Core Davidson Model

Scope

This reference defines the shared prediction model: separate coach and roster strength, a Davidson likelihood with win/draw/loss outcomes, Gaussian state uncertainty, and chronological prediction before every update.

Draw tendency and the joint state update are developed in Draw Tendency and the Joint Update. Team-value-specific extensions are separate references.

Strength coordinate

For a game between team 1 and team 2, the pregame strength difference is

$$D=(C{R}_1-C{R}_2)+(R{R}_1-R{R}_2).$$

Each coach strength and roster strength state is Gaussian:

$$C{R}_i\sim \mathcal{N}({\mu }_{C{R}_i},v_{C{R}_i}),R{R}_r\sim \mathcal{N}({\mu }_{R{R}_r},v_{R{R}_r}).$$

The team-1 coach and roster enter positively; the team-2 coach and roster enter negatively. If both teams use the same scalar roster, its strength contribution cancels exactly.

Davidson probabilities

Let the Davidson scale be $F$, define $c=\ln (10)/F$, and set $x=cD/2$. For equal-strength draw fraction $F_{\mathrm{D}\mathrm{R}\mathrm{A}\mathrm{W}}$, define

$$U_0=\frac{2F_{\mathrm{D}\mathrm{R}\mathrm{A}\mathrm{W}}}{1-F_{\mathrm{D}\mathrm{R}\mathrm{A}\mathrm{W}}}.$$

The draw coordinate $\eta$ changes the draw parameter through

$$U=U_0{e}^{\eta }.$$

With $Z=e^x+e^{-x}+U$, the pregame probabilities are

$$P_W=\frac{e^x}{Z},P_D=\frac{U}{Z},P_L=\frac{e^{-x}}{Z}.$$

At $D=0$ and $\eta =0$, the draw probability is exactly $F_{\mathrm{D}\mathrm{R}\mathrm{A}\mathrm{W}}$.

Outcome and strength score

Encode the observed result as $S=1$ for a team-1 win, $1/2$ for a draw, and $0$ for a team-1 loss. The Davidson expected score is

$$E=P_W+\frac{1}{2}{P}_D.$$

The exact likelihood score and curvature in the strength direction are

$$\frac{\partial \ell }{\partial D}=c(S-E),-\frac{{\partial }^2\ell }{\partial D^2}=c^2{V}_S,$$

where

$$V_S=P_W+\frac{1}{4}{P}_D-E^2.$$

So $S-E$ is an exact Davidson likelihood residual, not an Elo-style heuristic.

Allocating a strength correction

The joint inference step produces a posterior strength displacement $\delta D$. With pregame strength variance $V_D$, the correction is allocated by prior variance. For distinct rosters,

$$\mathrm{\Delta }C{R}_1=v_{C{R}_1}\frac{\delta D}{V_D},\mathrm{\Delta }C{R}_2=-v_{C{R}_2}\frac{\delta D}{V_D},$$$$\mathrm{\Delta }R{R}_1=v_{R{R}_1}\frac{\delta D}{V_D},\mathrm{\Delta }R{R}_2=-v_{R{R}_2}\frac{\delta D}{V_D}.$$

More uncertain states therefore receive more of the evidence. A same-roster game gives no scalar roster-strength update because that state has coefficient zero in $D$.

State evolution and evaluation

After posterior contraction, an evolving strength state receives process variance:

$$v^{\mathrm{n}\mathrm{e}\mathrm{x}\mathrm{t}}=v^{\mathrm{p}\mathrm{o}\mathrm{s}\mathrm{t}}+Q.$$

Games are processed chronologically. For each game the model reads only the current pregame state, records its win/draw/loss prediction, observes the result, updates the states, then proceeds to the next game. No result enters its own prediction.

Boundaries

The Davidson likelihood scores are exact under the stated model. Representing the posterior as independent scalar Gaussian states is an approximation; the details are in Draw Tendency and the Joint Update.