All-pay competition with captive consumers

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.