= Solution
Under the <binomial distribution> model's assumption of independent games with a common win <probability> $p$, the <probability> of winning a match is
$$
h(p)=\sum_{j=6}^{11}\binom{11}{j}p^j(1-p)^{11-j}.
$$
This is also the <cumulative distribution function> $F_{6,6}(p)$ of a <Beta distribution> with parameters $(6,6)$: differentiation of the binomial tail gives $h'(p)=11!\,p^5(1-p)^5/(5!)^2$, and $h(0)=0$. In particular $h$ is strictly increasing on $(0,1)$ and maps $[0,1]$ bijectively onto itself.
The new match <probability> is $\widetilde P_{ab}=h(P_{ab})$, while the desired linear predictor remains the game-level <log odds>. Therefore the <link function> is
$$
\boxed{g(u)=\log\frac{F_{6,6}^{-1}(u)}{1-F_{6,6}^{-1}(u)}},\qquad0<u<1.
$$
Its endpoint values are $-\infty$ and $+\infty$. This is not the ordinary match-level <log odds> link: it first recovers the underlying game <probability>. Symmetry of the binomial tail gives $h(1-p)=1-h(p)$, hence $g(1-u)=-g(u)$, as required by swapping the players. Dependence or varying game probabilities within a match would invalidate the stated binomial-tail conversion.
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