  We consider the problem, as suggested by Dembo ($2003$, $2006$), of late
points of a simple random walk in two dimensions. It has been shown that the
exponents for the numbers of pairs of late points coincide with those of nearly
favorite points and high points in the Gaussian free field, whose exact values
are known. We estimate the exponents for the numbers of a multipoint set of
late points in average. While there have been observed certain similarities
between among three classes of points, our results exhibit a difference.
