The heart of the equilibrium selection theory of Harsanyi and Selten (1988, A General Theory of Equilibrium Selection in Games, Cambridge, MA: MIT Press) is given by the tracing procedure, a mathematical construction that adjusts arbitrary prior beliefs into equilibrium beliefs. Although the term "procedure" suggests a numerical approach, the tracing procedure itself is a nonconstructive method. In this paper we propose a homotopy algorithm that generates a path of strategies. By using lexicographic pivoting techniques, it can be shown that for the entire class of noncooperative N-person games, the path converges to an approximate Nash equilibrium, even when the starting point or the game is degenerate. The outcome of the algorithm is shown to be arbitrarily close to the equilibrium beliefs proposed by the tracing procedure. Therefore, the algorithm does not compute just any Nash equilibrium, but one with a sound game-theoretic underpinning. Like other homotopy algorithms, it is easily implemented on a computer.