Delta-TV Matchup Correction

Scope

This reference defines the deterministic team-value-difference term used by the team-value extension. It builds on Roster Strength by Team Value.

Let

$$d=T{V}_1-T{V}_2.$$

Correction function

The implemented correction is

$$F_{TV}(d)=\mathrm{sign}(d)[C(1-e^{-(|d|/T{)}^Q})+A|d|e^{-(|d|/S{)}^P}].$$

The saturating first component and transient second component are empirical model structure. Their fitted parameters are fixed when the online rating updates run.

Symmetry and equal-value behavior

Because the magnitude depends on $|d|$ and the outer sign changes when the teams are swapped,

$$F_{TV}(-d)=-F_{TV}(d).$$

The correction is therefore exactly team-label symmetric. It also satisfies

$$F_{TV}(0)=0.$$

Equal team value receives no deterministic strength correction.

Local and large-difference behavior

For $x=|d|$ near zero,

$$F_{TV}(d)\sim \mathrm{sign}(d)[C{\left(\frac{x}{T}\right)}^Q+Ax].$$

For the fitted case $Q<1$, the function is continuous at zero but does not have a finite derivative there. This is harmless to the online inference because TV is observed before the game and is not a latent state differentiated by the update.

At large differences, the transient term vanishes and the correction is bounded:

$$\underset{d\to +\mathrm{\infty }}{lim}{F}_{TV}(d)=C,\underset{d\to -\mathrm{\infty }}{lim}{F}_{TV}(d)=-C.$$

Role in inference

The pregame strength mean is

$$D_0=D_{\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{n}\mathrm{t}}+F_{TV}(T{V}_1-T{V}_2).$$

The term contributes no latent uncertainty:

$$V_{TV}=0.$$

It changes the pregame probabilities and therefore the likelihood residual, joint MAP solution, and local curvature. It does not add a variance term to the pregame latent covariance. The coach and roster states therefore learn from the residual remaining after the modelled TV effect has been applied.

Domain policy

The deterministic correction can be disabled for configured roster classes where diagnostics do not support applying it. In that case

$$F_{TV}(T{V}_1-T{V}_2)=0,$$

while all other parts of the model remain active: coach strength, roster strength at TV, draw tendency, Davidson probabilities, and Gaussian updating.

What is model-dependent

The odd symmetry, zero correction at equal TV, and absence of added latent variance follow from the stated construction. The functional family, fitted parameters, and any roster-class suppression policy are empirical model decisions.