I was listening to NPR on the way home today when a story about college entrance essays got me thinking. How do admissions officers judge the merits of an entrance essay? Is it based on some fixed scale? I can’t imaging that something as subjective as an essay can be effectively judged on a fixed scale. Is it judged relative to the other essays? This makes more sense to me, but I don’t understand the logistics. So I thought of a simpler problem that has the same elements, but is easier to identify with.

Suppose I were to come to you and offer you 100 boxes. In each box is a random amount of money between $0.01 and $100. After each box is offered to you, you must decide to keep it or to return it to me. If you return it to me, I will show you the amount of money in the box, but you will never see it again (and you won’t get the money). You are allowed to keep the first 10 boxes you choose. If you have not taken all of your 10 boxes by the time I offer the last boxes, you must take the remaining boxes to round out your total of 10. When you pick your tenth box, I will not offer any more boxes. What strategy optimizes your earnings?

Or suppose you are in a fishing tournament. At the end of the day, you must weigh in with your 10 biggest fish. If the game warden catches you with more than 10 fish in your boat, you will be disqualified from the competition. What strategy will assure your highest total weight?

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