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Path-following algorithms have proved practical for the solution of fixed-point problems arising in economics and game theory. But the value of the methodology is diminished by the possibility of multiple solutions, since there is no guarantee of finding more than one solution even if many paths are traced. This Note describes a way of transforming problems so that the paths are modified while the solutions remain unchanged. As a result, previously inaccessible solutions may become accessible. It is shown that for any piecewise-linear problem in general position there is a solution-preserving transformation that makes all solutions simultaneously accessible.

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