  We introduce a model of a controlled random graph process. In this model, the
edges of the complete graph $K_n$ are ordered randomly and then revealed, one
by one, to a player called Builder. He must decide, immediately and
irrevocably, whether to purchase each observed edge. The observation time is
bounded by parameter $t$, and the total budget of purchased edges is bounded by
parameter $b$. Builder's goal is to devise a strategy that, with high
probability, allows him to construct a graph of purchased edges possessing a
target graph property $\mathcal{P}$, all within the limitations of observation
time and total budget. We show the following: (a) Builder has a strategy to
achieve $k$-vertex-connectivity at the hitting time for this property by
purchasing at most $c_kn$ edges for an explicit $c_k<k$; and a strategy to
achieve minimum degree $k$ (slightly) after the threshold for minimum degree
$k$ by purchasing at most $(1+\varepsilon)kn/2$ edges (which is optimal). (b)
Builder has a strategy to create a Hamilton cycle at the hitting time for
Hamiltonicity by purchasing at most $Cn$ edges for an absolute constant $C>1$;
this is optimal in the sense that $C$ cannot be arbitrarily close to $1$. This
substantially extends the classical hitting time result for Hamiltonicity due
to Ajtai--Koml\'os--Szemer\'edi and Bollob\'as. (c) Builder has a strategy to
create a perfect matching by time $(1+\varepsilon)n\log{n}/2$ while purchasing
at most $(1+\varepsilon)n/2$ edges (which is optimal). (d) Builder has a
strategy to create a copy of a given $k$-vertex tree if $t\ge
b\gg\max\{(n/t)^{k-2},1\}$, and this is optimal; (e) For $\ell=2k+1$ or
$\ell=2k+2$, Builder has a strategy to create a copy of a cycle of length
$\ell$ if $b\gg\max \{n^{k+2}/t^{k+1},n/\sqrt{t}\}$, and this is optimal.
