We study a game in which two firms compete in quality to serve a market consisting of consumers with different initial consideration sets. If both firms invest below a certain threshold, they only compete for those consumers already aware of their existence. Above this threshold, a firm is visible to all and the highest investment attracts all consumers. On the one hand, the existence of initially captive consumers introduces an anti-competitive element: holding fixed the behavior of its rival, a firm with a larger captive segment enjoys a higher payoff from not investing at all. On the other hand, the fact that a firm’s initially captive consumers can still be attracted by very high quality introduces a pro-competitive element: a high investment becomes more profitable for the underdog when the captive segment of the dominant firm increases. The share of initially captive consumers therefore has a non-monotonic effect on the investment levels of both firms and on consumer surplus. We relate our findings to competition cases in digital markets.

我们研究一种博弈，其中两家公司在质量上竞争以服务具有不同初始考虑因素的消费者组成的市场。如果两家公司的投资都低于某个特定的门槛，它们只会竞争那些已经意识到自己存在的消费者。超过此阈值，公司对所有人都是可见的，而最高的投资吸引了所有消费者。一方面，最初被俘虏的消费者的存在引入了一种反竞争因素：固定其竞争对手的行为，拥有更大的俘虏市场的公司由于完全不进行投资而获得了更高的回报。另一方面，一个公司最初被俘虏的消费者仍然可以被非常高质量吸引的事实引入了一个有利于竞争的因素：当主导公司的俘虏部分增加时，对失败者的高投资变得更有利可图。因此，最初被俘虏的消费者的份额对企业的投资水平和消费者剩余都没有单调的影响。我们将调查结果与数字市场中的竞争案例联系起来。