Random Almost-Popular Matchings

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 peoples 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 persons 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 $alpha(1-e^{-(1+e^{1/alpha})/alpha}) < 1-2epsilon$, then an $epsilon$-popular matching exists with $o(1)$ probability (lower bound).