I've been familiar to the BraleyTerry model proposed in 1952 .And In 1970 Davidson promoted a generalized model with ties.
`your text` '$$ P(\text{ties}) =\frac{\nu\sqrt{\alpha_i\alpha_j}{\alpha_i+\alpha_j+\nu\sqrt{\alpha_i+\alpha_j}} $$
$$ P(\text{i beats j}) =\frac{\alpha_i}{\alpha_i+\alpha_j+\nu\sqrt{\alpha_i+\alpha_j}}$$
And Generalized Davidson $logit P(\text{ties}) = log \frac{delta}{c} p^{\sigma\pi}(1-p)^{\sigma(1-\pi)}$'
Header Tuner developed the BradleyTerry2 package in R with a gnm model to deal with ties . He claims the log probability with the formula below. Model log expected counts for each match $k$ (log probabilities)
`'$$
\begin{aligned}
\log \left(\operatorname{pr}(\mathrm{i} \text { beats } \mathrm{j})_k\right) & =\theta_{i j k}+\log \left(\mu \alpha_i\right) \\
\log \left(\operatorname{pr}(\text { draw })_k\right) & =\theta_{i j k}+\log \delta-\log c \\
& +\sigma\left(\pi \log \left(\mu \alpha_i\right)-(1-\pi) \log \left(\alpha_j\right)\right) \\
& +(1-\sigma) \log \left(\mu \alpha_i+\alpha_j\right) \\
\log \left(\operatorname{pr}(\mathrm{j} \text { beats } \mathrm{i})_k\right) & =\theta_{i j k}+\log \left(\alpha_j\right)
\end{aligned}
$$'`
where $\theta_{i j k}$ fixes the total count for each match to 1 . With gnm: How could I manage to joint the original form with his nonlinear model
I think it may be something hiding on $\theta$.The author does not provide the derivation .
