Mathematics of tennis ranking : dynamical aspects and game outcome prediction by optimization methods

This master thesis investigates the potential of the Elo rating system in tennis in three parts. The first part describes the mathematical model used in the Elo rating system, which can be expressed as an autonomous discrete-time dynamical system. Therefore, we can analyze the stability to determine the system's behavior when time goes to infinity. There is an infinity of equilibria, invariant by shifts of the ratings. Moreover, this analysis provides some conditions on the K-factor of the system. If these conditions are satisfied, we can ensure that there is no eigenvalue outside the unit circle. However, the invariance by shifts forces an eigenvalue always to be equal to one. The system converges to a unique equilibrium using normalization of the ratings and a K-factor respecting the stability conditions. Using the continuous version of the system, we come to the same conclusion. Some values for an adaptive K-factor that respect the stability conditions are proposed to implement an Elo rating system in tennis. This takes into account the stage and the prestige of the tournament. The second and third chapters use two error criteria that are optimized in order to find the ratings that are the best estimates of the real probabilities inside a competition. The two functions used are the least-square and the log-likelihood approaches. The gradients of the two error criteria correspond to the update formula. Moreover, the gradient descent algorithm preserves the sum of the ratings. Thus, optimizing these functions leads to the same global minimum, which is the system's equilibrium if the matrix of the real probabilities is transitive. However, if the matrix of the real probabilities is not transitive, the global minimum of the two functions is different. Finally, we use the optimal ratings returned by the optimization to predict the outcome of some professional tennis games between 2000 and 2016. Several ratings are computed, considering the players' performances on the whole season, shorter periods, and specific surfaces. The conclusion is that the accuracy of predictions of the well-known ATP ranking and ratings from optimization methods based on the same period is the same. However, we can refine the accuracy of the predictions by adjusting the ratings with the level of each player on specific surfaces. It is, therefore, possible to implement an Elo rating system in tennis if the K-factor is well-chosen and if we consider that the level of the players varies according to the surface.