Chapter 14 Exercises: Expected Possession Value (EPV)

Section A: Conceptual Understanding (Exercises 1-8)

Exercise 1: EPV Fundamentals

A possession begins with the point guard bringing the ball up court. The initial EPV is 1.02 points. Answer the following:

a) What does an EPV of 1.02 mean in practical terms?

b) If the offense runs a play and the EPV rises to 1.15 after 10 seconds, what can we infer about the quality of the offensive execution?

c) If the possession ends with a missed three-pointer, does that mean the offensive execution was poor? Explain using EPV concepts.

d) How might the EPV change if the shot clock violation were imminent (2 seconds remaining)?

Exercise 2: EPV Curve Interpretation

Consider the following EPV values recorded during a single possession:

a) Calculate the value added by each action (excluding the shot outcome).

b) Which action created the most value?

c) What does the EPV of 0.28 after the miss represent?

d) If the shooter had a 38% three-point percentage on this type of shot, was 1.31 a reasonable EPV before the shot? Show your calculation.

Exercise 3: Markov Property

The Markov property assumes that the future depends only on the current state, not on how we arrived at that state.

a) Give two examples of basketball situations where the Markov property might be violated.

b) For each example, explain what additional state information would be needed to restore the Markov property.

c) Why might strict adherence to the Markov property be impractical in a real EPV model?

Exercise 4: State Space Dimensionality

A basketball tracking system captures positions at 25 Hz for all 10 players and the ball.

a) How many spatial dimensions are in the raw state space (positions only)?

b) If we add velocities for all tracked objects, how many continuous dimensions do we have?

c) List five discrete state variables that should also be included.

d) Explain why dimensionality reduction is necessary for practical EPV computation.

Exercise 5: Action Value Comparison

A player catches the ball in the corner with the following action values:

• $Q(s, \text{shoot}) = 1.14$
• $Q(s, \text{pass to post}) = 1.08$
• $Q(s, \text{drive baseline}) = 0.96$
• $Q(s, \text{pass to point guard}) = 1.02$

a) What is the optimal action?

b) If the player passes to the point guard, what is their Decision Value Added (DVA)?

c) If the player chooses to drive and ultimately scores on a layup (2 points), was it a good decision? Explain the difference between outcome and decision quality.

Exercise 6: Shot Value Calculation

A player takes a shot with the following parameters: - Location: 22 feet from basket (three-pointer) - Player's three-point percentage at this distance: 36% - Offensive rebound probability if missed: 22% - Expected value of possession after offensive rebound: 1.05 points

Calculate the expected value of taking this shot.

Exercise 7: Pass Value Analysis

A point guard is considering a pass to a teammate in the post. The relevant parameters are: - Pass completion probability: 88% - Expected EPV after successful pass: 1.18 - Expected value if pass is intercepted (turnover leading to transition): -0.95

Calculate the expected value of attempting this pass.

Exercise 8: Spatial Value Concepts

Explain, using EPV concepts, why:

a) Corner three-pointers are considered high-value shots despite being farther from the basket than some two-pointers.

b) There exists a "dead zone" of low-value shots inside but near the three-point line.

c) The paint area has the highest expected shot value but isn't the right location for every shot attempt.

Section B: Mathematical Foundations (Exercises 9-16)

Exercise 9: Bellman Equation Application

Consider a simplified basketball state space with three states: {Open Shot, Contested Shot, No Shot Available}. The action space is {Shoot, Pass, Dribble}. Transition probabilities and rewards are given below.

From "Open Shot" state: - Shoot: 50% chance of 2 points, 50% chance of 0 points (possession ends) - Pass: 80% to "Open Shot", 20% to "Contested Shot" - Dribble: 60% stay "Open Shot", 40% to "Contested Shot"

From "Contested Shot" state: - Shoot: 30% chance of 2 points, 70% chance of 0 points - Pass: 40% to "Open Shot", 50% to "Contested Shot", 10% to "No Shot" - Dribble: 20% to "Open Shot", 60% to "Contested Shot", 20% to "No Shot"

From "No Shot Available": - Only option: Force difficult shot with 20% success rate

a) Write the Bellman equations for this system.

b) Solve for the EPV of each state using value iteration (show at least 3 iterations).

c) What is the optimal policy (action to take in each state)?

Exercise 10: Shot Probability Model

Given a logistic regression model for shot success probability:

$$\log \frac{p}{1-p} = 2.1 - 0.08 \cdot d_{\text{basket}} - 0.15 \cdot d_{\text{defender}} + 0.3 \cdot I_{\text{catch\&shoot}}$$

Where $d_{\text{basket}}$ is distance to basket in feet, $d_{\text{defender}}$ is distance to nearest defender in feet, and $I_{\text{catch\&shoot}}$ is 1 for catch-and-shoot opportunities.

a) Calculate the probability of making a catch-and-shoot three-pointer (24 feet) with the nearest defender 5 feet away.

b) Calculate the probability for the same shot but off the dribble.

c) How close must the defender be to reduce the catch-and-shoot probability to 30%?

Exercise 11: Multiresolution Modeling

Explain the difference between macrotransitions and microtransitions in the Cervone et al. framework.

a) Classify each of the following as macro or micro: - Player cuts from corner to wing - Ball handler drives past defender - Pass completed to open shooter - Defensive rotation toward paint - Shot attempt - Screen set by center

b) Why is this multiresolution approach advantageous compared to treating everything as continuous or everything as discrete?

Exercise 12: EPV Decomposition

The EPV decomposition formula states:

$$EPV(\mathcal{S}_t) = \sum_{a \in \mathcal{A}} P(a \mid \mathcal{S}_t) \cdot V(a, \mathcal{S}_t)$$

A point guard is at the top of the key. Based on historical data: - P(shoot) = 0.15, V(shoot) = 0.92 - P(pass right wing) = 0.25, V(pass right) = 1.08 - P(pass left wing) = 0.20, V(pass left) = 1.05 - P(drive right) = 0.18, V(drive right) = 1.12 - P(drive left) = 0.12, V(drive left) = 1.04 - P(dribble in place) = 0.10, V(dribble) = 0.98

a) Calculate the current EPV.

b) If the player always chose the optimal action, what would the EPV be?

c) What is the "decision inefficiency" (difference between actual and optimal EPV)?

Exercise 13: Offensive Rebound Value

A team has the following offensive rebound characteristics: - Offensive rebound rate: 28% - Average EPV after offensive rebound: 1.08 points - Average EPV after defensive rebound (transition opportunity for opponent): -1.02 points

For a shot attempt with make probability $p$:

a) Write the formula for expected value of the shot attempt (2-point shot).

b) Calculate the expected value for a shot with 45% make probability.

c) What make probability makes this shot equivalent in value to a league-average possession (1.0 points)?

Exercise 14: Value Gradient Analysis

The spatial value function around the basket can be approximated as:

$$V(r, \theta) = 2.0 - 0.06r + 0.002r^2 - 0.1|\theta - 90|/90$$

Where $r$ is distance from basket (feet), $\theta$ is angle (degrees, 90 = straight on, 0/180 = baseline).

a) Calculate $\frac{\partial V}{\partial r}$ at $r = 5$ feet. What does the sign tell you?

b) Calculate the value at (r=8, $\theta$=90) and (r=8, $\theta$=45).

c) Write an expression for the gradient $\nabla V$ in polar coordinates.

Exercise 15: Continuous Time EPV

In continuous time, EPV satisfies:

$$\frac{dEPV}{dt} = \lambda_{\text{shot}} \cdot (V_{\text{shot}} - EPV) + \lambda_{\text{pass}} \cdot (V_{\text{pass}} - EPV) + \lambda_{\text{TO}} \cdot (0 - EPV)$$

Where $\lambda$ terms are event intensities (events per second).

If $\lambda_{\text{shot}} = 0.2$, $\lambda_{\text{pass}} = 0.5$, $\lambda_{\text{TO}} = 0.05$, $V_{\text{shot}} = 1.05$, $V_{\text{pass}} = 1.02$:

a) Find the steady-state EPV (where $\frac{dEPV}{dt} = 0$).

b) If current EPV is 0.95, is EPV increasing or decreasing?

c) How does this model relate to the shot clock constraint?

Exercise 16: Decision Value Added Aggregation

A player has the following decision log over 100 possessions:

a) Calculate the average DVA for each situation type.

b) Calculate the overall weighted average DVA.

c) In which situation type does this player have the most room for improvement?

d) If the player improved their pick-and-roll decisions to optimal, how much would their total value added increase per 100 possessions?

Section C: Applied Analysis (Exercises 17-25)

Exercise 17: Play Comparison

Two offensive plays are run 50 times each with the following results:

Play A (Horns Pick-and-Roll): - Average starting EPV: 1.00 - Average EPV at first pass: 1.06 - Average EPV at shot decision: 1.14 - Points per possession: 1.08

Play B (Motion Offense): - Average starting EPV: 1.00 - Average EPV at first pass: 1.03 - Average EPV at shot decision: 1.09 - Points per possession: 1.12

a) Which play generates more expected value at the shot decision point?

b) Which play has better execution (actual points vs. expected at shot decision)?

c) Calculate the "play design value" and "execution value" for each play.

d) Which play would you recommend running more, and why?

Exercise 18: Defender Impact

You have data on two defenders:

Defender A: In 200 possessions guarded, the opponent team's average EPV was 0.96.

Defender B: In 180 possessions guarded, the opponent team's average EPV was 1.04.

The league average EPV is 1.02.

a) Calculate each defender's "EPV Suppression" relative to league average.

b) If both defenders play 1,500 possessions per season, how many points does each save (or cost) compared to league average?

c) What additional context would you want before concluding that Defender A is clearly better?

Exercise 19: Shot Clock Analysis

Analyze how EPV changes with shot clock time. Given data:

a) At what shot clock range is EPV maximized?

b) Why might EPV be slightly lower in the 20-24 second range than 15-19 seconds?

c) Calculate the expected points for a team that takes all its shots in the 0-4 second range vs. a team that takes all shots in the 15-19 second range.

d) What does this analysis suggest about optimal shot selection timing?

Exercise 20: Player Comparison Using EPV Metrics

Two point guards have the following EPV statistics:

Player X: - Touches per game: 85 - Average EPV when receiving ball: 1.01 - Average EPV when releasing ball (pass/shot): 1.08 - Turnovers per game: 2.8

Player Y: - Touches per game: 72 - Average EPV when receiving ball: 1.02 - Average EPV when releasing ball: 1.06 - Turnovers per game: 1.9

a) Calculate EPV added per touch for each player.

b) Calculate total EPV added per game for each player.

c) Adjust for turnovers (assume each turnover costs 1.0 EPV).

d) Which player provides more value, and what are the tradeoffs?

Exercise 21: Passing Network Value

A team's passing data shows:

a) Calculate the value of each passing connection (EPV change $\times$ frequency).

b) Which passing connection creates the most total value per game?

c) Which passing connection has the highest per-pass value?

d) What does the negative value on SG-to-PG passes suggest?

Exercise 22: Drive Value Analysis

A player's driving statistics:

Assume: Made layups = 2 points, Foul = 1.6 expected points (free throws), and non-kick-out/non-foul drives that miss = 0.2 EPV (offensive rebound chance).

a) Calculate the expected value of a left-hand drive.

b) Calculate the expected value of a right-hand drive.

c) Should this player drive more often to the left or right?

d) What might explain the difference in kick-out rates?

Exercise 23: Defensive Scheme Evaluation

A team experiments with two pick-and-roll defensive schemes:

Drop Coverage: - Possessions tested: 150 - Ball handler shot frequency: 45% - Ball handler shot value: 0.88 - Roll man shot frequency: 20% - Roll man shot value: 1.08 - Other shot frequency: 35% - Other shot value: 1.02

Switch Coverage: - Possessions tested: 140 - Ball handler shot frequency: 52% - Ball handler shot value: 0.94 - Roll man shot frequency: 8% - Roll man shot value: 1.25 - Other shot frequency: 40% - Other shot value: 1.00

a) Calculate the expected points allowed under each scheme.

b) Which scheme is more effective overall?

c) Against what type of offensive player would you switch schemes?

Exercise 24: Transition EPV

Compare transition vs. half-court offense:

Transition (first 8 seconds): - Frequency: 18% of possessions - Average EPV at start: 1.18 - Average points scored: 1.15

Half-court (after 8 seconds): - Frequency: 82% of possessions - Average EPV at start: 0.98 - Average points scored: 1.00

a) Calculate the overall expected points per possession for this team.

b) How many additional points per game (100 possessions) would this team score if they increased transition frequency to 25% while maintaining the same efficiency?

c) What is the EPV "transition premium"?

Exercise 25: End-of-Game Decision Analysis

With 30 seconds left and down by 2 points, a team has the following options:

Option A: Quick shot (2-pointer) - Time to execute: 8 seconds - Success probability: 48% - If successful: Overtime with 50% win probability - If unsuccessful: Defensive rebound gives opponent ball with 22 seconds

Option B: Run clock and shoot three - Time to execute: 25 seconds - Success probability: 35% - If successful: Win - If unsuccessful: Offensive rebound 20% of time with 5 seconds left

Option C: Run clock and shoot two - Time to execute: 25 seconds - Success probability: 52% - If successful: Overtime with 50% win probability - If unsuccessful: Game over (no time for offensive rebound)

For simplicity, assume the opponent scores 50% of the time when they get the ball with significant time remaining.

a) Calculate the win probability for Option A.

b) Calculate the win probability for Option B.

c) Calculate the win probability for Option C.

d) Which option maximizes win probability? Discuss any simplifying assumptions.

Section D: Advanced Problems (Exercises 26-32)

Exercise 26: Multi-Agent EPV

Standard EPV treats offense as a single agent. Consider extending to value off-ball offensive players.

a) Define what "off-ball EPV contribution" might mean for a player setting a screen.

b) Propose a method to allocate EPV changes among the ball handler, screener, and shooter in a pick-and-pop play.

c) What data would be required to implement this allocation?

Exercise 27: Counterfactual Analysis Design

Design a counterfactual analysis to answer: "How much value does Player X's three-point shooting threat create for teammates' driving opportunities?"

a) Describe the counterfactual scenario you would construct.

b) What data would you need?

c) What confounding factors might bias your estimate?

d) Propose a method to control for at least one confounder.

Exercise 28: EPV Uncertainty Quantification

Point estimates of EPV ignore uncertainty.

a) List three sources of uncertainty in EPV estimates.

b) Explain how you would use bootstrap methods to construct confidence intervals for EPV.

c) A player has an estimated EPV contribution of +0.05 per touch with a 95% CI of [-0.02, 0.12]. How should a team interpret this for contract decisions?

Exercise 29: Model Calibration

You've built an EPV model and want to assess its calibration.

a) Describe a procedure to test whether your EPV estimates are well-calibrated.

b) You find that possessions with EPV predictions between 1.0 and 1.1 average 1.05 actual points, while possessions with EPV between 1.1 and 1.2 average 1.18 points. What does this suggest about your model?

c) Propose a recalibration procedure.

Exercise 30: Feature Importance Analysis

Your EPV model uses 50 features. Describe methods to assess which features are most important for predictions.

a) Explain permutation importance and how you would apply it.

b) What is the difference between feature importance for prediction accuracy vs. feature importance for decision-making?

c) A feature has high predictive importance but changes rarely during possessions. Is it useful for real-time decision support? Explain.

Exercise 31: Temporal Discounting

Standard EPV uses no temporal discounting ($\gamma = 1$). Consider cases where discounting might be appropriate.

a) Give a basketball scenario where $\gamma < 1$ makes sense.

b) If $\gamma = 0.99$ per second, how would EPV of a play requiring 10 seconds to develop compare to an immediate shot of equal terminal value?

c) How might late-game situations call for different discounting than early-game?

Exercise 32: Cross-League Transfer

You've built an EPV model on NBA data. Discuss applying it to:

a) NCAA men's basketball (different three-point line, shot clock, court size)

b) WNBA (same rules as NBA, different player capabilities)

c) EuroLeague (different rules, playing style)

For each, identify: - Which model components transfer directly - Which require recalibration - Which require fundamental reconstruction

Section E: Computation and Implementation (Exercises 33-40)

Exercise 33: State Discretization

Design a state discretization scheme for EPV computation.

a) Divide the half-court into regions suitable for EPV modeling. Justify your choices.

b) How many discrete states result from your court regions combined with 3 shot-clock buckets and 2 defensive pressure levels?

c) What is lost by this discretization compared to continuous state modeling?

Exercise 34: Value Iteration Implementation

Write pseudocode for value iteration to compute EPV in a discretized state space.

a) What is the stopping criterion?

b) How do you handle terminal states (shot made, shot missed, turnover)?

c) Estimate the computational complexity in terms of number of states $|S|$ and actions $|A|$.

Exercise 35: Real-Time EPV Requirements

For real-time EPV computation during a live game:

a) What latency is acceptable for coaching applications? For broadcast?

b) If your model requires 50ms to compute EPV from features, but feature extraction takes 200ms, what is your update rate?

c) Propose optimizations to improve real-time performance.

Exercise 36: Data Pipeline Design

Design a data pipeline for EPV computation.

a) Draw a diagram showing data flow from tracking cameras to EPV output.

b) Identify potential failure points and how you would handle them.

c) How would you handle missing tracking data (e.g., occluded player)?

Exercise 37: Model Validation Framework

Design a comprehensive validation framework for an EPV model.

a) What is your training/validation/test split strategy?

b) How do you prevent temporal leakage?

c) Define three quantitative metrics for model performance.

Exercise 38: Simulation-Based Testing

You want to test whether your EPV model would identify optimal decisions in known situations.

a) Design a simple simulation environment for pick-and-roll plays.

b) What ground truth would you use to evaluate model recommendations?

c) How would you handle the reality that "optimal" depends on player abilities?

Exercise 39: Feature Engineering

List 10 features you would engineer from raw tracking data for EPV prediction. For each:

a) Define the feature precisely.

b) Explain why it should be predictive of possession value.

c) Note any computational challenges in real-time extraction.

Exercise 40: Sensitivity Analysis

Design a sensitivity analysis for your EPV model.

a) What parameters would you vary?

b) What outputs would you examine?

c) How would you present results to non-technical stakeholders?

Answer Key

Exercise 1

a) On average, possessions starting from this position result in 1.02 points for the offense.

b) The offense has improved its position, likely through ball movement, defensive breakdowns, or advantageous positioning.

c) Not necessarily. A missed three-pointer from a 1.15 EPV situation means the play created good value; the outcome was simply unlucky. The expected value was actually 0.15 points above average before the shot.

d) EPV would typically decrease significantly (to perhaps 0.7-0.8) as shot clock pressure forces lower-quality attempts.

Exercise 6

$$E[\text{shot}] = 0.36 \times 3 + 0.64 \times (0.22 \times 1.05) = 1.08 + 0.148 = 1.228 \text{ points}$$

Exercise 7

$$E[\text{pass}] = 0.88 \times 1.18 + 0.12 \times (-0.95) = 1.038 - 0.114 = 0.924 \text{ points}$$

Exercise 12

a) $EPV = 0.15(0.92) + 0.25(1.08) + 0.20(1.05) + 0.18(1.12) + 0.12(1.04) + 0.10(0.98) = 1.039$

b) Optimal EPV = max action value = 1.12 (drive right)

c) Decision inefficiency = 1.12 - 1.039 = 0.081 points

Exercise 16

a) DVA by situation: - Open C&S: 0.00 - Contested: -0.07 - P&R: -0.07 - Post-up: -0.03 - Transition: -0.03

b) Weighted DVA = $(25 \times 0 + 15 \times (-0.07) + 30 \times (-0.07) + 10 \times (-0.03) + 20 \times (-0.03))/100 = -0.0405$

c) Contested arc and P&R handler (both -0.07)

d) 30 possessions $\times$ 0.07 improvement = 2.1 points per 100 possessions

Note: Complete solutions for all exercises are available in the instructor's manual.