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The question is similar to the original one, The expected payoff of a dice game, but now we can keep $m>1$ numbers (dice).

Formally, let $X_1,...,X_n\sim \text{Unif }[0,1]$ be a sequence of i.i.d random variables coming successively. We can keep at most $m$ numbers in hand where $1<m<n$. After seeing $X_i$, you may choose to stop, or you can discard one of the $m$ numbers and see the next random variable $X_{i+1}$.

What's the best strategy to maximize the sum of these $m$ values?

(For simplicity I take $X_i\sim \text{Unif }[0,1]$. Instead you may consider a dice if you want.)

RobPratt
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True Light
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  • As stated it looks like you discard any $x_i\lt \frac{1}{2}$ as long as you can. If it is Bernoulli, discard $X_i=0$. – herb steinberg May 05 '21 at 21:47
  • But in the original dice game (m=1), we discard the first dice if it is less than some threshold $\alpha_1$, and we discard the second dice if it is less than another threshold $\alpha_2$, where $\alpha_1>\alpha_2>E[X]$. So I'm guessing we also have similar thresholds here. – True Light May 05 '21 at 23:48
  • You need to clarify the game you are describing. – herb steinberg May 06 '21 at 00:52

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