Draw Tendency and the Joint Update

Scope

This reference describes the draw coordinate and the inference step shared by strength and draw tendency. It builds on the Core Davidson Model.

Draw coordinate

Draw tendency is separate from relative strength:

$$\eta =C{D}_1+C{D}_2+R{D}_1+R{D}_2.$$

Coach and roster draw states are Gaussian:

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

Unlike strength, draw tendency is symmetric between teams: every coefficient is positive. If both teams use the same roster, its contribution is $2R{D}_r$, so that roster reinforces rather than cancels in $\eta$.

Exact draw score

Let $Y_D=1$ for a draw and $0$ otherwise. The exact Davidson likelihood score and curvature in the draw direction are

$$\frac{\partial \ell }{\partial \eta }=Y_D-P_D,-\frac{{\partial }^2\ell }{\partial {\eta }^2}=P_D(1-P_D).$$

The residual $Y_D-P_D$ is therefore both the natural draw-update signal and the diagnostic that motivated explicit coach and roster draw states.

Why the update is joint

Strength and draw are coupled in the Davidson likelihood. Their exact cross-curvature is

$$-\frac{{\partial }^2\ell }{\partial D\partial \eta }=c{P}_D(\frac{1}{2}-E).$$

The full negative Hessian is

$$H=\left[\begin{array}{cc}{c}^2{V}_S& c{P}_D(\frac{1}{2}-E)\\ c{P}_D(\frac{1}{2}-E)& P_D(1-P_D)\end{array}\right].$$

The cross term vanishes only when $E=1/2$. This is why the model does not perform separate strength and draw updates.

Pregame uncertainty in observation space

Let $y_0=(D_0,{\eta }_0{)}^{\mathrm{\top }}$. Before the game, the retained state families are independent, so

$$K=\left[\begin{array}{cc}{V}_D& 0\\ 0& V_{\eta }\end{array}\right].$$

For distinct rosters,

$$V_{\eta }=v_{C{D}_1}+v_{C{D}_2}+v_{R{D}_1}+v_{R{D}_2}.$$

For a same-roster game, the roster coefficient is two, so

$$V_{\eta }=v_{C{D}_1}+v_{C{D}_2}+4v_{R{D}_r}.$$

Joint MAP solution

Let $\delta =(\delta D,\delta \eta {)}^{\mathrm{\top }}$ be the movement from the prior mean to the posterior mode. The exact MAP stationarity equation is

$$\delta =Kg(y_0+\delta ),$$

with

$$g(D,\eta )=\left[\begin{array}{c}c(S-E)\\ Y_D-P_D\end{array}\right].$$

Newton's method solves this equation numerically; it does not introduce a different update rule.

Mapping the draw correction back

Once $\delta \eta$ is known, an ordinary draw state receives

$$\mathrm{\Delta }C{D}_i=v_{C{D}_i}\frac{\delta \eta }{V_{\eta }},\mathrm{\Delta }R{D}_r=v_{R{D}_r}\frac{\delta \eta }{V_{\eta }}.$$

For a shared roster, the coefficient is two:

$$\mathrm{\Delta }R{D}_r=2v_{R{D}_r}\frac{\delta \eta }{V_{\eta }}.$$

Posterior uncertainty

The posterior is locally approximated around the MAP using a Laplace approximation:

$${\mathrm{\Sigma }}_{\mathrm{p}\mathrm{o}\mathrm{s}\mathrm{t}}\approx {({\mathrm{\Sigma }}^{-1}+A^{\mathrm{\top }}{H}_{\mathrm{M}\mathrm{A}\mathrm{P}}A)}^{-1},$$

where $A$ maps the retained latent states into the observation coordinates $(D,\eta )$.

The implementation then keeps marginal variances and discards persistent cross-covariances. That projection is a deliberate approximation; the MAP mean update is not.

Draw states are currently persistent traits: $Q_{CD}=Q_{RD}=0$. Their uncertainty contracts with evidence but is not re-inflated between games.