Mathematical modelling in hierarchical games with specific reference to tennis

This thesis investigates problems in hierarchical games. Mathematical models are used in tennis to determine when players should alter their effort in a game, set or match to optimize their available energy resources. By representing warfare, as a hierarchical scoring system, the results obtained in tennis are used to solve defence strategy problems. Forecasting in tennis is also considered in this thesis. A computer program is written in Visual Basic for Applications (VBA), to estimate the probabilities of players winning for a match in progress. A Bayesian updating rule is formulated to update the initial estimates with the actual match statistics as the match is progressing. It is shown how the whole process can be implemented in real-time. The estimates would provide commentators and spectators with an objective view on who is likely to win the match. Forecasting in tennis has applications to gambling and it is demonstrated how mathematical models can assist both punters and bookmakers. Investigation is carried out on how the court surface affects a player's performance. Results indicate that each player is best suited to a particular surface, and how a player performs on a surface is directly related to the court speed of the surfaces. Recursion formulas and generating functions are used for the modelling techniques. Backward recursion formulas are used to calculate conditional probabilities and mean lengths remaining with the associated variance for points within a game, games within a set and sets within a match. Forward recursion formulas are used to calculate the probabilities of reaching score lines for points within a game, games within a set and sets within a match. Generating functions are used to calculate the parameters of distributions of the number of points, games and sets in a match.