Two players go on a hot new game show called “Higher Number Wins.” The two go into separate booths, and each presses a button, and a random number between zero and one appears on a screen. (At this point, neither knows the other’s number, but they do know the numbers are chosen from a standard uniform distribution.) They can choose to keep that first number, or to press the button again to discard the first number and get a second random number, which they must keep. Then, they come out of their booths and see the final number for each player on the wall. The lavish grand prize — a case full of gold bullion — is awarded to the player who kept the higher number. Which number is the optimal cutoff for players to discard their first number and choose another? Put another way, within which range should they choose to keep the first number, and within which range should they reject it and try their luck with a second number?
This is either a very weird auction problem with symmetric players (I also assume the players are risk-neutral) or a very odd lotteries/game-theory game.
How would you approach this question mathematically speaking and what answer do you get for it? There's no prize for me getting the right answer to the site's riddle, I'm just curious. My intuition tells me that the optimal cutoff is 0.5, since you have a 50-50 chance of being higher or lower than your opponent's number, regardless of whether he/she repicks their random number or not, but I am not sure.