[ C H R O N I C L E ]
The US basketball season is in full swing. Some teams are clocking up good results; others are thinking of transferring players to enhance the team. Unfortunately (or not), transactions are governed by the powerful NBA, which limits the total sum available to pay a team. Directors therefore need to recruit intelligently. So, the teams invest in probability and statistics consultants.
The first idea consists of aggregating visible data (point scored, passes, faults, etc.) into a numerical indicator. A player’s efficiency rating (PER) is one of the simple tools designed in this way. But this indicator is not sufficient to detect a talented player: some very good team members know how to make their partners better without filling out statistics.
Advanced modelling starts with the description of a succession of game configurations, called possession: one team attacks and another defends; the ball is located as are all the players on the court. Then, a position is determined to be advantageous or not for the attacking team. All the configurations that can follow this position are listed and all possible issues are explored; score two or three points, lose the ball. Typically, if the player with the ball finds himself in one of his favourite positions and there is no defender between him and the basket, it is probable that the team will score points. However, if this player is surrounded by talented defenders, he will most probably lose possession of the ball; the likelihood of scoring points is low. It only remains to average out the various results according to their probability of occurring. The probabilistic model is used to describe how a situation may change, and at each instant to characterise the “value” of a possession by the expected number of points scored (expected points valuation or EPV). Mathematically, this is an integral calculation of on the distribution of possible future possessions. To evaluate the choices of a player, you only need to see whether they are increasing or lowering the EPV. A poor strategic choice will put the team in a more delicate position and lower their chances of scoring, which will lower their EPV. Inversely, a clever game or a well-executed action will provide a good scoring opportunity and lead to an increase in the EPV.
By calculating this over the long term, it is possible to tell whether a player is enhancing his team game and whether he deserves to be hired. This temporal game description is what is known in probabilities as a Markov process. It consists of a continuous succession of situation, with random transitions between these situation that only depend on the present. To apply it, you need to calculate from a given position, the probability of transition to another position, taking account of macro-changes visible to the naked eye (like a pass or a move), or micro-changes apparently more neutral (such as a player changing his front foot).
To do so, we use all the available parameters to estimate these transitions: the probability of a successful pass from one player to another in given positions, of a successful shot... according to the player’s habits, but also those of similar players - because sometimes the statistics for a single player are not sufficient. Not all teams publish their method, but the whole power of recent statistics is used to evaluate the configuration of play, to deduce the expected potential of points scored, study the variations and finally, know whether a given player is improving the game. When will we have mathematicians commenting on matches?
Roger is a teacher at the Louis-le-Grand secondary school in Paris. Evenry month, he explains a mathematical issue in a public talk.