Suppose Alice and Bob are playing a game with a fair coin. Every time the coin comes up heads, the score (initially 0) increases by 1. If the coin comes up tails, the score decreases by 1. If the score reaches -2 or 2, the game is over. Alice wins if the score is -2, and Bob wins if the score is 2.
The stake starts out as one dollar, but every turn, before the coin is flipped, Bob has the option of doubling the stake. What's his optimal strategy, and what's the EV of this strategy?
The payout is done by the loser, so this is a zero sum game.
One simple (but likely suboptimal) strategy for Bob is to double the stake every time the score is +1. I've done the calculation, and his expected value is $1.
To win as much as he can, I believe Bob should always double his stake when he has positive EV. Thus an optimal strategy appears to doubling his stake when the score is 0 and when the score is 1. But doing the calculation (using recursive states) seems to give a negative expected value--a bizarre contradiction I can't wrap my head around. Perhaps I've done the math wrong, but I can't help but think this is similar to the St. Petersburg paradox, where once in a while, Bob can win an insurmountably large amount of money. The only difference is that he might also lose it all--he's never guaranteed a positive payout.
I appreciate any thoughts on the problem.