Brad DeLong Solves the Balls in the Hat Game

Brad DeLong has solved the balls in the hat game, and in his comments section provides the solution, which is a psychologically horrible one: you basically have to run your losses and cut your winnings.

Keep playing until you get to (0,1) or (1,2) or (1,3) or (2,4) or (3,5) or (3,6). At those points the game is no longer worthwhile.

If you recall, there are 4 white balls and 6 black balls in the hat, which means the initial state is (4,6). You get $1 for every white ball you pick out, and lose $1 for every black ball you pick out. You can stop picking out balls at any time, and you want to play the game so as to have a positive expected value.

The interesting thing about this solution is that there's no chance of getting the best-case scenario (picking out four white balls in a row and then stoppping, for a profit of $4) but there is a chance of getting the worst-case scenario (picking out six black balls in a row and then recouping some of your losses with the remaining white balls, for a loss of $2). In fact, if the first ball out of the hat is white, then you stop then and there: you cash in your $1 and you're happy.

And I think I'm right in saying that's the only way you actually walk away with a profit in this game. If the first ball out of the hat is black, then your best-case scenario is to break even.

So your maximum upside is $1, your maximum downside is $2, and most of the time the game consists of a desperate struggle to get back up to $0 after picking out a black ball initially. You can take your +$0.07 EV, I'm not playing this game.