The Cambridge Lawn Tennis Club organises a tournament in which every match consists of 11 games, all of which are played. The player who wins 6 or more games is declared the winner.

For players $a$ and $b$, let $n_{a b}$ be the total number of games they play against each other, and let $y_{a b}$ be the number of these games won by player $a$. Let $\tilde{n}_{a b}$ and $\tilde{y}_{a b}$ be the corresponding number of matches.

A statistician analysed the tournament data using a Binomial Generalised Linear Model (GLM) with outcome $y_{a b}$. The probability $P_{a b}$ that $a$ wins a game against $b$ is modelled by

$$\log \left(\frac{P_{a b}}{1-P_{a b}}\right)=\beta_{a}-\beta_{b},$$

with an appropriate corner point constraint. You are asked to re-analyse the data, but the game-level results have been lost and you only know which player won each match.

We define a new GLM for the outcomes $\tilde{y}_{a b}$ with $\tilde{P}_{a b}=\mathbb{E} \tilde{y}_{a b} / \tilde{n}_{a b}$ and $g\left(\tilde{P}_{a b}\right)=$ $\beta_{a}-\beta_{b}$, where the $\beta_{a}$ are defined in $(*)$. That is, $\beta_{a}-\beta_{b}$ is the log-odds that $a$ wins a game against $b$, not a match.

Derive the form of the new link function $g$. [You may express your answer in terms of a cumulative distribution function.]