Probability

Value of a repeatable game with a fee

Prices the re-roll family in one line once you know the threshold: keep the $n-k$ highest faces, re-roll the rest.

V = \frac{\sum_{i=k+1}^{n} i - kc}{n - k}

Where

$n$: Number of faces.
$k$: How many of the lowest faces you re-roll.
$c$: Cost of a re-roll.

Assumptions

The game after a re-roll must be identical to the one you started with.
The threshold must satisfy $k \approx V - c$ , which is what makes the equation consistent.

Sanity check. With $c = 0$ the answer is the maximum face; with $c$ large enough it collapses to a single roll.

Where this is taught

Recursive expected value and the re-roll family · PROB · Expectation, variance and the big tricks