  For a set $A$ of $n$ people and a set $B$ of $m$ items, with each person
having a preference list that ranks all items from most wanted to least wanted,
we consider the problem of matching every person with a unique item. A matching
$M$ is called $\epsilon$-popular if for any other matching $M'$, the number of
people who prefer $M'$ to $M$ is at most $\epsilon n$ plus the number of those
who prefer $M$ to $M'$. In 2006, Mahdian showed that when randomly generating
people's preference lists, if $m/n > 1.42$, then a 0-popular matching exists
with $1-o(1)$ probability; and if $m/n < 1.42$, then a 0-popular matching
exists with $o(1)$ probability. The ratio 1.42 can be viewed as a transition
point, at which the probability rises from asymptotically zero to
asymptotically one, for the case $\epsilon=0$. In this paper, we introduce an
upper bound and a lower bound of the transition point in more general cases. In
particular, we show that when randomly generating each person's preference
list, if $\alpha(1-e^{-1/\alpha}) > 1-\epsilon$, then an $\epsilon$-popular
matching exists with $1-o(1)$ probability (upper bound); and if
$\alpha(1-e^{-(1+e^{1/\alpha})/\alpha}) < 1-2\epsilon$, then an
$\epsilon$-popular matching exists with $o(1)$ probability (lower bound).
