  We have synthesized magnesium cage compounds $R$Ni$_{2}$Mg$_{20}$ ($R$ = Pr
and Nd). We report the measurements of electrical resistivity $\rho$, magnetic
susceptibility $\chi$, isothermal magnetization $M$, and specific heat $C$.
Polycrystalline samples with $R$ = Pr and Nd were obtained by annealing, while
the counterparts for $R$ = La, Ce, and Y were found to be absent. The
${\rho}(T)$ data for $R$ = Pr and Nd monotonically decrease on cooling from 300
K to 40 K and exhibit shoulders at around 13 K and 15 K, respectively. The
Curie-Weiss behaviors of the ${\chi}(T)$ data indicate the trivalent states of
both the Pr and Nd ions. For $R$ = Pr, the maximum in $C(T)$ at around 7 K is
reproduced by a doublet-triplet two-level model with an energy gap of 14 K. The
peak in $C(T)$ at 0.7 K is attributed to a short-range order of quadrupolar
degrees of freedom in the non-Kramers ground doublet. On the other hand, for
$R$ = Nd, a maximum in $C(T)$ at around 9 K is explained by thermal excitation
from a ground state doublet to an excited quartet separated by 23 K. Upon
further cooling, $C(T)$ shows a lambda-shaped peak at 1.5 K. By applying
magnetic fields up to 2 T, the peak becomes broad and shifts to higher
temperatures, which is a characteristic of a ferromagnetic order.
