Simultaneous Search, Reservation Fees, and Sequential Outcomes

This paper studies a simultaneous-search problem in which a player observes the outcomes sequentially, and must pay reservation fees to maintain eligibility for recalling the earlier offers. We use postgraduate program applications to illustrate the key ingredients of this family of problems. We develop a parsimonious model with two categories of schools: Reach schools, which the player feels very happy upon joining, but the chance of getting into one is low; and safety schools, which are a safer choice but not as exciting. The player first decides on the application portfolio, and then the outcomes from the schools applied to arrive randomly over time. We start with the extreme case wherein the safety schools always admit the player, and show that it suffices to focus on the last safety school. This allows us to conveniently represent the player’s value function by a product form of the probability of entering the last safety period and the expected payoff from then on. We show that the player’s payoff after applications is increasing and discrete concave in both the numbers of reach and safety schools, and the optimal number of reach schools increases in the reservation fee. The proof technique utilizes stochastic coupling, stochastic dominance, and directional monotone comparative statics arguments. We also develop a recursive dynamic programing algorithm when admissions to safety schools are no longer guaranteed. We demonstrate instances in which the player applies to more safety schools when either the reservation fee gets higher or the admission probability drops lower, and articulate how these arise from the portfolio optimization consideration.