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There are infinite cards numbered 1-5. You construct a deck. Someone guesses a value and draws a card. If they guessed correctly, they get the value. What composition of the deck that minimizes their expected value?

One way to proceed in this question is to aim to make the opponent's expected value from choosing any of the numbers 1-5 the same. If you set these expected values to be equal, you get that the proportion of each card in the deck that you construct should be inversely proportional to its value. E.g the number of cards with the number 4 should be proportional to 1/4 etc. However, I'm not clear on how we get this condition. How do we know that the composition that minimizes the opponent's expected value is also one where the opponent is indifferent between all the cards?

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  • $\begingroup$ Note that all strategies are convex combinations of the strategies: Always pick $x$, for $x\in\{1,2,3,4,5\}$. Use the pigeonhole principle to show that for any distribution, there is at least one strategy among these $5$ that have an expected output $\geq1/(1+1/2+1/3+1/4+1/5)$. What does that tell you? $\endgroup$
    – Rushabh Mehta
    Commented Dec 27, 2020 at 17:02

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$\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5} =\frac{137}{60}$

The equal expectation approach makes each outcome $x$ happen with probability $\frac{60}{137 x}$, so if the opponent choses $x$ their expected value is a constant $\frac{60}{137}$; this makes the opponent indifferent between the card numbers.

If you choose any other distribution, then for at least one $x$ the probability is greater than $\frac{60}{137 x}$ and for another $x$ the probability is less than $\frac{60}{137 x}$; the opponent will prefer the former over the latter as it gives a higher expected value.

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