128 12 9 3000