How pricey must a re-roll be before you never bother?

Question

You roll a fair six-sided die, paid its face value, and may re-roll as often as you like at a cost of $\$ c $each. As$ c$ rises, re-rolling gets less attractive.

Find the critical cost $c^*$ at or above which you should simply keep your very first roll, whatever it is. How does the game's value behave as $c$ approaches that threshold?

Accepted Answer

Let G(c) be the game's value. The re-roll option is worth G - c, so you keep a face x iff x \ge G - c. "Never re-roll" means even the worst possible face, a 1, is worth keeping: 1 \ge G - c. If you truly never re-roll, then G = E[X] = 3.5, and the condition becomes 1 \ge 3.5 - c \implies c \ge 2.5. So the critical cost is c^ = \2.50. For c \ge 2.50, re-rolling even a 1 is not worth it: the re-roll is worth 3.5 - c \le 1, no better than keeping the 1 you already hold. The game is then worth exactly E[X] = \3.50. Just below the threshold. For c slightly under 2.5 you re-roll only a 1. Then G = …

How pricey must a re-roll be before you never bother?

Your answer

More Market Making questions