With BRM you of course do maximise the chance that you will actually make money long term and wont go broke.

But from a pure maxev standpoint, wouldnt it make sense to put literally all you have on the line on something that has the biggest edge?

This applies to investing too. Not putting all your eggs into one basket. But again wouldnt the maxev option be to put all your money into the one basket that has the highest expected return, rather than diversifying into 'worse' assets?

Is there a theory that shows why bankroll management/diversification is rational?

You wouldn't put all your money in lottery tickets even if the grand prize was a billion dollars, would you?

Your statement about maximising your edge would be correct if you never went broke.

What is ev? EV is defined as the weighted average of all outcomes. With proper bankroll management, the probability you go broke is near 0, so your ev will be similar to your winrate. if you win at 5 cents/100, you can expect to make $500 after a million hands, give or take.

By moving up in stakes, lets say you make 14 cents/100. But because you're not properly rolled, you go broke 80% of the time. Your EV after a million hands will be ($1400 * 20%) + (0 * 80%), which is actually lower at $280. Being properly rolled decreases your variance, which stops you from going broke.

Your assumption of highest expected return in regards to investments is also incorrect. For example, if Tesla had an expected return of, say 10% per year but with a risk (variance) of 20%, your risk reward ratio would be 0.5 (this is oversimplified). If the S&P 500 returns 6% but only has a risk of 6%, your absolute return is lower, but your risk reward ratio more than doubles.

With any risk tolerance, it's actually better to buy the basket with the best risk-return ratio. You want 6% returns? S&P risks 6% of your portfolio. If you want 10% return? Don't buy tesla. You borrow money to double your investment in the S&P, returning maybe 10% after interest but still only risking 12%.

To tie this back into poker, if you're playing online, it might be better to multitable rather than move up in stakes underrolled. This spreads out your risk since you're playing more hands, rather than playing with more money in each hand.

https://en.wikipedia.org/wiki/Kelly_criterion

Essentially you want to maximize your long term bankroll growth.

Imagine an opportunity where you can bet a variable amount to double up with a 55% probability and you can repeat this bet any number of times until you run out of money or reach 100k total winnings. You have 100$ in your pocket to play with.

If you want to maximize the expected $ after a single bet then putting the entire 100$ in at once is correct. But that's a terrible choice if you want to maximize your overall winnings, any reasonable strategy will have an EV close to 100k here. Risking all at once will frequently leave you locked out of making additional +EV bets.

Great post Fish+Chips.

From your investment example, the best decision is then to not to go all in on the investment (or bet) with the highest expected return (say, tesla) but to put everything on the investment with the highest risk/reward ratio?

What I dont understand is why the risk, when measured as variance or std dev, is this important.

Because the variance can work both ways, you can go broke below the ev of the investment, but also for example make double than the ev of the investment because of the variance.

Shouldnt the net result of variance therefore be net 0? Because what can happen on the negative side (below ev)can also happen on the positive on the positive side (above ev).

With the poker example you mentioned, where we have an 80% risk of ruin due to the variance, shouldnt we also adjust for the event where we run significantly above the expected 14 cents/100, also due to the higher variance?

The higher variance and more reckless BRM with ev of 14 cent /100, should go both ways, so wouldnt that keep our expected EV after x hands the same?

So therefore the only difference that a higher variance would make, is that it would take more hands to play before we can expect our actual winrate/100 to be around our EV winrate of 14 cent per 100?


Edit very interesting plexiq, will read that thoroughly

The net result of variance is only 0 if you're properly bankrolled. If you never go broke, then it makes sense that eventually variance will balance itself out.

If you don't have a proper bankroll though, it's possible that you go broke, which means you don't have a chance to recover from bad variance. this is why in my example I calculated EV to be $280: because you either make $1400 in the long term assuming you don't go broke (no variance in the long term), but there's an 80% chance you go broke, which means you can't recover.

In terms of investing, to many people, the extra upside is less important than safety from their savings losing value. On top of that, the companies most likely to go bankrupt are the riskier ones, and when you go bankrupt you lose the chance for good variance to even out the bad.

Many people look at returns before looking at risk, when you should really determine your risk profile first, then find the investment that provides the maximum return for your level of risk. Some (presumably younger) people can stomach a 20% standard deviation, while people near retirement can only handle 6%. This is because as you get closer to retirement, your investment horizon becomes short term instead of long term.

For our young person in the example, leveraging 3x the S&P would give you 18% (minus some interest) for 20% of risk, whereas Tesla would only give you 10%.

One thing I remember reading on another poker forum (RIO?) is that no BRM will save you if you´re a loser in the game. This is one o the main building blocks o my nitty approach to BRM

Anyway, I´d rather think of having 2 cushions, sizes depending on your personal preferences: learning BR, that you´d be ok losing in order to learn the game (preferably you´ll put the effort and will learn the game quickly enough to not lose much) and variance BR.

If you know you´re a winner in the stakes, and is just rebuilding or doing a BR challenge, than aggro obv always better. If you can redeposit, no need to have a BRM either.

It depends on how the betting scenario is constructed. If you aren't applying betting limits, betting it all actually will have a higher EV than using good bankroll management, mainly because the equation bakes in those ultra rare times you keep doubling your money and become a super-gazillionaire. Since the difference between being a super-gallionaire and just a regular gazillionaire probably isn't relevant to you (decreasing marginal utility), you should be willing to sacrifice EV (by betting smaller) in order to reduce the likelihood of the much more relevant scenario of going broke or not.

With betting limits, whether its how much you're allowed to bet or how much you can possibly win, there is no longer a gazillionaire possibility so no real justification at all for betting allin all the time. Using BRM is just mathematically better.

Literally all great posts.

Glad to actually know the theory now behind why BRM is important long term.

The kelly criterion is actually very cool too.

Thanks Fish+, plexiq, fazendeiro and mcnasty

One thing that I think was left out is opportunity cost. In the hypothetical example, we have identified a great game or investment that is high EV but carries risk and uncertainty. The OP proposes that investing as much as possible is the best decision (I realize he might not think this anymore). Yet, this one opportunity might not be as good as some future opportunities. In investing and poker, a massive loss can prevent us from making future investments either by limiting available funds, or because we are out of the game permanently.

This is what risk of ruin and the Kelly Criterion are about in my opinion. If we +EV opportunities, we can’t let an individual bad outcome ruin future good bets.

^ Yeah I also think that's a very important part.

The example that plexiq gave was actually experimented in a study, where bettors had an edge, but still a huge amount of them went broke by betting too big, and thus missed out on winning any money with that +ev opportunity. Really interesting read.

Its under 'experiment' on the kelly criterion wiki page

Let's say you are at $1000 bankroll. If you lose 500, you move down stakes. But to move up you need to win 1000. So If I were to offer you a coinflip bet for $500 it'd be wise to turn it down

This thread is old but if you want to make it more quantitative the issue is utility of money. All insurance is -ev in dollar terms, but still it's almost always worth it because we reduce risk. Basically the theory is twofold, but both prongs discourage risk:

1. Money won is worth less than money lost
2. The first dollar won is worth more than the nth dollar won

This is because money isn't just pieces of paper or a number in your bank account, you use it to buy things. The dollars you use to pay the rent and buy food are worth more than money you use to save, buy luxury items, or whatever you do with the extra money you'd have once you have enough to afford necessities. Even if that money allows you to leverage higher ev investments (aka higher stakes games), it still is going to be worth less than the money you use to put food on your table.

So yes, if the long-term objective is maximize the number of dollars you have, and you're allowed to go negative, bankroll management is -ev. But the reality is going to 0 or negative has disasterous consequences and going from having a 10k net worth to a $0 net worth is drastically different from going from a 200k net worth to a 190k net worth. That's why you would make drastically different bankroll decisions if you have 10k to your name compared to if you have 200k to your name.