  In this paper we consider the online Submodular Welfare (SW) problem. In this
problem we are given $n$ bidders each equipped with a general (not necessarily
monotone) submodular utility and $m$ items that arrive online. The goal is to
assign each item, once it arrives, to a bidder or discard it, while maximizing
the sum of utilities. When an adversary determines the items' arrival order we
present a simple randomized algorithm that achieves a tight competitive ratio
of $\nicefrac{1}{4}$. The algorithm is a specialization of an algorithm due to
[Harshaw-Kazemi-Feldman-Karbasi MOR`22], who presented the previously best
known competitive ratio of $3-2\sqrt{2}\approx 0.171573 $ to the problem. When
the items' arrival order is uniformly random, we present a competitive ratio of
$\approx 0.27493$, improving the previously known $\nicefrac{1}{4}$ guarantee.
Our approach for the latter result is based on a better analysis of the
(offline) Residual Random Greedy (RRG) algorithm of
[Buchbinder-Feldman-Naor-Schwartz SODA`14], which we believe might be of
independent interest.
