Consider the following 2-player game: Each player has some score. Taking turns, each player gets added to his score a uniform random on (0,1). If after this addition that player is ahead by at least 1, then he wins and his opponent must pay the game stakes, which are initially 1 unit. The game starts out with player II some small amount $x$ ahead of player 1.
It is not completely easy to find the expectation in the game as stated thus far: It is isomorphic to a random walk in the space of the difference between the two scores, with each step chosen uniformly on $(-\frac{1}{2}, +\frac{1}{2})$, player 1 winning if a step goes to at least +1, and player 2 winning if a step goes to -1. And in this isomorphism, the starting point maps to $\frac{1}{4}-x$. So you might think the value of this easy game can be computed based on an unbiased random walk, and is in this unbiased $\left(\frac{1+.25-x}{2}\right)-\left(\frac{1-.25-x}{2}\right)$ which is $\frac{1}{4}-x$.
Wrong. The "unbiased movement" is a mirage, because if a player is within winning distance, his expectation "wastes" the expected amount of overshoot. Look at it this way: If player 1 is .75 ahead when it is his turn to score, that corresponds to $x = -1$. But the game from that starting position is not a sure win for player 1. So the first question is, what is the value of the game as a function of x?
But now the real question: This game is played with a doubling cube. If the cube is neutral or in a player's possession, then before a that player's turn she may offer to double the stakes. And her opponent has two choices: Decline to double the stakes, thus forfeiting the current stakes, or take possession of the cube and continue the game, playing for twice the current stakes.
Find the correct strategy for an arbitrary score difference (including whether to double and whether to accept the double) and the game value as a function of the score difference and the possession of the cube.
For the first question: numerical experimentation (an iterative updating scheme with a mesh size of $1/100$) suggests that the following is the graph of the value of the game, based on the difference in scores at the start of one's turn. It's consistent with the assertion that the value of a tie game at the start of my turn is $1/4$, and with the assertion that the value when my turn begins with me $1/4$ behind is $0$. But that has to be balanced with the observation that the graph doesn't look truly linear in any section.