Duopoly Price Competition with Reference Effects Under Logit Demand

We consider a duopoly price competition with each firm selling one substitute product to strategic consumers, who assess their willingness to pay based on the reference price and make purchase decisions through the MNL model. The influence of reference price on consumer demand, known as the reference effect, can be potentially asymmetric, which incurs the non-smoothness in revenue functions. In the one-shot game, we show the existence, uniqueness, and structure of Nash equilibrium (NE) in loss-neutral and loss-averse scenarios (our proof also extends to the oligopoly setting). By contrast, in the gain-seeking scenario, the NE exists but may not be unique, where we provide a sufficient condition for the occurrence of multiple equilibria. Additionally, our one-shot game can be naturally extended to a repeated game with consecutive periods linked by reference price updates. We derive theoretical insights on the long-run market dynamics in this multi-period competitive framework under two practical pricing policies. Notably, the market stability (i.e., convergence behavior) hinges on the presence of a gain-seeking product. In the absence of gain-seeking products, we prove the convergence of price and reference price paths to the stationary Nash equilibrium (SNE) when firms possess full rationality and adopt the equilibrium pricing policy. We further characterize the structure of SNE and show that it is unique if and only if both products are loss-neutral. Meanwhile, when firms only exhibit bounded rationality and adhere to the best-response pricing policy, we draw the same conclusion in a special case where consumers have short-term memories. Conversely, when any gain-seeking product is introduced, the SNE ceases to exist, implying that the market never stabilizes in the long run under either equilibrium or best-response pricing policy