Chapter 13 Exercises: Win Shares and Wins Above Replacement

Section A: Conceptual Understanding (Exercises 1-8)

Exercise 1: Foundations of Win Shares

Explain in your own words why Win Shares starts with team wins rather than aggregating individual statistics. What problem does this approach solve, and what trade-offs does it create?

Exercise 2: Marginal vs. Absolute Value

Consider two players: - Player A averages 18 PPG on 45% shooting (league average efficiency) - Player B averages 14 PPG on 60% true shooting (well above average)

Which player likely has higher Offensive Win Shares relative to their minutes played? Explain the marginal value concept that drives this comparison.

Exercise 3: Replacement Level Philosophy

The Win Shares system uses 0.92 for offensive replacement level and 1.08 for defensive replacement level.

a) What do these numbers represent in practical terms? b) Why might the defensive replacement level be set higher (worse) than the offensive replacement level? c) How would Win Shares change across the league if replacement level was set at 0.85 offense and 1.15 defense instead?

Exercise 4: Defensive Win Shares Limitations

Identify and explain three specific scenarios where Defensive Win Shares might significantly misvalue a player's true defensive contribution. For each scenario, describe what the statistic would show versus the player's actual impact.

Exercise 5: Context Dependence

A player is traded mid-season from a slow-paced, defensive-oriented team to a fast-paced, offense-heavy team. Assuming his individual performance remains constant:

a) How might his raw Win Shares change? b) How might his WS/48 change? c) What does this tell us about the context-dependence of Win Shares?

Exercise 6: Historical Comparisons

Wilt Chamberlain averaged 50.4 PPG in 1961-62, while Michael Jordan's career high was 37.1 PPG in 1986-87. Yet their peak Win Shares seasons are comparable. Explain the factors in the Win Shares calculation that account for this apparent paradox.

Exercise 7: Floor Spacing and Win Shares

Modern basketball values three-point shooting for spacing purposes. A player who shoots 38% from three on 5 attempts per game may provide significant offensive value through gravity and spacing that doesn't show up in their counting stats.

How well does Win Shares capture this type of value? What would need to change in the methodology to better account for spacing?

Exercise 8: The Sum Property

Win Shares is designed so that a team's total Win Shares equals their actual wins. Discuss the advantages and disadvantages of this design choice compared to metrics that don't have this constraint.

Section B: Calculation Exercises (Exercises 9-18)

Exercise 9: Basic OWS Calculation

Given the following player statistics for a single season: - Points: 1,500 - Field Goals Made: 500 - Field Goal Attempts: 1,000 - Free Throws Made: 300 - Free Throw Attempts: 400 - Assists: 250 - Offensive Rebounds: 80 - Turnovers: 150

And the following league/team context: - League PPP: 1.08 - League PPG: 108 - Team ORB weight: 1.15 - Team assists on FG: 58%

Calculate the player's approximate Offensive Win Shares using the simplified formula:

$$\text{OWS} = \frac{\text{Points Produced} - 0.92 \times \text{League PPP} \times \text{Possessions Used}}{0.32 \times \text{League PPG}}$$

Show all intermediate calculations.

Exercise 10: Possessions Used

Calculate the possessions used for each of the following players:

Use the formula: Possessions = FGA + 0.44 × FTA + TOV - ORB × 0.034

Exercise 11: Individual Offensive Rating

A player has the following statistics: - Points Produced: 1,800 - Possessions Used: 1,400

Calculate their Individual Offensive Rating. If the league average ORtg is 110, how does this player compare?

Exercise 12: Marginal Offense

Using the data from Exercise 11 and a league PPP of 1.10, calculate:

a) The player's marginal offense b) Their Offensive Win Shares (assume league PPG = 110) c) If they played 2,500 minutes, their OWS/48

Exercise 13: Defensive Win Shares Components

A player has the following defensive statistics: - Minutes: 2,400 - Steals: 120 - Blocks: 80 - Defensive Rebounds: 350 - Personal Fouls: 180

Team statistics: - Team Minutes: 19,680 - Team Possessions: 8,000 - Team DRtg: 105 - Opponent FGA: 7,200 - Opponent FGM: 3,200 - Team Blocks: 400

Using FMwt = 0.45 (given), calculate: a) Stops1 (individual stops) b) Approximate total stops c) Stop percentage

Exercise 14: Complete DWS Calculation

Using the results from Exercise 13 and: - League PPP: 1.10 - League PPG: 110

Calculate the player's Defensive Win Shares.

Exercise 15: Total Win Shares Comparison

Two players have the following statistics:

a) Calculate total Win Shares for each player b) Calculate WS/48 for each player c) Which player is more valuable? Under what circumstances might your answer change?

Exercise 16: Marginal Points per Win

The "Marginal Points per Win" formula is: 0.32 × League PPG

a) Calculate this value for leagues with PPG of 100, 110, and 120 b) What is the intuition behind the 0.32 coefficient? c) A team outscores opponents by 500 points over a season (110 PPG league). How many wins does this predict above .500?

Exercise 17: WAR Calculation

A player has 12.0 Win Shares in 2,600 minutes played. Assuming replacement level WS/48 is 0.000:

a) Calculate their WAR b) If replacement level WS/48 was 0.025 instead, what would their WAR be? c) Discuss how sensitive the WAR calculation is to replacement level assumptions.

Exercise 18: Contract Value Analysis

Given: - Player's Win Shares: 8.5 - Player's Salary: $18,000,000 - Market rate: $3,200,000 per Win Share

Calculate: a) The player's market value b) Their surplus value c) At what Win Shares level would this contract represent neutral value?

Section C: Data Analysis Exercises (Exercises 19-28)

Exercise 19: Season Analysis

Using the following simplified team data:

The team won 50 games. Estimate how the 50 Win Shares should be distributed among these five players (ignoring bench players for simplicity). Explain your reasoning.

Exercise 20: Era Comparison

Consider these two MVP seasons:

1987-88 Michael Jordan: - 35.0 PPG, 5.5 RPG, 5.9 APG - 53.5% FG, 84.1% FT - 82 games, 40.4 MPG - League Pace: ~100 possessions/game - League PPG: ~108

2015-16 Stephen Curry: - 30.1 PPG, 5.4 RPG, 6.7 APG - 50.4% FG, 90.8% FT, 45.4% 3PT - 79 games, 34.2 MPG - League Pace: ~96 possessions/game - League PPG: ~106

Without doing exact calculations, analyze which season likely had higher: a) Total Win Shares b) WS/48 c) Offensive Win Shares as a percentage of total

Justify your reasoning.

Exercise 21: Efficiency vs. Volume

Create a framework for analyzing the trade-off between efficiency (WS/48) and volume (total WS).

A team needs to decide between: - Player A: 0.180 WS/48, can play 36 MPG - Player B: 0.220 WS/48, can only play 24 MPG

Over an 82-game season, which player provides more Win Shares? At what minutes threshold would Player B become more valuable?

Exercise 22: Defensive Specialist Valuation

A defensive specialist has these statistics: - 6.5 PPG, 4.0 RPG, 1.5 APG - 1.8 SPG, 0.5 BPG - 28 MPG

Using the limitation that DWS is based largely on these box score stats, estimate what this player's Win Shares profile might look like. Then discuss what defensive value they might provide that Win Shares cannot capture.

Exercise 23: Regression to the Mean

A player had the following WS/48 over three seasons: - Year 1: 0.180 - Year 2: 0.095 - Year 3: 0.210

a) Calculate the mean and standard deviation b) Project Year 4 using 50% regression to a prior of 0.120 c) What factors might explain the volatility in this player's WS/48?

Exercise 24: Team Composition

A team has the following Win Shares distribution:

a) Calculate surplus value for each category b) Identify which roster segment provides the most surplus value c) Recommend a general strategy for improving this team's efficiency

Exercise 25: Draft Pick Value

Historical data suggests the following expected Win Shares production over a 4-year rookie contract:

A team is offered pick #7 for pick #15 and pick #22.

a) Calculate the expected WS from each option b) Calculate the variance of each option c) Under what risk preferences should the team accept this trade?

Exercise 26: Aging Curve Analysis

Model a player's projected Win Shares using this aging curve:

A 24-year-old produced 10.0 Win Shares last season. Project his Win Shares for ages 25-30, and calculate total projected career Win Shares over this period.

Exercise 27: Trade Deadline Decision

Your team is 8th seed and considering trading for a rental player who will be a free agent after the season.

Target player statistics: - Current WS: 6.0 (through 50 games) - WS/48: 0.150 - Expected minutes ROS: 1,200

Cost: A future first-round pick (expected value: 6.0 WS over 4 years)

a) Project the target's Win Shares for the remainder of the season b) Compare the immediate and long-term value c) What playoff expectations would justify this trade?

Exercise 28: Multi-Year Contract Analysis

A player is being offered a 4-year, $120M contract. His current statistics: - Age: 27 - Win Shares last season: 11.0 - WS/48: 0.165 - Minutes per game: 34

Using the aging curve from Exercise 26 and the market rate of $3.4M per Win Share:

a) Project Win Shares for each year of the contract b) Calculate expected market value for each year c) Calculate total surplus value (positive or negative) of the contract d) Would you recommend signing this contract?

Section D: Advanced Problems (Exercises 29-35)

Exercise 29: Building a Win Shares Model

Design a simplified Win Shares model that could be calculated from basic box score statistics only (PTS, REB, AST, STL, BLK, TOV, FG%, FT%, 3P%, MP).

a) Write out the formula with coefficients you would assign to each statistic b) Explain the reasoning behind each coefficient c) What key elements of the full Win Shares calculation are you sacrificing?

Exercise 30: Uncertainty Quantification

Win Shares provides a point estimate but no confidence interval. Propose a methodology for estimating the uncertainty in a player's Win Shares.

Consider: a) Sample size (minutes played) b) Game-to-game variance c) Defensive measurement uncertainty d) Team context effects

Write a formula or algorithm for calculating a 95% confidence interval for Win Shares.

Exercise 31: Win Shares vs. Plus/Minus

Compare Win Shares to Adjusted Plus/Minus (APM) on the following dimensions:

For each criterion, explain which metric performs better and why.

Exercise 32: Optimal Roster Construction

You have a $130M salary cap and need to build a team that maximizes expected wins.

Available player types:

Constraints: - 15-player roster - At least 5 players earning $10M+

Find the optimal roster composition that maximizes total Win Shares while staying under the cap.

Exercise 33: Win Shares Decomposition

A player improved from 6.0 to 10.0 Win Shares between seasons. Their statistics changed as follows:

Decompose the 4.0 WS improvement into contributions from: a) Increased minutes b) Improved scoring efficiency c) Improved playmaking d) Improved defensive statistics

Exercise 34: Cross-Era Adjustment

Develop a methodology for comparing Win Shares across eras that accounts for: - Different pace of play - Different schedule lengths - Different three-point line rules - Different talent concentration (fewer teams in earlier eras)

Apply your adjustment to compare: - Wilt Chamberlain's best season (1963-64) - Michael Jordan's best season (1987-88) - LeBron James's best season (2012-13)

Exercise 35: Machine Learning Enhancement

Propose a machine learning approach to improve Win Shares predictions. Your proposal should include:

a) What additional features would you include beyond traditional box score stats? b) What would be your training target (actual wins? Plus/minus? Some combination?) c) What model architecture would you use and why? d) How would you handle the interpretability vs. accuracy trade-off? e) How would you validate your model?

Section E: Programming Exercises (Exercises 36-40)

Exercise 36: Win Shares Calculator

Implement a complete Win Shares calculator in Python that: - Takes player and team statistics as input - Calculates OWS and DWS separately - Returns total WS and WS/48 - Handles edge cases (0 minutes, missing data)

Test your implementation against known Win Shares values from Basketball-Reference.

Exercise 37: Historical Database Query

Write SQL queries to answer the following questions from a basketball statistics database:

a) Find the top 10 seasons by Win Shares since 1980 b) Find players whose WS/48 exceeded 0.250 for at least 3 seasons c) Calculate the correlation between Win Shares and MVP voting shares d) Find the most "undervalued" seasons (high WS but no All-Star selection)

Exercise 38: Visualization Dashboard

Create a Python visualization that includes: - Scatter plot of OWS vs DWS for all players in a season - Histogram of WS/48 distribution - Time series of a player's career Win Shares - Comparison bar chart of selected players

Use matplotlib or a similar library, with appropriate labels and legends.

Exercise 39: Monte Carlo Simulation

Build a Monte Carlo simulation to estimate the uncertainty in Win Shares:

a) Assume each component statistic has a standard error of 10% of its value b) Run 10,000 simulations varying each input c) Calculate the 5th and 95th percentile of Win Shares d) Visualize the distribution of outcomes

Exercise 40: API Integration

Write a Python script that: - Fetches current season statistics from a basketball statistics API - Calculates Win Shares for all players - Compares your calculations to published Win Shares - Identifies discrepancies and possible sources of error

Document your methodology and any assumptions made.

Answer Key for Selected Exercises

Exercise 10: Possessions Used

Player A: $$\text{Poss} = 1200 + 0.44 \times 400 + 180 - 100 \times 0.034 = 1200 + 176 + 180 - 3.4 = 1552.6$$

Player B: $$\text{Poss} = 800 + 0.44 \times 600 + 120 - 40 \times 0.034 = 800 + 264 + 120 - 1.36 = 1182.64$$

Player C: $$\text{Poss} = 1500 + 0.44 \times 200 + 250 - 150 \times 0.034 = 1500 + 88 + 250 - 5.1 = 1832.9$$

Exercise 11: Individual Offensive Rating

$$\text{ORtg} = 100 \times \frac{1800}{1400} = 128.6$$

This player is producing at 128.6 points per 100 possessions, compared to league average of 110. They are +18.6 points above average per 100 possessions.

Exercise 12: Marginal Offense

a) Marginal Offense: $$\text{Marg Off} = 1800 - 0.92 \times 1.10 \times 1400 = 1800 - 1416.08 = 383.92$$

b) Offensive Win Shares: $$\text{OWS} = \frac{383.92}{0.32 \times 110} = \frac{383.92}{35.2} = 10.9$$

c) OWS/48: $$\text{OWS/48} = \frac{10.9}{2500} \times 48 = 0.209$$

Exercise 16: Marginal Points per Win

a) Calculations: - 100 PPG: 0.32 × 100 = 32.0 marginal points per win - 110 PPG: 0.32 × 110 = 35.2 marginal points per win - 120 PPG: 0.32 × 120 = 38.4 marginal points per win

b) Intuition: The 0.32 coefficient is derived from the Pythagorean expectation for basketball. It represents the approximate conversion rate between point differential and wins, scaled to single games.

c) Expected wins above .500: $$\text{Wins above .500} = \frac{500}{35.2} = 14.2 \text{ wins}$$

So a team that outscores opponents by 500 points should expect approximately 41 + 14.2 = 55 wins.

Exercise 17: WAR Calculation

a) With 0.000 replacement level: $$\text{WAR} = 12.0 - \frac{2600}{48} \times 0.000 = 12.0 - 0 = 12.0$$

b) With 0.025 replacement level: $$\text{WAR} = 12.0 - \frac{2600}{48} \times 0.025 = 12.0 - 54.17 \times 0.025 = 12.0 - 1.35 = 10.65$$

c) Sensitivity: WAR is highly sensitive to replacement level assumptions. A change from 0.000 to 0.025 in replacement WS/48 reduced this player's WAR by 1.35 wins (11.25%). For a player with lower total Win Shares, the percentage impact would be even larger.

Exercise 18: Contract Value Analysis

a) Market Value: $$\text{Market Value} = 8.5 \times \$3,200,000 = \$27,200,000$$

b) Surplus Value: $$\text{Surplus} = \$27,200,000 - \$18,000,000 = +\$9,200,000$$

c) Neutral Value Win Shares: $$\text{Neutral WS} = \frac{\$18,000,000}{\$3,200,000} = 5.625 \text{ WS}$$

At 5.625 Win Shares, the contract would represent neutral value.