We study a new game of price competition amongst firms selling identical goods, whose defining property is that the revenue of a firm at a price is independent of any competing prices that are strictly lower. This game is motivated by a variety of customer choice models that induce this property, prominently those stemming from the well-known satisficing heuristic for decision-making. Under a mild condition, we show that every pure-strategy local Nash equilibrium in this game corresponds to a set of prices generated by the firms sequentially setting local best-response prices in a fixed order. In other words, despite being simultaneous-move games, equilibria have a sequential-move equilibrium structure. We thus call these games pseudo-competitive games. We find that these games are often plagued by strictly-local Nash equilibria, in which the price of a firm is only a local best-response to the competitor's price, when a globally optimal response with a potentially unboundedly higher payoff is available. We show that price dynamics resulting from the firms utilizing gradient-based pricing algorithms may often converge to such undesirable outcomes. We propose a new online learning approach to address this concern under certain regularity assumptions on the revenue curves.