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Airline booking data have shown that the fraction of customers who choose the cheapest available fare class often is much greater than that predicted by the multinomial logit choice model calibrated with the data. For example, the fraction of customers who choose the cheapest available fare class is much greater than the fraction of customers who choose the next cheapest available one, even if the price difference is small. To model this spike in demand for the cheapest available fare class, a choice model called the spiked multinomial logit (spiked-MNL) model was proposed. We study a network revenue management problem under the spiked-MNL choice model. We show that efficient sets, that is, assortments that offer a Pareto-optimal tradeoff between revenue and resource use, are nested-by-revenue when the spike effect is nonnegative. We use this result to show how a deterministic approximation of the stochastic dynamic program can be solved efficiently by solving a small linear program. The solution of the small linear program is used to construct a booking limit policy, and we prove that the policy is asymptotically optimal. This is the first such result for a booking limit policy under a choice model, and our proof uses an approach that is different from those used for previous asymptotic optimality results. Finally, we evaluate different revenue management policies in numerical experiments using both synthetic and airline data.