Kelly Can't Fail

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

> particularly poker players

The Kelly criterion seems excellent for many forms of gambling, but poker seems like it could be an exception: in poker, you’re playing against other players, so the utility of a given distribution of chips seems like it ought to be more complicated than just the number of chips you have.

(I’m not a poker player.)

You are right that Kelly criterion deals with binary results. This won't work for poker. In poker, we use expected value because wins and losses are not binary because of the amount you win or lose. Once you figure out your approximate EV, you use a variance calculator in addition to that (example: https://www.primedope.com/poker-variance-calculator/) to see how likely and how much it is you will be winning over a certain number of hands in the long run.

Could this work with roulette betting on color? Seems like you could spend a lot of time not winning or losing

Roulette results are uncorrelated and you have the exact same chance of winning each time, so the Kelly criterion isn’t applicable. Betting on a color has a negative edge and you don’t have the option of taking the house’s side, so it just tells you the obvious thing that you should bet zero.

It's used for bankroll management (basically deciding what stakes to play) rather than for sizing bets within a particular game.

> exact same chance of winning each time, so the Kelly criterion isn’t applicable.

Actually, the main assumption that leads to the Kelly criterion is that you will have future opportunities to bet with the same edge, not constrained by the amount.

For example, if you knew this was your last profitable betting opportunity, to maximise your expected value you should bet your entire stake.

I'm slightly surprised it leads to such a nice result for this game - I don't see a claim that this is the optimal strategy for maximizing EV zero variance is great, but having more money is also great.

Of course you are right about roulette and, if you are playing standard casino roulette against the house, the optimal strategy is not to play. But that's not because bets are uncorrelated, it's because they are all negative value.

What makes 0 better than the other numbers?

Can't bet negative in that kind of game. If a game is expected to lose you money, don't play.

Chris "Jesus" Ferguson "proved" an application of this theory back in ~2009 [1]. He was a the time promoting Full Tilt and commited to turn $1 dollar bankroll to $10000 by applying a basic strategy of never using more than a low % of his bankroll into one tournament or cash game session.

So, if one's skill would turn your session probability to +EV, by limiting your losses and using the fact that in poker the strongest hands or better tourney positions would give you a huge ROI, it would be just a matter of time and discipline to get to a good bankroll.

Just remember that for the better part of this challenge he was averaging US$ 0.14/hour, and it took more than 9 months.

[1] https://www.thehendonmob.com/poker_tips/starting_from_zero_b...

> Just remember that for the better part of this challenge he was averaging US$ 0.14/hour, and it took more than 9 months.

But consider the rate of return! He turned $1 into $10,000 in 9 months. Could he then turn that $10k into $100M in another 9 months?

Or if he'd started with $100 instead of $1, could he have turned that into $1M in 9 months? That would still be incredibly impressive.

Certainly the game changes as the bets and buy-ins get higher, but even if he couldn't swing the same rate of return with a higher starting point and larger bets (though still only betting that same certain low percent of his bankroll), presumably he could do things like turning $5k into $1M. Even $100k into $1M would be fantastic.

I think the challenge is that the larger you bet, the harder it is to find people who are bad at basic strategy poker but willing to bet against you for a long series of bets.

$0, not 0 on the wheel.

> The problem is kelly deals with binary results,

Incorrect. https://entropicthoughts.com/the-misunderstood-kelly-criteri...

The Kelly criterion generalises just fine to continuous, simultaneous, complicated allocations.

All it takes is a list of actions which we are choosing from (and these can be compound actions with continuous outcomes) and the joint probability distribution of wealth outcomes after each action.

> Actually, the main assumption that leads to the Kelly criterion is that you will have future opportunities to bet with the same edge, not constrained by the amount.

Not the same edge -- any edge! And this condition of new betting opportunities arriving every now and then is a fairly accurate description of life, even if you walk out of the casino.