This paper develops a computational algebra approach for comparing the Cournot and Bertrand competition. This approach addresses the limitations of traditional backward induction, which becomes infeasible in multi-stage games lacking closed-form solutions and geometric methods, which fail in multi-stage settings and do not scale beyond two players. Building on implicit differentiation and triangular decomposition, our method produces exact results without requiring closed-form solutions to the equilibrium equations, thereby enabling rigorous proofs of economic theorems. Illustrative applications demonstrate our method's effectiveness in nonlinear settings and reveal how nonlinearities can fundamentally alter standard Cournot-Bertrand rankings.