The fraction of customers who choose a particular item from among a set of available items can be increased significantly by the inclusion of a related inferior (and apparently irrelevant) item in the choice set. This violation of the independence from irrelevant alternatives and the regularity properties is called the decoy effect, dominance effect, or attraction effect. The decoy effect is one of the robust cognitive biases in the decision-making processes of customers. We propose a discrete choice model that is simple and that captures decoy effects. A monopolist may take advantage of the decoy effect to increase profit. However, exploitation of the decoy effect in a competitive setting requires closer investigation. To understand the effect of decoys on competition, we study product assortment competition in a duopoly in which each seller may choose whether to include a decoy in the seller's product assortment. We provide a complete characterization of the Nash equilibria and their dependence on choice model parameters. We study the evolution of assortment competition and we evaluate the stability of the equilibria in the context of sellers learning about the behavior of their competitors. Our results indicate under what conditions it is beneficial for a seller to include a decoy into the seller's assortment, and under what conditions the seller obtains a free ride from the competitor's decoy. Our results also show that every pure-strategy Nash equilibrium is stable and every mixed-strategy Nash equilibrium is unstable.