  We propose two asymptotic expansions of two interrelated integral-type
averages, in the context of the fractional $\infty$-Laplacian $\Delta_\infty^s$
for $s\in (\frac{1}{2},1)$. This operator has been introduced and first studied
in [Bjorland, C., Caffarelli, L. and Figalli, A., \textsl{Nonlocal Tug-of-War
and the inifnity fractional Laplacian}, Comm. Pure Appl. Math., \textbf{65},
pp. 337--380, (2012)]. Our expansions are parametrised by the radius of the
removed singularity $\epsilon$, and allow for the identification of
$\Delta_\infty^s\phi(x)$ as the $\epsilon^{2s}$-order coefficient of the
deviation of the $\epsilon$-average from the value $\phi(x)$, in the limit
$\epsilon\to 0+$. The averages are well posed for functions $\phi$ that are
only Borel regular and bounded.
