We present a general framework to find ε-equilibrium solutions of oligopolistic markets in which demand is modeled at the disaggregate level using discrete choice models. Consumer choices are modeled according to random utility theory, and the choice probabilities are linearized and embedded as lower-level constraints in the supply optimization problems. To model competition, we introduce a mixed integer optimization model based on the fixed-point iteration algorithm, which can find an optimal equilibrium or near-equilibrium solution of a finite game with small strategy sets. To solve larger equilibrium problems, a model-based algorithmic approach is proposed. First, a heuristic reduction of the search space is performed. Then, an iterative procedure solves a subgame equilibrium problem with restricted strategy sets using the fixed-point optimization model, compares the optimal solution against the best responses of all suppliers over the original strategy sets, and adds best response strategies to the restricted problem until all ε-equilibrium conditions are satisfied simultaneously. Numerical experiments show the applicability of our algorithm to find oligopolistic ε-equilibria for two transportation case studies.