Kelly criterion, EV, ruin and running it multiple times

You ask some interesting questions, for which I don't really feel qualified to answer.

It seems that it is more of an academic debate though as Kelly criterion is typically too aggressive for real life bankroll management.

It does make sense that you should be able to bet more aggressively given that you are running it twice as each board is effectively half the bet size. So if you're aiming to maximize geometric growth I think you should be betting bigger/ more aggressively and thus growing faster given that you are running it twice.

A caveat is that occasionally you might get it all in with close to 100% equity or 0% equity, so I'm not sure how that effects the Kelly equation. The run-out on the two boards is only part of the variance, the other part is where each player happens to be within their own ranges. Hope that makes sense. Intuitively I would guess that running it twice we could bet more aggressively than if we were running it once for the full all-in amount, but would still have to bet less aggressively compared to if we were making completely separate bets for effectively 1/2 all-in size.

Anyway regarding your other question 1, if theoretically the casino had an infinite bankroll and could guarantee that players would bet following the hypothetical pattern then yes I do think the casino would profit in the long run even if each individual wager was slightly -EV for the casino.

The limited bankroll/ gambling budget of players contributes to casino profits because of the variance distribution "never" causing the casino to go broke while players run out of money and have to quit.

Anyway interesting thoughts.

1. The Kelly criterion maximizes the growth of the final bankroll, so if you have an infinite bankroll, it’s not applicable. In the case of a casino, both the casino and the players (since there are infinitely many of them) have infinite bankrolls.

2. If variance is lower while EV stays the same, then according to the Kelly criterion, we could invest a higher percentage of our bankroll. Therefore, "run it twice" reduces variance, which means we would need fewer buy-ins to play at a given limit. So, a player who uses "run it twice" would, in theory, move up to the next limit faster and also increase what you refer to as linear EV.

Just for the "what if," it does seem possible for the casino to profit from -EV bets given a specific betting pattern from patrons and assuming the casino had an infinite bankroll (in reality they don't).

For a more extreme example assume guests bet 100% of their individual bankrolls on a bet with 51% equity to double up or go broke. However they continue "letting it ride," continuously betting 100% of their bankroll until they go broke.

If the casino truly had an infinite bankroll and the guests had to continue betting their entire bankroll indefinitely, then the casino could not lose even though each individual bet is -EV.

In reality though casinos do have an effective bankroll, it's just much higher than most individuals.

I still think the effective bankrolls/budgets of guests likely increases a casino's profit.

For example suppose a guest has a $100 gambling budget for the night. Over the course of time they will lose due to the casino's edge.

But even if there was 0 edge for the casino, the $100 will go up and down. However even if they're having a great night and they're up to $1,000 they can then go on a losing streak and lose it all back. However as soon as they hit -$100 they are effectively out of action and can no longer benefit from the positive side of variance.

So it seems that a hypothetical casino with 0 edge on bets but an infinite bankroll could still make money due to guests being more likely over time to quit when they're down than to quit when they're up.

In that example, you can view all the players in the casino as a single player with a very large bankroll. The fact that one player loses their roll doesn't matter, because another one takes their place and the game continues. So, if anyone in that example has an infinite roll, it's the customers, not the casino. Just imagine if a casino actually offered a +EV bet and unlimited stakes—it would be out of business in a day.

So you would contend that in my 0% edge example the result would converge on 0, with no profit or loss for the casino? Maybe add in an additional caveat that there is an effective max bet for customers.

Surely the distribution of wins and losses amidst the effective bankrolls plays some roll?

Maybe it's best to go back to the original OP's example. If a player uses a -EV betting strategy with their bankroll (negative growth rate) while placing individual+ EV bets there are scenarios where they are basically "guaranteed" to lose.

Surely if they are losing than their opponent (the casino) is profiting from that loss even if theoretically the individual bets favored the player?

Mathematical nitpick: careful when talking about infinite bankrolls.

A casino with an infinite bankroll does not make any money. If it wins a bet, its bankroll is still infinite, and if it loses, its bankroll is still infinite.

All bankrolls are finite. You can’t just assume infinite bankrolls to simplify computations, or you get some absurd stuff.

What do you think about the OP's question 1? Maybe instead of infinity substitute some humongous number that it is effectively like infinity against a player using a negative logarithmic betting pattern.

I think this sort of mental exercise can shed a lot of light on wealth maximization/ bankroll management strategies.

The OP's example is actually fairly closely analogous to a situation where a good poker player has poor bankroll management habits. A good player with an edge can bet too aggressively and basically guarantee they go broke as the logarithmic function of their wagers becomes negative.

They will just move up in stakes until they lose it all, rinse and repeat.

I once heard Matt Berkey give an interesting perspective on bankroll management. He advocated that newer players who had not "made it" yet use very aggressive bankroll management (maybe betting according to Kelly criterion to maximize gains, assuming they can get a job and start over if they go broke).

Then he advocated becoming increasingly conservative as someone's level of wealth increases (not sure he practices what he preaches, lol).

Anyway this makes sense given the diminishing returns inherent in the value of money. There's a phase when wealth building is the greatest comcern, especially since at lower wealth levels the cost of basic living can exert a strong downward influence on someone's finances. Once you achieve a certain level of wealth conserving your purchasing power will take precedent over maximizing gains.

Anyway I'm interested to read other perspectives.

Don't know why this is funny. The answer is unironically yes, but of course the reason why is because you used the term "infinite bankroll," and also because you are forcing each player to play ad infinitum, using a fixed bankroll percentage, and not allowing them to stop when they wish. These things are actually super important and can't be glossed over.

The source of the apparent paradox is that Kelly Criterion betting assumes you bet a fixed percentage of your bankroll, and keep that percentage fixed as your bankroll changes. When we bet 5% of our bankroll on this 51-49 game each time, what happens is that when we lose, our bankroll and our bet size decrease so much that we don't make back enough to cover the initial loss. This is in stark contrast to the more intuitive strategy of picking one bet size based on our initial bankroll and then sticking with that constant bet size throughout the repeated game. If we do that, then we will eventually win as long as we don't go broke first--which brings us to bankroll considerations.

"Bankroll edge" is a real thing. Let's use a simpler example that is easier to understand. Let's say I own a casino and you are trying to beat me at this 51-49 game using a Martingale strategy. You come to the casino and you plan to win $1 at a time by starting with a $1 bet and doubling it each time you lose until you eventually win.

Is this a guaranteed win for you? The answer is no, and the reason the answer is no is because you have a finite amount of money. If you go on a losing streak that is so bad that you exhaust your bankroll, you are done and you lose a ton. But as long as this losing streak does not manifest, you will indeed win $1 at a time.

So the question of whether you will win long-term is actually a question of how many times I can pay you $1 while waiting for you to bust out. It's purely a bankroll question, and NOT an EV question. The smaller my bankroll, the more likely you will bust me $1 at a time; and if my bankroll is large enough, I can force you to bust out with probability approaching 1.

This Kelly problem is very similar. If you're betting too much of your bankroll each time, then long-term it will exponentially decay away. The only question is whether my bankroll allows me to wait this out.

By the way, poker players love to talk about Kelly criterion, but in researching this topic just now I learned that this is NOT what actual poker players do. If you are a poker player you do not get to tailor your buy-in or bets to your exact bankroll size. You pick one level and you play the same level over and over (unless you move up/down discretely, not continuously as in the Kelly conditions). This actually allows arithmetic repeated bet sizing which is not the same as when you're talking about Kelly strategy.

These are really good questions!

Yes lol. But only because there's no upper stopping point. Essentially what happens at the limit is that you have an infinitesimally small chance of the player going on a crazy hot streak that ruins the house. But that never happens with an infinite bankroll. That's why Vernon was comparing it to martingale strategy.

Heisemberg gave a good answer imo. Lowering variance increases your Kelly fraction so you can move up stakes faster. You can achieve the same thing buying in short-stacked or selling action.

"Beyond optimal Kelly" wording is problematic. You need to use 3-outcome Kelly formula to incorporate chops. RIT doesn't increase beyond optimal Kelly, rather optimal Kelly bet increases.

I mean you can still utilize Kelly as a poker player though. For example, if you compare the Kelly growth rate for different stakes you can decide when to move up or down by always choosing the highest growth. And you can sell action to fine tune your risk.

I would guess the most sensible way to utilize Kelly is as a warning for when NOT to play stakes that are so high that too much of your roll is at risk. But there is also a question of not knowing how big your edge is when you move up, which means that when trying to move up you may wind up totally ignoring Kelly and being a bit more aggressive.

One thing most people seem to forget when calculating bankroll requirements is cashflow -- including monthly withdrawals, taxes and potential for new deposits.

If you make withdrawals from winnings (including taxes), your risk of going on a downswing is now increased.

If you have another source of income and can easily replenish your bankroll if needed, you risk is decreased and you can be more aggressive with your bankroll.

i found out that most people don't know the maths concept of infinity

1st is just martingale, but no casino has infinite bankroll. In fact casinos are extremely careful with their bankrolls. It's a paradox of martingale that with infinity it is a +ev strategy
2nd question, optimal Kelly fraction is a function of volatility. In fact if you imagine a situation with 0 volatility (an arbitrage) the optimal kelly fraction is 100% of your bankroll, which is the exact same as arithmetic growth. Geometric growth is always lower than arithmetic growth due to volatility. The function of Kelly is to bring geometric growth as close to arithmetic growth as possible, and this in turn is modulated by volatility of returns. So the more you lower your standard deviation, the higher the optimal Kelly bet is. Which is to say, all else equal, you always want to minimize volatility (as long as you're not giving up any arithmetic EV in the process). So yes, running it twice is free volatility reduction, thus optimal for compound growth.

Now there's some very big caveats to this. Namely that you never actually know what your arithmetic EV is. It changes with your mood, with your opponents, with your relative seat at the table, with the stakes you are playing for etc. So you have to be really really careful with optimizing around Kelly bets, especially when dealing with very volatile or very small returns (as is often the case in poker). The margin for error is extremely slim in such cases and you can easily overbet your edge.

BUT all this being said, it is VERY rare that any player actually uses a correct bankroll number. The reason for this is that when you say bankroll in a Kelly sense, that means everything. Your entire net worth + whatever money you expect to make in the future discounted to the present. So as you see your real Kelly bankroll is much much larger than what you have deposited on poker sites.

It seems like the bots would have to know each other's bankrolls. If you're well rolled for a given stake and a winning villain is not, you'd obviously want to run it once in the interest of your long term profitability. Increasing a skilled player's variance and thus the likelihood of him either quitting or moving down would have to make the game softer.