Here is an idealized draw poker game whose solution is extremely interesting mathematically. (I made this up myself but would not be surprised if someone has anticipated me.)
2 players each draw a real number uniformly from the unit interval [0,1]. They can’t see each other’s numbers. Player 1 decides to either keep his number or throw it away and draw another. Player 2, after hearing which decision player 1 made; decides to keep his number or throw it away and draw another. Players then reveal their numbers, higher number wins.
What is the best strategy for each player, and how often does each player win? To put this another way, if the players are wagering $100 on the outcome of the game, what is the value of the extra information player 2 gets by hearing player 1’s decision?
Hint: if your answer is simple, then it’s wrong.
(Added 11/6: solution now given in the Comments.)