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 = p 2 ( 3 – 2 p ) where p = probability of a player winning a set 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 p 2 ( 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: I PM = I PG I GS I SM where : I PM = importance of point in a match I PG = importance of point in a game I GS = importance of game in a set I SM = 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 ). 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 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

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 p 2 ( 3 – 2 p ) + ε 2 ( 2 p – 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 3 rd 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 3 rd set played and a proportion of the time on the 2 nd set played. For example: p =0.6, ε = 0.1 A B Table 4: Chances of players winning a match Chances of winning match with: 1 no increase/decrease in effort 2 increase on 3 rd set and decrease on 1 st set 3 increase on 3 rd set and decrease on 1 st set and ½ the time on 2 nd set

Optimizing a game 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 [ X I ( 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 [ X I ( 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 [ X I ( 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