We introduce and solve a new class of static portfolio choice problems, where only the best realized alternative matters. A decision maker must simultaneously choose among independent ranked options, and the better alternatives have a lower chance of panning out. Each choice is costly, and just one option may be exercised. This often emerges in practice: • A student must make a costly and simultaneous application to many colleges, and is accepted with smaller chances by the better schools. • An economics department must decide which of several PhD job candidates to fly out, and the better recruits will be available with smaller probability. We show that such portfolio choice problems quite generally entail maximizing a submodular function of finite sets - which is NP hard in general. Still, we develop a marginal improvement algorithm that produces the optimal set for our binary option structure in a quadratic number of steps. Applying it, we then show that the optimal choices are less risky than the sequentially optimal ones in Weitzman (1979), but riskier than the best singleton college choices. We also give practical rules of thumb, such as: (i) don't insure, choosing a safety school; instead, take risks - unless success rates are positively correlated; (ii) apply to an upwardly diverse portfolio of schools. We also provide comparative statics on the chosen set.
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