  There exists a significant body of work on determining the acquisition number
$a_t(G)$ of various graphs when the vertices of those graphs are each initially
assigned a unit weight. We determine properties of the acquisition number of
the path, star, complete, complete bipartite, cycle, and wheel graphs for
variations on this initial weighting scheme, with the majority of our work
focusing on the expected acquisition number of randomly weighted graphs. In
particular, we bound the expected acquisition number $E(a_t(P_n))$ of the
$n$-path when $n$ distinguishable "units" of integral weight, or chips, are
randomly distributed across its vertices between $0.242n$ and $0.375n$. With
computer support, we improve it by showing that $E(a_t(P_n))$ lies between
$0.29523n$ and $0.29576n$. We then use subadditivity to show that the limiting
ratio $\lim E(a_t(P_n))/n$ exists, and simulations reveal more exactly what the
limiting value equals. The Hoeffding-Azuma inequality is used to prove that the
acquisition number is tightly concentrated around its expected value.
Additionally, in a different context, we offer a non-optimal acquisition
protocol algorithm for the randomly weighted path and exactly compute the
expected size of the resultant residual set.
