A duopolistic market is analyzed in which firms engage in price competition over consumers who respond only to large price differences. When prices are approximately the same, consumers randomize. It is shown that the market has an equilibrium with bounded support if, and only if, consumers’ sensitivity to price differences is sufficiently high. If an equilibrium exists, then it is unique, firms play a mixed pricing strategy, their profits are positive, and consumers pay, with probability 1, a price that is higher than marginal cost. Moreover, the less sensitive consumers are to price differences, the higher are the firms’ profits.