Roll a die, keep it or re-roll once, what's it worth?
Asked at SIG, Optiver
You roll a fair six-sided die. You may either keep the result and get paid its face value in dollars, or discard it and roll once more, in which case you're paid the second roll, whatever it is.
What is the optimal strategy, and what is the fair value of the game?
Show a hint
The second roll, if you take it, is worth a known amount in expectation. Compare each first-roll outcome to that number.