A Mathematical, Statistical, Probabilistic and Strategic Analysis of Tennis
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A MATHEMATICAL, STATISTICAL, PROBABILISTIC AND STRATEGIC ANALYSIS OF TENNIS Geoff Pollard A thesis for the fulfilment of the requirements for the degree of Doctor of Philosophy at Swinburne University of Technology 2017 1 Abstract The relatively complicated tri‐nested scoring system used in tennis presents many opportunities for the mathematical, statistical, probabilistic and strategic analysis of the game. Summary data is now available for nearly all matches in professional tennis, but point‐by‐point data required for studies of independence between points are much harder to obtain and analyse. Fortunately, data were available for the 2011 Grand Slams and these provide the statistical data to go with the mathematical and probabilistic analysis. My own background in tennis as a player and administrator provides the accompanying strategic analysis. The opportunity to have a second serve, if the first serve fails, invites an investigation into the optimal use of both serves. It is shown that if you don’t serve some faults, and even some double faults, you are not taking enough risk on service to maximise the chance of winning the point. Not all points are equally important and an alternative definition of importance is presented to challenge the traditional definition. If you can lift (increase the probability of winning the point) on one or two points in a game, the two definitions produce quite different solutions of when to lift. Assuming all points in a game are independent is a basic assumption for mathematical and probabilistic analysis. This assumption is challenged. Four direct and four indirect tests of independence are developed and applied to the point‐ by‐point data. There is one clear difference between successive points and that is that one point is served to the first court and the other point is served to the second court. This 2 suggest that tennis could be considered as point pairs and this approach is discussed and applied to the data. Tennis players have always endured the indefinite nature of the scoring system, but with tennis now professional, administrators have gradually introduced modifications to the scoring system to improve the efficiency of tennis as presented to spectators, especially on television. The efficiency of all the three and five set systems are calculated and discussed, as well as the efficiency of tournament structures. But efficiency in scoring or tournament structure may not actually improve a tournament, so the final chapter looks at other strategic ways tournaments could improve. Lessons from the Australian Open can be modified for other tournaments. 3 Acknowledgements I would like to acknowledge the contribution of my brother Professor Graham Pollard who encouraged me to get involved in academic research again after an interval of approximately twenty years when my full‐time role as President, Chairman and CEO of Tennis Australia changed to Executive President and Chairman and I employed a CEO to run the day‐to‐day business. Graham and I co‐authored 21 articles with an overall even split of responsibility, he being named first 11 times and me ten times. He also encouraged me to stick with the thesis, even at times when I could not understand why a person approaching retirement would bother. I would also like to acknowledge the help and guidance from my supervisor Professor Denny Meyer and thank her for being so tolerant with my challenges and time and travel restraints that went with my national and International roles in tennis administration. She offered great encouragement throughout the long ordeal and never wavered in her commitment. My other supervisor Professor Stephen Clarke retired from Swinburne University, but was always there and was of great help as a new set of eyes to my original draft of this thesis. Finally I would like to thank my wife of 47 years, Eleanor, who could never really understand why I was still studying, what interest there was in numbers and whenever would I finish and be able to clear the desk. We already have a list of things to do together now we have more time and we are looking forward to it. We have two great children and three lovely grandchildren and will also be able to spend more time with them. Thank you Eleanor for your love and support. 4 Declaration of Authorship This thesis is presented to fulfil the requirements of a Doctor of Philosophy at Swinburne University of Technology. This thesis is a complete update and rewrite of ten years of part‐time academic research, much of which has already been published in academic journals and conference papers and one is a chapter in a major encyclopaedia. Altogether there are 39 published articles and 21 of these were in conjunction with my brother, Graham. We each made substantial contributions to all of our joint publications. Four chapters are completely my own work and four chapters are the result of the collaboration with my brother. All the data analysis, which was substantial, is mine. With this qualification, I hereby declare that this thesis and the work presented in it is entirely my own. No other person’s work has been used without due acknowledgement in this thesis. All references and verbatim extracts have been quoted and the sources of the data used in the analyses have been specifically acknowledged. Geoff Pollard December 8, 2017 5 6 Table of Contents ............................................................ Error! Bookmark not defined. CHAPTER 1. ................................................................................................................... 15 INTRODUCTION AND OUTLINE ..................................................................................... 15 1.1. Introduction ....................................................................................................... 15 1.2. Outline................................................................................................................ 18 CHAPTER 2 .................................................................................................................... 23 AN OVERVIEW OF OPERATIONS RESEARCH AND MANAGEMENT SCIENCE IN TENNIS 23 2.1. Current and Potential use of Operations Research and Management Science in Tennis. ....................................................................................................................... 25 2.2. Developing Alternative and Optimal Scoring Systems. ...................................... 27 2.3. Developing Optimal Match Strategies. .............................................................. 30 2.4. Surface Optimization. ........................................................................................ 32 2.5. Optimization of Ball Characteristics. .................................................................. 35 2.6. Optimization of Racquet Characteristics. .......................................................... 36 2.7. Optimization from a Coaching Perspective. ...................................................... 38 2.8. Optimizing Player Technique. ............................................................................ 40 2.9. Minimizing Medical Risk. ................................................................................... 41 2.10. Efficient Tournament Management. ............................................................... 45 2.11. Strengthening the Mental Side of the Game. .................................................. 46 2.12. Accurate rankings of Players. ........................................................................... 47 2.13. Improving Officiating. ...................................................................................... 49 2.14. Integrity and Betting ........................................................................................ 51 2.15. Summary and Implications .............................................................................. 60 CHAPTER 3. ................................................................................................................... 63 FIRST AND SECOND SERVICE ...................................................................................... 63 3.1. The basic equation for serving ........................................................................... 65 3.2 Literature on first and second serves ................................................................. 66 3.3. Optimal risk taking on first and second serve. ................................................... 69 3.4. Penalty for non‐optimal serving ........................................................................ 74 3.5. Extension to a quadratic model and other curves ............................................. 76 3.6. Where the server might focus attention on increasing the probability of winning a point ......................................................................................................... 82 7 3.7. General tennis discussion .................................................................................. 88 3.8. Summary and implications ................................................................................. 89 CHAPTER 4 .................................................................................................................... 92 IMPORTANCE ................................................................................................................ 92 4.1. The classical definition and analysis of Importance........................................... 95 4.2. A different approach to importance .............................................................