  In order to clarify the mechanism of the enhancement of superconducting
transition temperature $T_{\rm c}$ due to anharmonic local oscillation of a
guest ion in a cage composed of host atoms, i.e., {\it rattling}, we analyze
the anharmonic Holstein model by applying the Migdal-Eliashberg theory. From
the evaluation of the normal-state electron-phonon coupling constant, it is
found that the strong coupling state is developed, when the bottom of a
potential for the guest ion becomes wide and flat. Then, $T_{\rm c}$ is
enhanced with the increase of the anharmonicity in the potential, although
$T_{\rm c}$ is rather decreased when the potential becomes a double-well type
due to very strong anharmonicity. From these results, we propose a scenario of
anharmonicity-controlled strong-coupling tendency for superconductivity induced
by rattling. We briefly discuss possible relevance of the present scenario with
superconductivity in $\beta$-pyrochlore oxides.
