Roll a die with 100 faces, labeled from 1 to 100.
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You get to roll once and receive the amount of dollars labeled on the face. How much would you like to pay for this roll?
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How much would you pay if you can re-roll the dice if you are unsatisfied with the first outcome?
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You can roll the dice infinitely many times, and each roll costs 1 dollar except the first one. What is your strategy?
My thoughts:
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The expectation of a 100 faces dice is 50.5, so this is the fair value.
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If I can re-roll once, I would re-roll if I rolled between 1 and 50. Now the expectation would be $0.5 * 50.5 + 0.5 * 75.5 = 63$.
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I guess my strategy would be keep re-rolling until the expectation improvement would be less than the cost of 1 dollar to re-roll? Can someone give me a closed form solution ?
Best Answer
Your answers for $1$ and $2$ are right.
For $3$, say you reroll if you roll at most $x$. Then your expected payoff is
$$ E=\frac x{100}(E-1)+\left(1-\frac x{100}\right)\frac{x+1+100}2\;, $$
and solving for $E$ yields
$$ E=\frac{10100-3x-x^2}{2(100-x)}\;. $$
Then setting the derivative with respect to $x$ to zero yields
$$ 10100-3x-x^2=(2x+3)(100-x) $$
with solution $x=10(10\pm\sqrt2)$. Only the smaller solution is feasible, $x=10(10-\sqrt2)\approx85.86$, and substituting the two adjacent integers, $85$ and $86$, into the expression for $E$ yields $\frac{262}3\approx87.33$ for $x=85$ and $\frac{1223}{14}\approx 87.36$ for $x=86$.
Thus a maximal profit of about $\$87.36$ is achieved if you reroll whenever you have at most $86$.