Very cool to see no variance in the outcome. But that also makes it feel like there should be a strategy with better expected return due to the unique problem structure. Do we know if the Kelly strategy is optimal here?
replies(6):
I was thinking these are very different strategies, but they’re not exactly. The Kelly strategy does the same thing when there’s only one color left. The difference is this strategy does nothing before that point.
Still, they feel like limit cases. Betting it all with only one color left is the only right move, so it’s what you do before that. Nothing and Kelly seem like the only good strategies.
It would be interesting to do the math and show why they're equal. It seems like you should be able to make the same sort of portfolio probability argument.
I guess the number of possible arrangements of cards with N of one color remaining is... The number of permutations of N times 2 times the number of permutations of 52 minus N times 26 choose N?
Ah, yes this works, you can see it here: https://www.wolframalpha.com/input?i=%28summation+of+N%21+*+....
That is: (summation of N! * (52 - N)!* (26 choose N) * 2^N/52! from N=0 to 26 (for some reason the * 2 for different suits was over counting, so I removed it. Not sure why? Also it seems like it should be from 1 to 26, but that also doesn't give the right answer, so something is whack)