I think the portfolio argument is an unnecessary detour though. There's a two-line proof by induction.
1. The payoff in the base case of (0,1) or (1,0) is 2.
2. If we are at (r,b), r >=b , have $X, and stake (r-b)/(r+b) on red, the payoff if we draw red and win is X * (1+(r-b)/(r+b)) * 2^(r+b-1) / (r+b-1 choose r-1) = X * 2^(r+b) * r / ((r+b) * (r+b-1 choose r-1)) = X * 2^(r+b) / (r+b choose r).
Similarly, if we draw black and lose, the payoff is X * (1-(r-b)/(r+b)) * 2^(r+b-1) / (r+b-1 choose r) = X * 2^(r+b) * b / ((r+b) * (r+b-1 choose r)) = X * 2^(r+b) / (r+b choose r). QED
In probability theory, Proebsting's paradox is an argument that appears to show that the Kelly criterion can lead to ruin. Although it can be resolved mathematically, it raises some interesting issues about the practical application of Kelly, especially in investing. It was named and first discussed by Edward O. Thorp in 2008.[1] The paradox was named for Todd Proebsting, its creator.
A very similar card game played by deciding when to stop flipping cards from a deck where red is $1 and black is −$1 as described in Timothy Falcon’s quantitative-finance interview book (problem #14). Gwern describes it and also writes code to prove out an optimal stopping strategy: https://gwern.net/problem-14
Kelly criterion is one of my favorite game theory concepts that is used heavily in bankroll management of professional gamblers, particularly poker players. It is a good way to help someone understand how you can manage your finances and stakes in a way that allows you to climb steadily forward without risking too much or any ruin, but is frequently misapplied in that space. The problem is kelly deals with binary results, and often situations in which this is applied where the results are not binary (a criteria for applying this) you can see skewed results that look almost right but not quite so, depending on how you view the math
pcthrowaway ·1 days ago
For example, if the deck has 26 red cards on top, you'd end up dwindling your initial $1.00 stake to 0.000000134 before riding it back up to 9.08
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ilya_m ·1 days ago
I think the portfolio argument is an unnecessary detour though. There's a two-line proof by induction.
1. The payoff in the base case of (0,1) or (1,0) is 2.
2. If we are at (r,b), r >=b , have $X, and stake (r-b)/(r+b) on red, the payoff if we draw red and win is X * (1+(r-b)/(r+b)) * 2^(r+b-1) / (r+b-1 choose r-1) = X * 2^(r+b) * r / ((r+b) * (r+b-1 choose r-1)) = X * 2^(r+b) / (r+b choose r).
Similarly, if we draw black and lose, the payoff is X * (1-(r-b)/(r+b)) * 2^(r+b-1) / (r+b-1 choose r) = X * 2^(r+b) * b / ((r+b) * (r+b-1 choose r)) = X * 2^(r+b) / (r+b choose r). QED
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lordnacho ·1 days ago
In probability theory, Proebsting's paradox is an argument that appears to show that the Kelly criterion can lead to ruin. Although it can be resolved mathematically, it raises some interesting issues about the practical application of Kelly, especially in investing. It was named and first discussed by Edward O. Thorp in 2008.[1] The paradox was named for Todd Proebsting, its creator.
https://en.wikipedia.org/wiki/Proebsting%27s_paradox
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fancy_pantser ·1 days ago
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JohnMakin ·1 days ago
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