Various problems arising in game theory, mathematical economics, and optimization theory may be formulated as complementarity problems. For example, the necessary Kuhn-Tucker conditions for the general optimization problem produce a problem of this type. The list of contributions to the theory of the linear complementarity problem is very extensive, comprising Lemke and Howson [6], Cottle and Dantzig [ 11, Eaves [2], Ingleton [4], Karamardian [5], Murty [8,9], and Saigal [lo], although many computational problems in mathematical economics and game theory, e.g., the computation of equilibria of the generalized von Neumann model or Nperson games, are typical examples for nonlinear complementarity problems. For that reason the question arises of how far some aspects of the linear theory are appropriate for a generalization to the nonlinear case. Some classes of linear complementarity problems produce the constant parity property, i.e., the property that the number of solutions is either odd within the whole class or even (see [9, lo]). Problems of this type depend on some real parameters and are connected by homotopies preserving the parity of the number of solutions. Moreover, these homotopies generate paths, which connect different solutions. Thus, there are some chances to find new solutions if some solutions were known. The investigation of classes of nonlinear problems will lead to similar results in some cases. Of course, as in the linear case these results will not be true without the nondegeneracy assumption. Therefore the question arises whether this assumption will be satisfied within a sufficiently large subclass. It turns out that for special types of classes the assumption holds at least for almost all problems with respect to the Lebesgue measure on the parameter space.