= Solution
Let $x$ be the parameter-contrast vector for the specified game: it has entries $+1$ for $\beta_i,\beta_j,\beta_{\{i,j\}}$, entries $-1$ for $\beta_k,\beta_\ell,\beta_{\{k,\ell\}}$, and zero elsewhere. Put
$$
\eta=x^T\beta,
\qquad
p=g(\eta)=\frac{e^\eta}{1+e^\eta}.
$$
Under the stated regularity assumptions, <asymptotic normality of a maximum likelihood estimator> gives
$$
\sqrt N(\widehat\beta-\beta)
\xrightarrow{d}N\!\left(0,I(\beta)^{-1}\right),
$$
where $i_N(\beta)/N\to I(\beta)$. Since $g'(\eta)=p(1-p)$, the <delta method> gives
$$
\boxed{\sqrt N(\widehat p-p)
\xrightarrow{d}
N\!\left(0,\,[p(1-p)]^2x^TI(\beta)^{-1}x\right).}
$$
Equivalently, for large $N$,
$$
\boxed{\widehat p\ \dot\sim\
N\!\left(p,\,[p(1-p)]^2x^Ti_N(\beta)^{-1}x\right),}
$$
with $p=e^{x^T\beta}/(1+e^{x^T\beta})$.
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