37.A Hundred Cards Against a Guesser
★★★expected valueJane Street
You write a positive whole number on each of 100 blank cards, repeating values as often as you like. Your opponent then sees the full list of what you wrote, the deck is shuffled, and they name one number. If the top card of the shuffled deck shows that number they win that many dollars; otherwise they win nothing. Your opponent plays to maximise their expected winnings and you play to minimise it. What is the smallest expected payout, in dollars, that you can hold them to?
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If you use a value on several cards your opponent can always name it, so you are really choosing a collection of value-times-count products and only the largest one matters. Ask how many cards a given ceiling on that product lets you lay down.
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