The rising prevalence of algorithmic pricing in industry calls for a closer study of market outcomes that may arise when these algorithms interact in a competitive environment. In this paper, we study the impact of boundedly-rational customer choice on the landscape of joint demand curves in a competition, and the resulting consequences on the convergence properties of competing gradient-based pricing algorithms. In particular, we study price competition in a duopoly under a new customer behavior model motivated by platform competition for perfect substitutes, e.g., ride-hailing. In this model, a customer samples the firms in an idiosyncratically preferred order until she finds one that is priced below her willingness-to-pay. We exhaustively characterize the equilibrium profile of the resulting pricing game when the customers’ willingness-to-pay distribution has a monotone hazard rate. We show that these games are frequently plagued by a particular strictly-local Nash equilibrium, in which the price of the firm with a smaller market share is only a local best-response to the competitor’s price, when a globally optimal response with a potentially unboundedly higher payoff is available. Through numerical experiments, we show that distributed gradient-based learning dynamics may often converge to this undesirable outcome. Our results thus demonstrate that algorithms that adaptively rely on local information to learn good decisions may suffer from serious drawbacks in competitive environments.