  In this article, we study the no-boundary wave function in scalar-tensor
gravity with various potentials for the non-minimally coupled scalar field. Our
goal is to calculate probabilities for the scalar field - and hence the
effective gravitational coupling and cosmological constant - to take specific
values. Most calculations are done in the minisuperspace approximation, and we
use a saddle point approximation for the Euclidean action, which is then
evaluated numerically. We find that for potentials that have several minima,
none of them is substantially preferred by the quantum mechanical
probabilities. We argue that the same is true for the stable and the runaway
solution in the case of a dilaton-type potential. Technically, this is due to
the inclusion of quantum mechanical effects (fuzzy instantons). These results
are in contrast to the often held view that vanishing gravitation or
cosmological constants would be exponentially preferred in quantum cosmology,
and they may be relevant to the cosmological constant problem and the dilaton
stabilization problem.
