  We amplify previous discussions of the fine-tuning price to be paid by
supersymmetric models in the light of LEP data, especially the lower bound on
the Higgs boson mass, studying in particular its power of discrimination
between different parameter regions and different theoretical assumptions. The
analysis is performed using the full one-loop effective potential. The whole
range of $\tan\beta$ is discussed, including large values. In the minimal
supergravity model with universal gaugino and scalar masses, a small
fine-tuning price is possible only for intermediate values of $\tan\beta$.
However, the fine-tuning price in this region is significantly higher if we
require $b-\tau$ Yukawa-coupling unification. On the other hand, price
reductions are obtained if some theoretical relation between MSSM parameters is
assumed, in particular between $\mu_0$, $M_{1/2}$ and $A_0$. Significant price
reductions are obtained for large $\tan\beta$ if non-universal soft Higgs mass
parameters are allowed. Nevertheless, in all these cases, the requirement of
small fine tuning remains an important constraint on the superpartner spectrum.
We also study input relations between MSSM parameters suggested in some
interpretations of string theory: the price may depend significantly on these
inputs, potentially providing guidance for building string models. However, in
the available models the fine-tuning price may not be reduced significantly.
