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Optimal Use of Tennis
Resources
Tristan Barnett
Stephen Clarke
Alan Brown
Background
Mathematics in Industry Study Group 2003 (MISG)
Defence Science of Technology Organisation (DSTO)
Tennis: (points, games, sets, match)
Warfare: (skirmishes, battles, campaigns, war)
Analysis of Hierarchical Games
The value of a point depends on the current score.
Introduction

On which point should a player increase his effort to optimize his
chance of winning a game?

Are there situations where it is correct for a player to throw a
game, or even a set?

Does varying effort about an overall mean have an effect on the
chance of winning a game?

Applications to solving defence strategy problems.
Probabilities on winning a 3 set match
Chance of winning a match = p2 ( 3 – 2 p )
where p = probability of a player winning a set
Current match
score in sets
Match score at
Increase in
which an increase
probability of
is applied
winning match
( 0,0 )
ε 2 p (1-p)
(0,0)
( 1,0 )
ε p (1-p)
( 0,1 )
ε p (1-p)
( 1,1 )
ε 2 p (1-p)
Table 1: The increase in probability when effort is applied throughout the match
The chance of a player winning the match when he decides at the start of the
match to apply one increase in effort on the first, second or third set played is
equal to p2 ( 3 – 2 p ) + ε 2 p ( 1 – p ).
Importance
Reference: Morris, The Most Important Points in Tennis (1977)
If P(a,b) is the probability a player wins the game from game score (a,b), then the
importance of a point I(a,b) is represented by:
I(a,b) = P(a+1,b) - P(a,b+1)
Multiplication result for importance:
IPM = IPG IGS ISM
where: IPM = importance of point in a match
IPG = importance of point in a game
IGS = importance of game in a set
ISM = importance of set in a match
Weighted-Importance
Morris: If N(a,b|0,0) is the probability of reaching game score (a,b) from game
score (0,0), then the time-importance of a point T(a,b) is represented by:
T(a,b | 0,0) = I(a,b) N (a,b | 0,0)
If N(a,b|g,h) is the probability of reaching game score (a,b) from game score
(g,h), then the weighted-importance of a point W(a,b|g,h) is represented by:
W(a,b | g,h) = I(a,b) N (a,b | g,h)
Weighted importance for any point in the match is represented by:
W(a,b:c,d:e,f | g,h:i,j:k,l) = I(a,b:c,d:e,f) N(a,b:c,d:e,f | g,h:i,j:k,l)
Weighted-Importance Property
Suppose a player, who ordinarily has probability p of winning a set, decides that he will
try harder every time the set ( e,f ) occurs. If by doing so he is able to raise his
probability of winning from p to p + ε, ( p+ ε < 1 ) for that set alone, then he raises his
probability of winning the match from P ( k,l ) to P ( k,l ) +ε W( e,f | k,l ).
opponent score in sets
0
1
player score in sets 0 2 p (1- p)
p (1- p)
1 p (1- p)
2 p (1- p)
Table 2: The weighted importance of sets in a match from (0,0)
The optimal strategies for a player with 1 ≤ M ≤ 3 available increases,
is to apply an increase in effort on any M sets of the match.
Optimizing a game
player score in points
0
15
30
40
Ad
0
0.257
0.123
0.046
0.010
opponent score in points
15
30
40
0.134 0.057 0.016
0.154 0.124 0.063
0.107 0.154 0.157
0.040 0.100 0.122
0.048
Ad
0.075
Table 3: The weighted importance of points in a game from
(0,0), where the probability of the server winning a point is 0.6.
An optimal strategy for a player with M ≥ 1 available
increases, is to apply an increase in effort on the first M
points of the game.
Optimizing a set and a match
Suppose player A is leading in a set with a score of (5,3) (A score = 5, B score =3), with
player B to serve the next game.
It can be shown that player A should aim to win with a score (6,4) by conserving energy
while player B is serving. If it happens that the score reaches (5,4) he should increase his
effort to win his own serve and the set. This strategy dominates the alternative of
expending the energy to break B’s serve and trying to win the set with a score (6,3).
It can also be shown that a player ahead on sets, but behind in the current set, may be
better off to save energy to try and win the next set, rather than expend additional energy
in the current set.
Example: 2003 Davis Cup final: Philippoussis df Ferrero 7-5 6-3 1-6 2-6 6-0
Varying the Effort
An increase in effort by ε on a set played and a corresponding decrease in effort by ε in
another set played will preserve the overall mean of winning a set.
However the chance of a player winning the match by varying effort about the mean is
p2 ( 3 – 2 p ) + ε2 ( 2p – 1 ).
For p > ½ (i.e the stronger player), the chance of winning has increased.
The increase or decrease in probability of winning the match for a player
is caused by the variation about the mean probability of winning a set.
Varying the Effort
However, since the 3rd set is only played a proportion of the time the better player can
further increase his chance of winning the match by applying an increase in effort on the
3rd set played and a proportion of the time on the 2nd set played.
For example: p=0.6, ε = 0.1
1
2
3
A
B
0.648
0.650
0.675
0.352
0.350
0.377
Table 4: Chances of players winning a match
Chances of winning match with:
1 no increase/decrease in effort
2 increase on 3rd set and decrease on 1st set
3 increase on 3rd set and decrease on 1st set
and ½ the time on 2nd set
Optimizing a game
player score in points
0
15
30
40
Ad
0
0.27
0.21
0.13
0.05
opponent score in points
15
30
40
0.35
0.37
0.25
0.33
0.44
0.42
0.26
0.46
0.69
0.12
0.31
0.46
0.31
Ad
0.69
Table 5: The importance of points in a game, where the probability
of the server winning a point is 0.6.
A player can gain a significant advantage by increasing effort on the important
points and decreasing effort on the unimportant points.
For the better player, this gain is a result of both the variability
about the mean and also the importance of points.
Applications to warfare
Tennis: (points, games, sets, match)
Warfare: (skirmishes, battles, campaigns, war)
Applications to warfare
A team has M available increases in effort available for use in the war. Where should
they apply the increases to optimize their chances of winning the war?
A team has M available increases in effort available for use in the war. However
there are costs associated for applying an increase in effort at a particular skirmish
(and a reward for winning the war). Where should they apply the increases to
optimize their chances of winning the war?
A team has a “large” number of available increases in effort available for use in the
war. However there are costs associated for applying an increase in effort at a
particular skirmish (and a reward for winning the war). Where should they apply the
increases to optimize their chances of winning the war?
Applications to warfare
Let E[X(e,f)] = [pP(e+1,f) + (1-p) P(e,f+1)]r
E[XI(e,f)] = [(p+ε) P(e+1,f) + (1-p-ε) P(e,f+1)]r - c
where: E[X(e,f)] = expected payout at set (e,f) in a match with no increase
E[XI(e,f)] = expected payout at set (e,f) in a match with an increase
r = reward for winning the overall war
c = the cost of applying an increased effort
If E[XI(e,f)] – EX[(e,f)] > 0, then an increase should be applied at (e,f) or equivalently:
I(e,f) ε r – c > 0
Similarly an increase should be applied for points in a match for which:
I(a,b: c,d: e,f) ε r – c > 0
Further Research
 The effect on the probability of winning the match arising from
depleting available energy to win the point.
 The ability to generalise from tennis to a more complex game
structure.
 The definition of a model of match outcome into which the effect of
morale or other psychological effects can be incorporated.
Acknowledgements
Elliot Tonkes
Vladimir Ejov