  There has been a long history for studying randomized greedy matching
algorithms since the work by Dyer and Frieze~(RSA 1991). We follow this trend
and consider the problem formulated in the oblivious setting, in which the
algorithm makes (random) decisions that are essentially oblivious to the input
graph.
  We revisit the \textsf{Modified Randomized Greedy (MRG)} algorithm by Aronson
et al.~(RSA 1995) that is proved to be $(0.5+\epsilon)$-approximate. In
particular, we study a weaker version of the algorithm named \textsf{Random
Decision Order (RDO)} that in each step, randomly picks an unmatched vertex and
matches it to an arbitrary neighbor if exists. We prove the \textsf{RDO}
algorithm is $0.639$-approximate and $0.531$-approximate for bipartite graphs
and general graphs respectively. As a corollary, we substantially improve the
approximation ratio of \textsf{MRG}.
  Furthermore, we generalize the \textsf{RDO} algorithm to the edge-weighted
case and prove that it achieves a $0.501$ approximation ratio. This result
solves the open question by Chan et al.~(SICOMP 2018) about the existence of an
algorithm that beats greedy in this setting. As a corollary, it also solves the
open questions by Gamlath et al.~(SODA 2019) in the stochastic setting.
