Abstract: This project studies the problem of a team manager (or, a coach) in a competitive league, where the manager is required to make decisions about the overall effort (energy) level that the team exerts in each game, across the whole season. In particular, the study focuses on effort dynamics “late” in the game and the impact of score-difference up to that point in the game, on the effort choice. Central to the analysis is the trade-off between “saving energy for future games” and “winning the current game” in a long season. Decisions are modelled as a Markov Decision Process, and the model is solved computationally in Python. We find, indeed, saving energy is optimal for the team manager throughout most of the season. This conclusion is robust to several variations of the base model we consider, including against a field of teams whose coaches employ similar strategies.
The final minutes of a basketball game are when legends are made or hearts are broken. It is what Michael Jordan passionately deemed “Winning Time”. Coaching decisions made during this period are critical to a team’s chances of success. It is also the most heated topic of discussion among fans. In the course of sports leagues, usually lasting several months, it is expected that the abilities of teams taking part in the season (or, tournament) will change over time. Therefore, it is not hard to fathom why a coach would be inclined to hold back or rest his players in a particular game in consideration of a farsighted objective maximizing the success rate of the team in the whole season. Several considerations may play at the back of his mind take for example: the current fatigue level of the team, strength of opponent, current score in the game, energy level of star players, schedule of upcoming games, etc. In this project, we look at effort dynamics “late” in a game and the impact of information feedback (score-difference) up to that point in the game, on the effort choice thereafter. We ask: when is it optimal, if at all, for the coach to save energy for late in the current game in consideration of long-term team success?
To this end, we consider the problem of a coach (or, team manager; or, even the players) in a competitive league, where they are required to make decisions about the overall effort level that the team exerts in each game, across the whole season. The sports we have in mind are ones such as basketball, volleyball, handball, etc., where the score-difference between competing teams can change by large margins very rapidly and frequently. We will stick to the basketball terminology henceforth for ease of presentation. The sequence of opponents that the team manager (or, coach) faces over the season, is collectively called the field, as is common, e.g., in evolutionary game theory.
During a game, players are exposed to frequent high-intensity movements like sprinting, jumping, acceleration, deceleration, and abrupt changes of direction; which can lead to acute and accumulated fatigue. Fatigue may affect the ability of the players to perform over the course of a lengthy season. Oftentimes, several games are played on consecutive days. These factors render the need for not only monitoring fatigue level; but more generally warrants an effective management of the team energy level. This is particularly more relevant for the case of star players. Resting or limiting minutes of star players is indeed something coaches frequently consider. Therefore, it is no surprise that the coach may want to hold back the intensity of the team (or limit the minutes of his star players) early in the game, and make an informed scientific decision based on the scores at a later point in the game. However, not exerting sufficient effort may prove to be costly when it comes to winning the game. Our model explores this “saving energy for the long run” versus “winning the current game” dilemma in a long season.
The rest of star players resulting from coaches' optimal strategies is also of considerable interest to team owners, investors in the league, etc. Fans are more willing to pay for quality competition that features the star players, and hence revenue maximization considerations require designing the season's structure so that coaches optimizing will lead to more opportunities for the fans to see the best players. Hence it is important for those with a vested interest in the revenue from the sport's fans to know how the incentives set for the teams will affect the time the best players spend on the courts and pitches.
The project is organized in a series of jupyter notebooks. Each notebook provides a formal description of the model as well as Python codes for simulations. A synopsis of the models are provided below.
- The first notebook (base_model_notebook.ipynb) discusses the base model that lays down a framework to study the above mentioned trade-off. A short description is as follows. At the start of each game, effort (energy) reserves are realized for both the team and the field. Score-difference at the end of the third quarter (advantage) is probabilistically determined as a function of effort exerted by both teams during those quarters. At the beginning of the fourth quarter (Q4), the coach has to decide between how much more effort (energy) to have their players exert in the current game; and how much to save for future games; as a function of the advantage at the beginning of Q4. Effort reserves for the team and the field are replenished (stochastically) before the start of the next game. The coach's goal is to maximize the expected sum of winning in the whole season.
The key simplifications (that are later generalised) made in the base model are as follows:
- The field is assumed to split its energy equally in all quarters of every game, i.e., it does not act as a strategic agent, therefore making the base model a single-player decision problem.
- The coach/manager contemplates energy saving decisions only in the last quarter (Q4) of every game.
Findings: As one would expect, initial conditions are theoretically shown to be inconsequential in a long season: visualizing simulation results confirm the same. The Expected Win Rate (EWR), which is defined as “the average expected value (or, winning probability) in games remaining” and the optimal strategies are presented in detail. Visualizing optimal strategies from the simulations show that the manager finds it optimal to save energy throughout most of the season. The optimal strategies (i.e., energy saved) do not show noticeable difference across different stages of the league.
- The two other notebooks study extensions of the base model:
- In the first extension, the field behaves strategically, i.e., the managers of the field-of-opponents employ the optimal strategy that is evaluated in the base model. Simulation results show that the conclusions from the base model remain true. See the notebook field_is_strategic_notebook.ipynb for a detailed analysis.
- The second extension challenges the assumption made in the base model that an exogenously given constant fraction of the effort reserve is used up by the team, throughout the first three quarters combined. The manager is allowed to act strategically during the first three quarters (Q1 through to Q3 combined) as well. Interestingly, the (optimal) energy used in the first three quarters peaks at about 75%. This is not obvious to begin with: one might think perhaps to put in a greater effort in the first three quarters to get a large lead. Allowing the coach to have greater control can indeed be helpful (the EWR is higher compared to the base model), as it allows the manager to decide when facing a team that has high energy, if he/she may wish to “not bother” and save its energy instead. See the notebook strategic_all_quarters_notebook.ipynb for details.