323.Swap or Keep Your Number
★★★optimal stoppingCitadel Securities
You and another player each independently draw a number uniformly from the interval (0,1). You see your own number but not theirs, and may then either keep it or force a swap, taking their number while they take yours. They have no say. Playing optimally, what is the probability that you finish holding the higher number?
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Your rule will be a threshold: swap when your draw is low. Write the winning probability as a function of that threshold and maximise it.
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