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Two Opaque Boxes: Place or Take for Maximum Expected Payoff

Jane StreetQuantitative Trader · Onsite
MediumQuantitative TraderOnsiteCoding & Algorithms

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This question evaluates probabilistic reasoning, expected-value analysis, and sequential decision-making under uncertainty, testing competency in stochastic modeling, combinatorics, and optimization.

You are playing a one-player game with two opaque boxes, both of which start empty. The game lasts exactly 100 turns, and on every turn you must choose one of the following two actions:

  • Place: a third party puts $1 into one of the two boxes, chosen uniformly at random. (This money is not yours yet — it just sits in the box.)
  • Take: one of the two boxes is chosen uniformly at random; you receive all of the money currently inside that box, and the box is emptied.

The boxes are opaque: you can never see how much money is in either box, and you do not learn how much money you have collected until the game is over. In other words, you receive no feedback during the game, so your decisions cannot depend on any observed outcomes.

Assuming optimal play, what is the expected total payoff of this game? Describe the optimal strategy (which turns should be Place and which should be Take, and in what order) and compute the exact expected value it achieves.