Roster Strength by Team Value

Scope

This reference extends roster strength where pregame team value is available. It replaces one scalar official-roster rating with a locally learned curve. The deterministic effect of the difference between the two team values is covered separately in Delta-TV Matchup Correction.

Extended strength coordinate

The shared model becomes

D = (C R_{1} - C R_{2}) + R R_{1} (T V_{1}) - R R_{2} (T V_{2}) + F_{T V} (T V_{1} - T V_{2}) .

The $R R (T V)$ terms remain latent Gaussian quantities. The final term is an observed deterministic covariate and contributes no latent variance.

Official-roster control points

For roster $r$ , let $x_{r}$ be a vector of latent roster strengths at fixed TV control points. The retained state is

x_{r} \sim N (μ_{r}, Σ_{r}), Σ_{r} = diag (v_{r, 1}, \dots, v_{r, m}) .

Let $w_{r} (T)$ be the piecewise-linear interpolation weights at team value $T$ . At most two weights are nonzero, all are nonnegative, and they sum to one. Outside the configured range, the model clamps to the nearest endpoint.

R R_{r} (T) = w_{r} (T)^{⊤} x_{r} .

Therefore

E [R R_{r} (T)] = w_{r} (T)^{⊤} μ_{r}, V_{r} (T) = w_{r} (T)^{⊤} Σ_{r} w_{r} (T) .

With the retained diagonal covariance,

V_{r} (T) = \sum_{i} w_{r, i} (T)^{2} v_{r, i} .

The squared weights are required by the variance of a linear combination of independent Gaussian states. Halfway between equally uncertain points, the interpolated variance is $v / 2$ , not $v$ .

Matchup variance and mean update

For distinct official rosters, the latent strength variance is

V_{D} = v_{C R_{1}} + v_{C R_{2}} + w_{1}^{⊤} Σ_{1} w_{1} + w_{2}^{⊤} Σ_{2} w_{2} .

If the core joint update produces $δ D$ , the roster point means move by

Δ μ_{1} = Σ_{1} w_{1} \frac{δ D}{V_{D}}, Δ μ_{2} = - Σ_{2} w_{2} \frac{δ D}{V_{D}} .

Under diagonal covariance, only points with nonzero interpolation weight receive a direct update. Learning is therefore local in team value.

Same-roster games

If both sides use the same roster state, define the contrast vector

q = w_{1} - w_{2} .

Then the roster contribution, its variance, and its update are

D_{R} = q^{⊤} x_{r}, V_{R} = q^{⊤} Σ_{r} q, Δ μ_{r} = Σ_{r} q \frac{δ D}{V_{D}} .

At equal TV, $q = 0$ : the common roster state contributes neither strength nor roster-strength information to that game.

Uncertainty and state evolution

The local posterior creates covariance between active control points. The implementation retains only the diagonal marginal variances, deliberately discarding persistent cross-covariances. Under that retained diagonal approximation, future mean updates remain local to control points with nonzero interpolation weight.

After posterior contraction, active points may receive empirical process variance:

v_{i}^{n e x t} = v_{i}^{+} + Q_{R R, p o i n t} .

Inactive points retain their variance.

Custom rosters

Custom or non-official rosters use one scalar latent state rather than a TV curve:

R_{r} \sim N (μ_{r}, v_{r}) .

Its contribution is $+ R_{r}$ or $- R_{r}$ by team side, and its mean update is

Δ μ_{r} = \pm v_{r} \frac{δ D}{V_{D}} .

The number and placement of control points, interpolation choice, clamping, and process-variance policy are model choices rather than consequences of the Davidson likelihood.

Roster Strength by Team Value ​

Scope ​

Extended strength coordinate ​

Official-roster control points ​

Matchup variance and mean update ​

Same-roster games ​

Uncertainty and state evolution ​

Custom rosters ​