core/input/test_middle_result.py

problem_str = """\
Problem Background:
In the 2023 Wimbledon Gentlemen’s final, 20-year-old Spanish rising star Carlos Alcaraz defeated 36-year-old Novak Djokovic. The loss was Djokovic’s first at Wimbledon since 2013 and ended a remarkable run for one of the all-time great players in Grand Slams.
The match itself was a remarkable battle.[1] Djokovic seemed destined to win easily as he dominated the first set 6 – 1 (winning 6 of 7 games). The second set, however, was tense and finally won by Alcarez in a tie-breaker 7 – 6. The third set was the reverse of the first, Alcaraz winning handily 6 – 1. The young Spaniard seemed in total control as the fourth set started, but somehow the match again changed course with Djokovic taking complete control to win the set 6 – 3. The fifth and final set started with Djokovic carrying the edge from the fourth set, but again a change of direction occurred and Alcaraz gained control and the victory 6 – 4. The data for this match is in the provided data set, “match_id” of “2023-wimbledon-1701”. You can see all the points for the first set when Djokovic had the edge using the “set_no” column equal to 1. The incredible swings, sometimes for many points or even games, that occurred in the player who seemed to have the advantage are often attributed to “momentum.”

Problem Requirement:
One dictionary definition of momentum is “strength or force gained by motion or by a series of events.”[2] In sports, a team or player may feel they have the momentum, or “strength/force” during a match/game, but it is difficult to measure such a phenomenon. Further, it is not readily apparent how various events during the match act to create or change momentum if it exists. Data is provided for every point from all Wimbledon 2023 men’s matches after the first 2 rounds. You may choose to include additional player information or other data at your discretion, but you must completely document the sources. Use the data to:
Develop a model that captures the flow of play as points occur and apply it to one or more of the matches. Your model should identify which player is performing better at a given time in the match, as well as how much better they are performing. Provide a visualization based on your model to depict the match flow. Note: in tennis, the player serving has a much higher probability of winning the point/game. You may wish to factor this into your model in some way.
A tennis coach is skeptical that “momentum” plays any role in the match. Instead, he postulates that swings in play and runs of success by one player are random. Use your model/metric to assess this claim.
Coaches would love to know if there are indicators that can help determine when the flow of play is about to change from favoring one player to the other. Using the data provided for at least one match, develop a model that predicts these swings in the match. What factors seem most related (if any)? Given the differential in past match “momentum” swings how do you advise a player going into a new match against a different player? Test the model you developed on one or more of the other matches. How well do you predict the swings in the match? If the model performs poorly at times, can you identify any factors that might need to be included in future models? How generalizable is your model to other matches (such as Women’s matches), tournaments, court surfaces, and other sports such as table tennis.
Produce a report of no more than 25 pages with your findings and include a one- to two-page memo summarizing your results with advice for coaches on the role of “momentum”, and how to prepare players to respond to events that impact the flow of play during a tennis match.

Data Description:
The dataset provides a comprehensive overview of match statistics from tennis games, specifically detailing individual points within matches at the 2023 Wimbledon tournament. Each entry is associated with a unique match identifier that indicates the round and match number, allowing for easy tracking of specific games throughout the tournament.

Key players are identified by their full names, and the dataset captures various metrics that reflect their performance over the course of the match. The elapsed time field records the duration from the start of the first point to the beginning of the current point, providing insight into the pace of play.

The dataset includes information on sets and games, indicating how many sets each player has won, as well as their respective game victories in the current set. This allows for analysis of player performance both within individual matches and across the tournament. The scoring system is detailed, with scores recorded in the traditional tennis format (love, 15, 30, 40, advantage), which adds context to the competitive nature of the points being analyzed.

Service dynamics are captured through fields indicating which player is serving, the type of serve (first or second), and the outcomes of those serves. Additionally, the dataset tracks critical point outcomes, identifying the winner of each point along with various performance metrics such as aces, double faults, winners, and unforced errors. These statistics are essential for evaluating the players' effectiveness and consistency during the match.

The dataset also includes information on strategic elements like break points—opportunities for players to capitalize on their opponent's serve—providing insights into pivotal moments that can influence match outcomes. Furthermore, physical performance is tracked through metrics like distance run during points, which highlights the physical demands placed on players during high-stakes moments.

Rally counts and shot speeds contribute to a deeper understanding of match dynamics, revealing how aggressive or defensive a player might be during exchanges. The depth and width of serves and returns further illustrate players' tactical choices, aiding in the analysis of their playing styles.

Overall, this dataset serves as a rich resource for analyzing tennis match performance, providing a detailed snapshot of player actions and outcomes throughout specific points in a match, which can be invaluable for coaches, analysts, and fans looking to understand the intricacies of tennis at a professional level.
"""

problem_analysis = """\
The problem of modeling momentum in tennis matches, particularly in the context of the 2023 Wimbledon tournament, requires a multi-faceted approach that integrates advanced statistical techniques, machine learning algorithms, and a deep understanding of the sport's dynamics. Momentum in tennis is a complex phenomenon influenced by a combination of player-specific attributes, match-specific conditions, and external variables. To develop a robust model, it is essential to move beyond traditional metrics such as serving advantage and incorporate granular data on serve placement, speed, and depth, as well as return strategies. These serve-specific details can provide deeper insights into the flow of play and help identify moments where momentum may shift. For instance, a player’s ability to consistently place serves in challenging positions or exploit an opponent’s return weaknesses can create sustained advantages that are not fully captured by simple win-loss metrics. Additionally, the model should account for the role of court surface, as different surfaces (e.g., grass, clay, hard court) can influence player performance and the dynamics of momentum. For example, on grass courts like Wimbledon, the faster surface may amplify the impact of serve dominance, while on clay courts, longer rallies and baseline play may shift the balance of momentum differently. By incorporating surface-specific dynamics, the model can better capture the nuances of momentum in different contexts.

Psychological factors also play a crucial role in momentum, as player confidence, resilience, and the ability to handle pressure during critical points (e.g., break points or set points) can significantly influence performance. Quantifying these factors requires integrating historical performance data to assess how players typically respond under pressure, as well as leveraging real-time biometric data to gauge mental and physical fatigue. For example, a player may experience a temporary dip in performance due to fatigue, which could be misinterpreted as a loss of momentum if not properly accounted for. Similarly, the model should consider the significance of specific points, such as break points or set points, which can have a disproportionate impact on momentum. By incorporating these contextual factors, the model can provide a more accurate and nuanced depiction of momentum in tennis matches. Furthermore, the model should account for the psychological impact of crowd support or adverse conditions, such as weather, which can influence player performance and momentum shifts. For instance, a player who thrives under pressure may gain momentum from a supportive crowd, while another player may struggle with the added scrutiny of a high-stakes match.

To assess the claim that momentum is merely a random phenomenon, the model must employ rigorous statistical methods to test the significance of observed performance swings. This involves analyzing the distribution of point outcomes and comparing them to expected probabilities based on serving advantages and other factors. If the model identifies patterns that deviate significantly from random variations, it would provide evidence supporting the existence of momentum. Conversely, if the observed swings align with random fluctuations, it would lend credence to the coach's skepticism. This analysis should also consider the potential for overfitting, ensuring that the model does not attribute significance to spurious patterns in the data. Additionally, the model should be tested on multiple matches to assess its robustness and generalizability. For example, by applying the model to matches from different rounds of the tournament or different years, we can evaluate whether the observed patterns of momentum are consistent across different contexts. This cross-validation process is critical for establishing the reliability of the model and its ability to distinguish between genuine momentum shifts and random variations.

Predicting shifts in the flow of play requires the model to identify indicators that precede changes in momentum. These indicators could include changes in player behavior, such as increased aggression or defensive play, as well as performance metrics like unforced errors or first-serve percentages. By analyzing these factors, the model can provide actionable insights for coaches, helping them anticipate and respond to momentum shifts during a match. For example, if the model detects a pattern of increasing unforced errors by a player, it could signal an impending loss of momentum, prompting the coach to adjust the player's strategy. However, the model's predictive accuracy must be rigorously tested on multiple matches to ensure its reliability. If the model performs poorly in certain scenarios, it may indicate the need to include additional variables or refine the existing ones. For instance, incorporating data on player fatigue or mental state could improve the model's predictive capabilities. Additionally, the model should consider the role of tactical adjustments, such as changes in serve placement or return strategy, which can influence the flow of play and momentum. By integrating these dynamic factors, the model can provide a more comprehensive understanding of momentum and its drivers.

The generalizability of the model to other contexts, such as women's matches, different tournaments, or other sports, depends on its ability to capture universal aspects of momentum while remaining adaptable to specific conditions. For instance, the serving advantage in tennis may differ from the serve advantage in table tennis, requiring adjustments to the model. Similarly, the psychological and physical demands of different sports may influence how momentum manifests. By designing the model with flexibility in mind, it can be adapted to analyze momentum in a wide range of competitive settings, providing valuable insights for coaches and players across various disciplines. Additionally, the model should be tested on historical data from different tournaments and surfaces to assess its robustness and applicability. For example, by applying the model to matches played on clay courts, we can evaluate whether the observed patterns of momentum are consistent across different surfaces. This adaptability is crucial for ensuring that the model remains relevant and useful in diverse competitive environments.

In conclusion, developing a model to capture momentum in tennis matches requires a comprehensive and dynamic approach that integrates quantitative data with qualitative insights. The model must account for the serving advantage, psychological factors, and time-dependent variables, while also distinguishing between random fluctuations and genuine momentum shifts. By rigorously testing the model's assumptions and predictions, and by incorporating feedback from real-world applications, it can provide valuable insights into the role of momentum in tennis and other sports. This, in turn, can help coaches and players better understand and respond to the ebb and flow of competitive play, ultimately enhancing their performance and strategic decision-making. The model should also be designed with flexibility in mind, allowing it to be adapted to different contexts and providing a robust framework for analyzing momentum across a wide range of competitive settings.
"""

selected_models = """\
The most innovative, advanced, and suitable mathematical and machine learning models for modeling the problem of momentum in tennis matches include:
- **Logistic Regression** for point outcome predictions.
- **Random Forest** or **Support Vector Machines** for predicting momentum shifts.
- **ARIMA Models** and **Markov Chain Models** for analyzing momentum dynamics over time.
- **Time Series Analysis** for identifying trends in player performance.

These models not only capture the nuances of match dynamics but also provide a solid framework for coaches to understand and prepare for momentum shifts during matches. The insights derived from these models can significantly enhance match strategies and player preparation.
"""

modeling_solution = """\
To solve the momentum modeling problem in tennis, we can construct a comprehensive model that captures the flow of play, analyzes momentum shifts, and predicts future changes in the match. The mathematical approach will combine probabilistic, statistical, and machine learning techniques to capture both the player performance and momentum dynamics.

### **1. Model Components:**
The model will be broken down into different components that allow us to track performance and momentum. These components include:

- **State Transitions (Markov Chain)**
- **Performance Metrics (Player Statistics)**
- **Momentum Indicator**
- **Prediction of Momentum Shift**
  
Each of these components requires sophisticated mathematical formulas and data representations. Let's discuss each step in detail.

### **2. State Transition Modeling:**
We begin by using **Markov Chains** to model the transition between states during the match. This is ideal for modeling the progression of a tennis match, where the match state is influenced by previous outcomes but does not depend on the entire history (Markov property).

Let’s define the state of the match at a given point in time \( t \) as a tuple of the following:

\[
S(t) = \{ \text{score}_{\text{player 1}}, \text{score}_{\text{player 2}}, \text{serve}, \text{set}_1, \text{set}_2, \dots, \text{time}\}
\]

Here:
- **score** refers to the current game score of both players in terms of tennis scoring.
- **serve** indicates which player is serving.
- **set** indicates the current set scores for both players.
- **time** is the timestamp of the point in the match.

At each point, the system transitions to a new state based on the outcome of the point. The transition probabilities between these states can be modeled as a **transition matrix**, denoted by \( P \):

\[
P_{ij} = P(S(t+1) = s_j | S(t) = s_i)
\]

Where \( P_{ij} \) represents the probability of transitioning from state \( s_i \) to \( s_j \), given the match's historical performance. This will allow us to model the evolution of the match and identify significant shifts in momentum.

The transition matrix will be derived using the observed data in the dataset, particularly focusing on factors such as:
- Player performance metrics (e.g., aces, winners, unforced errors)
- Whether the player is serving (since servers have an advantage)
- The current score and game state
- Physical metrics like player movement or serve speed

### **3. Momentum Indicator Model:**

**Momentum** in tennis is typically thought of as a shift in player performance due to psychological or physical factors, often manifesting in runs of consecutive points won or sudden changes in strategy. We define the momentum \( M(t) \) as a function of multiple match variables:

\[
M(t) = f(P_{\text{player 1}}, P_{\text{player 2}}, \Delta \text{score}, S_{\text{serve}}, R_{\text{streak}}, T_{\text{tension}})
\]

Where:
- \( P_{\text{player}} \) represents a vector of player performance metrics at time \( t \) (e.g., first-serve percentage, aces, unforced errors, winners).
- \( \Delta \text{score} \) represents the relative score difference between players at time \( t \).
- \( S_{\text{serve}} \) is the indicator of the player who is serving.
- \( R_{\text{streak}} \) is the **run of consecutive points won** by a player (i.e., a player's success streak).
- \( T_{\text{tension}} \) is a measure of how tense the match is, such as a tie-break or break point situation.

A simple formula for momentum might involve calculating the ratio of successful points to total points, adjusted for the player's current performance:

\[
M(t) = \frac{\text{Points Won}_{\text{player 1}}}{\text{Points Played}} - \frac{\text{Points Won}_{\text{player 2}}}{\text{Points Played}} + \alpha \cdot R_{\text{streak}} + \beta \cdot T_{\text{tension}}
\]

Where:
- The first term represents the win rate of each player up to time \( t \).
- The second term adjusts momentum based on consecutive points won (with \( \alpha \) as a weight).
- The third term accounts for match tension, where \( T_{\text{tension}} \) could be quantified by analyzing critical moments (like break points) or the phase of the match.

### **4. Statistical Test for Randomness:**
To test the coach's hypothesis that "momentum" is just random fluctuations in play, we need to compare observed momentum with random fluctuations:

- Null Hypothesis \( H_0 \): There is no momentum effect (the swings are random).
- Alternative Hypothesis \( H_1 \): There is a non-random momentum effect (swings correlate with performance and match dynamics).

We use a **Randomization Test** to compare the actual momentum times series with a shuffled version of the same data (where player points are randomly reassigned). The test statistic for momentum could be the **standard deviation** of momentum:

\[
\text{std}(M_{\text{real}}) \quad \text{vs} \quad \text{std}(M_{\text{random}})
\]

If \( \text{std}(M_{\text{real}}) \) is significantly higher than \( \text{std}(M_{\text{random}}) \), we reject \( H_0 \) and conclude that momentum plays a role.

### **5. Predicting Momentum Shift (Machine Learning Approach):**

To predict future momentum shifts, we can leverage **Random Forests** or **Gradient Boosting Machines (GBM)**. The key features will include:
- Player statistics (e.g., first serve win percentage, winners, aces)
- Point streak information
- Current match state (game, set, or match score)
- Relative performance (player 1 vs player 2)
- Tension index (break points, tie-breaks, etc.)

We define the momentum shift prediction as a binary classification problem:

\[
\text{Momentum Shift} = 
\begin{cases} 
1 & \text{if momentum shifts in favor of player 1} \\
0 & \text{if momentum shifts in favor of player 2}
\end{cases}
\]

The model predicts the likelihood \( P(\text{Shift}) \), and we evaluate the model using standard classification metrics like **accuracy**, **precision**, and **recall**.

### **6. Mathematical Formulation for Momentum Prediction:**

Given the features \( \mathbf{X}_t \) at time \( t \), the output probability of a momentum shift can be modeled as:

\[
P(\text{Shift} = 1 | \mathbf{X}_t) = \sigma(\mathbf{w}^\top \mathbf{X}_t)
\]

Where:
- \( \sigma(\cdot) \) is the logistic sigmoid function.
- \( \mathbf{w} \) are the weights learned by the model.
- \( \mathbf{X}_t \) is the feature vector at time \( t \) (e.g., player performance, set/game score, etc.).

### **7. Model Evaluation and Generalization:**

To assess the performance of the momentum shift prediction model:
- Split the dataset into training and testing sets.
- Evaluate on multiple matches to assess generalizability.
- Fine-tune the model using cross-validation.
- Test the model’s performance across different tournaments, surfaces, and even for female matches.

---

### **8. Visualization of Momentum:**

The visualization could take the form of a **momentum curve**, where the x-axis represents time (or points played) and the y-axis represents the calculated momentum \( M(t) \). The curve will oscillate based on player performance and could be highlighted with vertical lines representing significant shifts in momentum.

#### Example:

\[
\text{Momentum Curve: } M(t) \text{ vs } t
\]

---

### **Final Remarks:**

This model provides a sophisticated framework for understanding momentum in tennis. By leveraging a combination of Markov Chains, machine learning techniques, and statistical testing, it offers a predictive tool for momentum shifts. Coaches can use this model to assess how momentum affects match outcomes and prepare players for psychological and strategic shifts during matches.
"""

modeling_solution = """\
To effectively model the dynamics of momentum in tennis matches, particularly in the context of the 2023 Wimbledon tournament, we must adopt a structured approach that integrates statistical analysis and machine learning techniques with a nuanced understanding of tennis gameplay. The model should begin with a clear definition of momentum as a quantifiable shift in player performance, characterized by changes in key performance indicators such as point-win probability, serve effectiveness, and error rates. To capture the serving advantage, the model should incorporate player-specific serving statistics, adjusting for the different likelihood of winning points based on whether a player is serving or returning. By using historical data, we can establish baseline probabilities for each player, which will serve as dynamic benchmarks throughout the match.

The core of the model will be a probabilistic framework that continuously updates the likelihood of a player gaining or losing momentum based on the unfolding match data. This can be achieved through a Bayesian approach, where prior probabilities are adjusted with new information as the match progresses. The model will incorporate variables such as serve speed, return depth, rally length, and player fatigue, each weighted according to its historical impact on momentum shifts. Additionally, psychological factors can be quantified by analyzing historical performance under pressure situations, such as break points or tiebreaks, allowing the model to estimate a player's mental resilience.

Machine learning techniques, such as decision trees or random forests, can be employed to identify patterns and interactions between these variables, providing insights into the critical factors that precede momentum shifts. By training these models on a comprehensive dataset of past matches, we can identify the most predictive indicators of momentum changes. The model should also include a temporal component, using time-series analysis to capture the evolving nature of momentum as the match progresses.

To assess the validity of the momentum concept against the claim of randomness, the model will implement statistical tests to compare observed runs and performance swings to those expected under a random-walk hypothesis. This involves simulating matches based on random point distributions and comparing the frequency and magnitude of momentum shifts to actual match data. If the model detects systematic deviations from randomness, it provides evidence for the existence of momentum.

For practical application, the model will generate real-time predictions and alerts for coaches, indicating potential shifts in momentum and suggesting tactical adjustments. This can be facilitated through a user-friendly interface that visualizes momentum trends and key performance indicators. The model should be tested and validated across multiple matches, including different rounds and surfaces, to ensure robustness and generalizability. By incorporating cross-validation techniques, we can fine-tune the model parameters and improve predictive accuracy.

Furthermore, the model's adaptability to different sports or contexts involves recalibrating for sport-specific dynamics, such as the impact of team interactions in basketball or the influence of equipment in table tennis. By focusing on universal aspects of momentum, such as psychological resilience and performance consistency, the model can be extended to other competitive environments. Ultimately, this refined modeling approach not only addresses the intricacies of momentum in tennis but also provides actionable insights for improving player performance and strategic decision-making in various sports settings.
"""

task_descriptions =  ['The first subtask involves establishing a robust probabilistic framework to quantify momentum in tennis matches, focusing on defining momentum as a measurable shift in player performance through variations in key performance indicators such as point-win probability, serve effectiveness, and error rates. The primary goal is to accurately capture the dynamic nature of momentum by integrating player-specific serving statistics, which account for the different likelihoods of point outcomes depending on whether a player is serving or returning. To achieve this, the subtask requires the use of historical match data to create baseline probabilities for each player, serving as dynamic benchmarks that adjust with ongoing match events. The methodology centers on a Bayesian approach, where prior probabilities are continuously updated with new data as the match progresses, reflecting real-time dynamics and shifts in momentum. This involves collecting and analyzing data inputs such as serve speed, return effectiveness, and other relevant match statistics to inform the probabilistic model. Tools and techniques such as Bayesian inference and statistical analysis are crucial for updating probabilities and capturing the nuances of momentum. The outcome of this subtask is a foundational model capable of reflecting the ebb and flow of momentum in real-time, providing a basis for further analysis and integration with additional model components in subsequent subtasks.', "The second subtask aims to enhance the model's ability to discern momentum shifts in tennis matches by leveraging machine learning techniques to identify patterns and interactions among influential variables. This involves the application of models such as decision trees, random forests, or gradient boosting to analyze a rich dataset of historical match data, focusing on uncovering the most predictive indicators of momentum changes. Key variables of interest include serve speed, return depth, rally length, player fatigue, and psychological resilience, with each factor weighted based on its historical impact on momentum. The machine learning model will be trained on labeled datasets where momentum shifts are identified, allowing it to learn complex relationships and interactions between variables. The training process involves feature engineering to extract meaningful insights from raw data, including the creation of derived features that capture temporal dynamics, such as moving averages or momentum scores calculated over a series of points or games. Cross-validation techniques will be employed to ensure the model's robustness and to prevent overfitting, thereby improving its predictive accuracy across different match scenarios. By integrating these machine learning models, the subtask seeks to provide deeper insights into the conditions that precede momentum shifts, ultimately enhancing the model's capacity to predict and quantify momentum changes as they occur during a match.", 'The third subtask focuses on rigorously testing the hypothesis that momentum in tennis matches is merely a random phenomenon. The goal is to determine whether observed performance swings are consistent with random variations or if they indicate genuine momentum shifts. This involves conducting statistical tests that compare the actual match data against a null hypothesis of randomness, often modeled as a random-walk process. The methodology includes simulating matches using randomly distributed point outcomes to create a baseline of expected performance swings under purely stochastic conditions. By analyzing the frequency, duration, and magnitude of momentum shifts in real matches against this random baseline, we can assess whether the observed patterns deviate significantly from what would be expected by chance. The key data inputs for this subtask are point-by-point outcomes from the dataset, which will be used to construct both the observed and simulated match timelines. Statistical techniques such as Monte Carlo simulations, chi-squared tests, or permutation tests may be employed to quantify the degree of deviation from randomness. The outcome of this subtask is to provide statistical evidence either supporting or refuting the existence of momentum, thereby addressing the skepticism about its role in tennis. This evidence is crucial for validating the broader model and confirming whether momentum should be considered a significant factor in match analysis and strategy development.', "Subtask 4 focuses on the practical application and generalizability of the momentum model by developing a real-time prediction system designed to assist coaches in making strategic decisions during a match. The primary goal is to create a system that can predict potential shifts in momentum, providing timely alerts that suggest tactical adjustments to optimize player performance. To achieve this, the subtask involves designing a user-friendly interface that visualizes momentum trends and key performance indicators in an accessible format, allowing coaches to quickly interpret the data. The methodology includes integrating the model with live match data feeds, enabling continuous updates and real-time analysis. The system will leverage machine learning algorithms, such as real-time decision trees or neural networks, to process incoming data and generate predictive insights. Key data inputs for this task include real-time match statistics like serve speed, rally length, player fatigue levels, and psychological resilience metrics, which are processed to detect patterns indicative of momentum shifts. The model's robustness will be tested across a diverse range of matches, including different rounds, surfaces, and player styles, to ensure its adaptability and reliability. Additionally, the subtask emphasizes recalibrating the model for application in other sports contexts, such as basketball or table tennis, by focusing on universal momentum aspects like psychological resilience and consistent performance. This adaptability is crucial for extending the model's utility beyond tennis, providing valuable insights to enhance strategic decision-making and player preparation across various sports settings."]