NBA Court Realty

NESSIS 2015

Work in collaboration with: Luke Bornn, Alex D'Amour, Alex Franks, Kirk Goldsberry, Andrew Miller

Specifically: divide the court into \( m \) equally sized cells

• \[ w^i_j(t) = \text{dist}(\text{player } i, \text{cell } j) \] at time \( t \)

\[ x^i_j(t) = \begin{cases} \frac{1}{1 + w^i_j(t)} & i = \text{argmin}_h w^h_j(t) \\ 0 & \text{otherwise} \end{cases} \]

• \( \beta \in \mathbb{R}^m \), where \( \beta_i \) is price/value of region \( i \)
• \( X^i(t) \) row vector, entry \( j \) is \( x^i_j(t) \).
• \( X^i(t) \beta \) is the total property value of player \( i \) at time \( t \)

Portfolio value differential: \[ V_t \beta = \left(\sum_{i: \text{ offense}} X^i(t) - \sum_{i: \text{ defense}} X^i(t)\right)\beta \]

Market capitalization: \[ M_t \beta = \left(\sum_{i: \text{ offense}} X^i(t) + \sum_{i: \text{ defense}} X^i(t)\right)\beta \]

• \( \beta_i > 0 \) for all \( i \), with \( \sum_i \beta_i = c \)
• \( \beta \) spatially smooth
• \( V_t \beta \) should behave like a random walk
• \( M_t \beta \) should be as high as possible

Minimize \[ \sum_t \frac{1}{2}\beta'(V_t - V_{t - 1})'(V_t - V_{t-1})\beta + \lambda_1 \frac{1}{M_t} \beta + \lambda_2\beta'\Omega_{\kappa}\beta \]

subject to:

• \( \sum_{i} \beta_i = c \)
• \( \beta_i > 0 \)
• \( \Omega_{\kappa} = \exp(- \kappa(\text{ distance matrix})) \)

Free parameters: \( \lambda_1, \lambda_2, \kappa \).

• We use \( 2 \times 2 \) boxes for court regions
• Approximately 34 million portfolio snapshots
• Only need inner products
• Solving for \( \beta \) is quadratic programming problem

Player' movement reveals spatial valuation

• New quantifications for team movement and positioning
• Valuations differ by team
• Allocations differ by team (e.g. on/off ball)

Next steps

• Validating court space valuation
• Player heterogeneity
• Improve property definition