  Performing (variance-based) global sensitivity analysis (GSA) with dependent
inputs has recently benefited from cooperative game theory concepts.By using
this theory, despite the potential correlation between the inputs, meaningful
sensitivity indices can be defined via allocation shares of the model output's
variance to each input. The ``Shapley effects'', i.e., the Shapley values
transposed to variance-based GSA problems, allowed for this suitable solution.
However, these indices exhibit a particular behavior that can be undesirable:
an exogenous input (i.e., which is not explicitly included in the structural
equations of the model) can be associated with a strictly positive index when
it is correlated to endogenous inputs. In the present work, the use of a
different allocation, called the ``proportional values'' is investigated. A
first contribution is to propose an extension of this allocation, suitable for
variance-based GSA. Novel GSA indices are then proposed, called the
``proportional marginal effects'' (PME). The notion of exogeneity is formally
defined in the context of variance-based GSA, and it is shown that the PME
allow the distinction of exogenous variables, even when they are correlated to
endogenous inputs. Moreover, their behavior is compared to the Shapley effects
on analytical toy-cases and more realistic use-cases.
