Optimal application behavior with incomplete information

This paper studies the application behavior of agents in an environment where agents with dierent underlying qualities but with identical preferences are competing for placements (in colleges, on sports teams, etc) and are allowed to submit multiple applications. Two aspects of incomplete information are considered that make the application behavior in these environments an interesting decision problem to study: uncertainty about one's own quality and noise in the evaluation of applications. First, I consider what happens when agents face uncertainty about their own quality (inter- preted as their ranking among the pool of applicants), need to form beliefs about their quality and make applications based on these beliefs. I show that with a fixed num- ber of applications, given that the distribution characterizing the belief of the agents satisfies the monotone likelihood ratio property (MLRP), there is assortative matching on the expected quality of the agent in the sense that all applications are increasing in the expected quality of the agent. When agents are allowed to decide how many applications to submit given some application cost, however, this result breaks down. It is possible to characterize the optimal application behavior locally and in the limit, but the optimal number of applications shows no regularity globally. I show that, even if MLRP holds, it is possible for an agent with a better underlying quality and a better mean belief to secure a worse placement, something that is not possible with a fixed number of applications. I also show that the number of applications — for high enough dispersion — is decreasing in the dispersion of beliefs, and for suciently high dispersion, agents submit no application or submit a single application for the best placement. Second, I study what happens when in the environment studied a random element is added to the evaluation of applications. I show that agents with low dispersion in their beliefs are the ones to submit multiple applications. The results thus show that both aspects of incomplete information imply that agents facing more uncertainty about their own quality submit fewer applications, and hence secure worse placements. In a companion paper I embed the application decision of the agents into a general equilibrium model of college applications where colleges optimally chose which students to admit, and study the properties of its equilibrium.