A bst ract. Review. m wal u; j woo pa.arizona.ed u

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1 A bst ract Review m wal ker@arizona.ed u; j woo ders@b pa.arizona.ed u

2 any

3 playing playing play well

5 i.e. distribution

6 2 2 1 T he 2 2 Point G a me 2 2 point ga me i.e. 2 2 i.e.

7 sr 1 sr sr 1 sr 2 2 sr sr 2 2 sr L L R L R R L R 1 match

8 i.e. 1 L L 1 L R 1 R R 1 R L L L L R R R R L M ixed St ra tegy C on dition 1 2 A ssu m p t ion 1: 2 2 T he T ennis M a t c h as a G ame games sets match infinite-horizon

9 two binary Markov games sr

10 , Va r ia t io n in P oin t G ames i.e. sr should not

11 A ssu m p t ion 2: 2 O n Testing t he T heor y i.i.d. sr

12 estimate are sr does win ra tes for his serves A l t e r n a t iv e M o de l s o f t he P oin t G ame

13 were

14 T he D a t a

15 sr 2 2 equilibrium i.e.

16 4 Testing for E quality of W inning P robabilities null hypothesis

17 H ypo t heses a nd T es t s f o r t he T ennis D a t a i.e. = 1 40 i j i.e. success failure i j S i j F i j i.e. null hy pot hesis i L = i R i

18 i L = i i R = i i = X j L, R " ( i j S i j i ) 2 i j i + ( j i F i j (1 i )) 2 # i j (1 i ) i i N L i S + N R i S i i n i L + n i R i i i.e. 1 according to the theory i.e. joint all forty i L = i R each one = 1 40 i.i.d.

19 i L i R i P i.e. 40 i = 1 i fewer i.e. i i L = i R i i i i [0 1] etc [0 1] ( ) ( ) = 1 ( ) 0 1 Pr[ ( ) ] = Pr[1 ( ) ] = Pr[ ( ) 1 ] = Pr[ 1 (1 )] = 1 ( 1 (1 )) = 1 (1 ) =

20 c.d.f i.e. ( ) = [0 1] 40 ˆ ( ) ˆ ( ) = P 40 i = 1 [0,x] ( i ) [0,x] ( i ) = 1 i [0,x] ( i ) = = 40 sup x [0,1] ˆ ( ) = viz. viz.

21 typical A pp l y in g o u r T es t s t o O N ei l l s D a t a

22 1 50 joint every P 50 i = 1 i i.e.

23 = T he P o w e r o f O u r T es t s % 3 i.e % viz.

24 = 2 3 L R L ( ) = (1 ) R ( ) = (1 ) i.e. = 2 3 P 40 i = 1 i 5% P 40 i = 1 i = 2 3 L = R P 40 i = 1 i i.e

25 0 = 2 3 = 5 i.e. 58 L = 685 R = 610) = 6 i.e Serial I ndependence i.e. i.i.d. H ypo t heses a nd T es t s f o r t he T ennis D a t a i = ( 1 i i n i L + n ) i R i n i L i R i i run i.e.

26 L R ( ; L R ) ( ; L R ) i.e. ( ; L R ) = P rk = 1 ( ; L R ) ( i ; i L i R ) ( i 1; i L i R ) 025 i.e. i 025 i 025 i i ( 1) ( ) i 1 i each = ( )

27 continuous ( 1) ( ) i i L i R i [ ( i 1; i L i R)( i ; i L i R )] i [0 1] 1 40 = i i.e. Se r ia l In dependen c e in O Nei l l s D a t a [0 1] L R ( ) ( ; L R ) Pr[ ] = i.e. ( 1) ( ) Pr[ ] = (2) + + ( 1) + ( ) x F ( r 1) F ( r ) F ( r 1) = ( ) ( 1) = ( ) (2) + + ( 1) = ( 1) i

28 i i =

29 6 C oncluding R emar ks some 2 2 viz

31 R eferences Econometrica 58 Thinking Strategically: The Competitive Edge in Business, Politics, and Everyday Life American Economic Review 88 Hard Courts Nonparametric Statistical Inference Introduction to the Theory of Statistics Proceedings of the National Academy of Sciences 84 Psychological Bulletin 77

32 Table 1: Mixtures and Win Rates in Tennis Data Serve Direction Total Mixture Points Won Win Rates Match Hnd Server Court L R C Srvs L R C L R C L R C 74Wimbldn R Rosewall Ad Wimbldn R Rosewall Deuce Wimbldn R Smith Ad Wimbldn R Smith Deuce Wimbldn R Borg Ad Wimbldn R Borg Deuce Wimbldn L McEnroe Ad Wimbldn L McEnroe Deuce USOpen L McEnroe Ad USOpen L McEnroe Deuce USOpen R Borg Ad USOpen R Borg Deuce Wimbldn L Connors Ad Wimbldn L Connors Deuce Wimbldn L McEnroe Ad Wimbldn L McEnroe Deuce French R Lendl Ad French R Lendl Deuce French L McEnroe Ad French L McEnroe Deuce Australn R Edberg Ad Australn R Edberg Deuce Australn R Cash Ad Australn R Cash Deuce Australn R Wilander Ad Australn R Wilander Deuce Australn R Cash Ad Australn R Cash Deuce Masters R Becker Ad Masters R Becker Deuce Masters R Lendl Ad Masters R Lendl Deuce USOpen R Sampras Ad USOpen R Sampras Deuce USOpen R Agassi Ad USOpen R Agassi Deuce USOpen L Korda Ad USOpen L Korda Deuce USOpen R Sampras Ad USOpen R Sampras Deuce Totals

33 Table 2: Testing for Equality of Winning Probabilities in Tennis Data Serves Total Mixture Pts Won Win Rates Pearson Match Hand Server Court L R Srvs L R L R L R statistic p-value 74Wimbldn R Rosewall Ad Wimbldn R Rosewall Deuce Wimbldn R Smith Ad Wimbldn R Smith Deuce Wimbldn R Borg Ad Wimbldn R Borg Deuce Wimbldn L McEnroe Ad Wimbldn L McEnroe Deuce USOpen L McEnroe Ad USOpen L McEnroe Deuce USOpen R Borg Ad USOpen R Borg Deuce Wimbldn L Connors Ad ** 82Wimbldn L Connors Deuce Wimbldn L McEnroe Ad Wimbldn L McEnroe Deuce French R Lendl Ad French R Lendl Deuce French L McEnroe Ad French L McEnroe Deuce Australn R Edberg Ad Australn R Edberg Deuce Australn R Cash Ad Australn R Cash Deuce Australn R Wilander Ad Australn R Wilander Deuce Australn R Cash Ad Australn R Cash Deuce Masters R Becker Ad Masters R Becker Deuce Masters R Lendl Ad Masters R Lendl Deuce USOpen R Sampras Ad USOpen R Sampras Deuce * 95USOpen R Agassi Ad USOpen R Agassi Deuce USOpen L Korda Ad USOpen L Korda Deuce USOpen R Sampras Ad USOpen R Sampras Deuce Totals * and ** indicate rejection at the 5% and 10% level of significance, respectively.

34 Table 3: Testing for Equality of Winning Probabilities in O'Neill's Data Mixtures Win Rates Pair Player Joker Non-J Joker Non-J Pearson Q p-value ** * ** * * * ** * * ** * ** * * * * 10 rejections at 5% ** 15 rejections at 10%

35 Table 4: Runs Tests on Tennis Data Serves Runs Match Hnd Server Court L R Total r i F(r i -1) F(r i ) U[F(r i -1),F(r i )] 74Wimbldn R Rosewall Ad Wimbldn R Rosewall Deuce Wimbldn R Smith Ad Wimbldn R Smith Deuce Wimbldn R Borg Ad Wimbldn R Borg Deuce Wimbldn L McEnroe Ad Wimbldn L McEnroe Deuce USOpen L McEnroe Ad USOpen L McEnroe Deuce USOpen R Borg Ad USOpen R Borg Deuce Wimbldn L Connors Ad * Wimbldn L Connors Deuce ** Wimbldn L McEnroe Ad Wimbldn L McEnroe Deuce French R Lendl Ad French R Lendl Deuce ** French L McEnroe Ad French L McEnroe Deuce * Australn R Edberg Ad * Australn R Edberg Deuce Australn R Cash Ad ** Australn R Cash Deuce Australn R Wilander Ad Australn R Wilander Deuce Australn R Cash Ad Australn R Cash Deuce * Masters R Becker Ad Masters R Becker Deuce Masters R Lendl Ad Masters R Lendl Deuce USOpen R Sampras Ad USOpen R Sampras Deuce * USOpen R Agassi Ad USOpen R Agassi Deuce USOpen L Korda Ad USOpen L Korda Deuce USOpen R Sampras Ad USOpen R Sampras Deuce * and ** indicate rejection at the 5% and 10% level of significance, respectively.

36 Table 5: Runs Tests on O'Neill's Data Choice Runs Pair Player J N r i F(r i -1) F(r i ) U[F(r i -1),F(r i )] * * ** * * * * ** ** * ** * * * * * * * * * and ** indicate rejection at the 5% and 10% levels of significance.

37 Figure 1: A Typical Point Game Receiver L R Server L π LL π LR R π RL π RR Outcomes (cell entries) are probability Server wins the point. Example Receiver Server's L R Minimax Server L /3 R /3 Rec's Minimax: 2/3 1/3 Value =.65

38 Figure 2: The Tennis Court Server's Side Deuce Ad Ad Court Deuce Court L R L R Ad Deuce (Odd; 1) (Even; 0) Receiver's Side

39 Figure 3 Win Rates in Tennis Data: Histogram of p-values p-values

40 Figure 4 Win Rates in Tennis: Kolmogorov Test k=.670 (p-value =.76)

41 Figure 5 Win Rates in O'Neill: Histogram of p-values p-values

42 Figure 6 Win Rates in O'Neill: Kolmogorov Test k=1.704 (p-value =.006)

43 Figure 7 The Power Function Probablity of rejecting null Receiver probability of Left

44 Figure 8 Runs in Tennis Data: Kolmogorov Test k=1.948 (p-value =.001)

45 Figure 9 Runs in O'Neill's Data: Kolmogorov Test k=2.503 (p=value = )

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