  The fundamental assignment problem is in search of welfare maximization
mechanisms to allocate items to agents when the private preferences over
indivisible items are provided by self-interested agents. The mainstream
mechanism \textit{Random Priority} is asymptotically the best mechanism for
this purpose, when comparing its welfare to the optimal social welfare using
the canonical \textit{worst-case approximation ratio}. Despite its popularity,
the efficiency loss indicated by the worst-case ratio does not have a constant
bound. Recently, [Deng, Gao, Zhang 2017] show that when the agents' preferences
are drawn from a uniform distribution, its \textit{average-case approximation
ratio} is upper bounded by 3.718. They left it as an open question of whether a
constant ratio holds for general scenarios. In this paper, we offer an
affirmative answer to this question by showing that the ratio is bounded by
$1/\mu$ when the preference values are independent and identically distributed
random variables, where $\mu$ is the expectation of the value distribution.
This upper bound also improves the upper bound of 3.718 in [Deng, Gao, Zhang
2017] for the Uniform distribution. Moreover, under mild conditions, the ratio
has a \textit{constant} bound for any independent random values. En route to
these results, we develop powerful tools to show the insights that in most
instances the efficiency loss is small.
