Equilibria Existence in Games: A Perspective from the Lefschetz-Nielsen Theory

Classical fixed-point theorems guaranteeing the existence of equilibria in noncooperative games, such as those of Brouwer and Kakutani, rely on convexity assumptions. Nevertheless, many games are inherently nonconvex, rendering such results inapplicable. In this work, we establish the existence of equilibria in games by employing tools from algebraic topology. We introduce a local equilibrium notion, the first-order Nash equilibrium, that captures stationary strategy profiles. Using the Lefschetz fixed-point theorem, we prove that at least one local equilibrium exists under mild assumptions, even when cost functions are nonconvex. We further refine this result through the Nielsen extension of the Lefschetz theorem, which characterizes the persistence of equilibria under continuous deformations of the game.