dice roll question

Question

How much would an individual pay to play a game where they get the payout of a number they roll on a standard 6 sided dice but get up to 6 re-rolls.

Answer 1

It seems obvious that you should stop if you have $4,5,6$ and reroll if you have $1,2$. Let us compute the value if we roll on $3$ or less. Let the value of the game be $v$. If we roll $1,2,3$ we lose $1$ and roll again, so we expect $v-1$. If we roll $4,5,6$ we are content and average $4$, so $$v=\frac 12(v-1)+\frac 12\cdot 4\\v=3$$ Now let our strategy be to stand on $3$ or above. Now we have $\frac 13$ chance of rolling again and getting $v-1$ and $\frac 23$ chance of standing and getting $3.5$, so $$v=\frac 13(v-1)+\frac 23\cdot 3.5\\v=3$$ In either case the value of the game is $3$, so it doesn't matter if you roll or stand on $3$.

Answer 2

If you have a $3$, it takes an expected $2$ rolls to get more, and then the expected value of the greater roll is $5$, so you're indifferent between stopping and continuing. Thus the value of the game is $3$.