Suppose you rolled a 6 sided fair dice and earn how much you roll. If you are unsatisfied, you may roll again (only once though). What is the expected pay-off?
So for the original game with no re-rolls, the payoff is 3.5, so it makes sense to re-roll when you get less than 3.5. Then the expected payoff $E$ satisfies $$E + E + E + 4 + 5 + 6 = 6E,$$ so $3E + 15 = 6E$, or $-3E = -15$, or $E = 5$.
Does this mean my stopping rule is not optimal? Since $E > 4$, we should roll again at 4 right?