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Existence of a perfect matching in a random bipartite digraph withbipartition $(V_1, V_2)$, $|V_i|=n$, is studied. The graph is generated in tworounds of random selections of a potential matching partner such that theaverage number of selections made by each vertex overall is below $2$. Moreprecisely, in the first round each vertex chooses a potential mate uniformly atrandom, and independently of all vertices. Given a fixed integer $m$, a vertexis classified as unpopular if it has been chosen by at most $m$ vertices fromthe other side. Each unpopular vertex makes yet another uniform/independentselection of a potential mate. The expected number of selections made by ageneric vertex $v$, i.e. its out-degree, is asymptotic to $1+\BbbP(\text{Poisson}(1)\le m)\in (1,2)$. Aided by Matlab software, we prove thatfor $m=1$, whence for all $m\ge 1$, the resulting bipartite graph has a perfectmatching a.a.s. (asymptotically almost surely). On the other hand, for $m=0$a.a.s. a perfect matching does not exist, and the graph consists of a singlegiant component of size $2n -O(n^{1/2+o(1)})$ and possibly some components ofsize $O(\log n)$. This is a thorough revision of the joint paper (JCT(B) 88(2003), 1-16) by the first author and the third author.

On a perfect matching in a random bipartite digraph with average out-degree below two