We study a game of price competition amongst firms selling homogeneous goods defined by the property that a firm’s revenue is independent of any competing prices that are strictly lower. This property is induced by any customer choice model involving utility-maximizing choice from an adaptively determined consideration set, encompassing a variety of empirically validated choice models studied in the literature. For these games, we show a one-to-one correspondence between pure-strategy local Nash equilibria with distinct prices and the prices generated by the firms sequentially setting local best-response prices in different orders. In other words, despite being simultaneous-move games, they have a sequential-move equilibrium structure. Although this structure is attractive from a computational standpoint, we find that it makes these games particularly vulnerable to the existence of strictly-local Nash equilibria, in which the price of a firm is only a local best-response to competitors' prices when a globally optimal response with a potentially unboundedly higher payoff is available. Our results thus suggest that strictly-local Nash equilibria may be more prevalent in competitive settings than anticipated. We moreover show, both theoretically and empirically, that price dynamics resulting from the firms utilizing gradient-based dynamic pricing algorithms to respond to competition may often converge to such an undesirable outcome. We finally propose an algorithmic approach that incorporates global experimentation to address this concern under certain regularity assumptions on the revenue curves.