  We study constraint effective potentials for various strongly interacting
$\phi^4$ theories. Renormalization group (RG) equations for these quantities
are discussed and a heuristic development of a commonly used RG approximation
is presented which stresses the relationships among the loop expansion, the
Schwinger-Dyson method and the renormalization group approach. We extend the
standard RG treatment to account explicitly for finite lattice effects.
Constraint effective potentials are then evaluated using Monte Carlo (MC)
techniques and careful comparisons are made with RG calculations. Explicit
treatment of finite lattice effects is found to be essential in achieving
quantitative agreement with the MC effective potentials. Excellent agreement is
demonstrated for $d=3$ and $d=4$, O(1) and O(2) cases in both symmetric and
broken phases.
