Strict dominance solvability without equilibrium

A game in strategic form is strict dominance solvable if iterative elimination of strictly dominated strategies yields a unique strategy profile (strict dominance solution). Textbook presentations of this material are framed in the context of finite games and it is argued that if a strict dominance solution exists, it must also be the unique Nash equilibrium. We construct a simple counter example demonstrating that strict dominance solutions need not constitute Nash equilibria in infinite games, even if each player has a unique undominated strategy. This conclusion has special pedagogical significance as the sensitivity of results to the finite game context can often be lost on those being introduced to the material for the first time. As an additional pedagogical exercise, we establish that the traditional textbook conclusion extends to settings in which strategy spaces are compact and utility functions are continuous.