I'm not sure where the profit = EV - a * variance formula was taken, and what is meant by variance there in the first place.

But anyway, if profit > EV for a player, it means by definition that s/he is risk-seeking, which implies that it makes sense for him/her to play -EV games and still 'profit' mentally. So I don't think that pro poker players are or should be risk-seeking - they're known to naturally detest -EV games. Neither do I think that they should artificially change their attitude to lower their risk aversion ('be less nitty') - it will eventually lead to a huge inner dissonance if they happen to lose a lot.

What they can do is controllably play LAG in order to increase the EV, but only if the variance is small enough and only if their profit - in terms of their inner preferences - becomes bigger this way.

Let's correct the formula, though.

If we use

the rigorous definition of variance and

the logarithmic utility function (i.e. if a person is indifferent to the choice between, say, being granted 20% of their net worth and a 50/50 flip between a zero reward and a reward of 44% of their net worth, as (log(1) + log(1.44))/2 = log(1.2) - note that the flip is worth less than 44%/2=22% that it would be worth to someone risk-neutral, so this person is risk-averse), then we arrive at the following approximate formula:

profit = EV - (0.5 * variance / networth)
(The formula is accurate only for games with payouts that are not much bigger than the amount risked, i.e. for cash games and small non-jackpot SnGs; for MTTs and jackpot SnGs, it significantly underestimates the risk-adjusted profit.)

That's why there's BRM - if someone is playing underrolled, i.e. has too low a net worth for the stake, then the second component is too large and drags the risk-aversion-adjusted profit into the minus, or at least makes it suboptimal.

In the general case of a person having an

isoelastic utility function, i.e. constant relative risk aversion 'rho', the formula becomes

profit = EV - (0.5 * rho * variance / networth)
Most people are thought to have rho ~ 1.

Also, rho usually depends on the amount being risked - people can be risk-seeking (rho<0) when buying a cheap lottery ticket and risk-averse (rho>0) when their livelihood is at stake.

Poker tables, imo, are a place where risk-seeking (recreational) players meet risk-averse (professional) ones, enabling a mutually beneficial games where recs transfer a bit of EV to regs, trading it for increased variance and entertainment. All-reg tables are hence unsustainable because they lower overall happiness.

The amount of EV that recs are willing to forfeit to get increased variance is so big that it allows poker rooms (that aren't really exposed to as big variance as regs - they're very rich) to leech on this partnership of regs and recs and take away a significant part of this EV as rake / fees

But there's little that regs can do with it - they're bad at organising games on their own (well, some do hold home games, but most of online ones prefer to have someone bring them the fish, whereas they can sit in their rooms in underwear and grind).

To conclude, let's derive the classical Kelly BRM formula, the caveat being that poker games become tougher as stakes go up, so it can't be applied directly to poker, except one case - selling action without markup or buying action with or without markup.

Let 'k' denote the share (between 0=0% and 1=100%) that a player leaves to him-/herself (and sells the rest).

His/her expectation then becomes k*EV, and the variance becomes k*k*variance.

So we have to solve the optimisation problem

profit = k*EV - (0.5*rho*k*k*variance/networth) -> max
Finding a maximum of a quadratic function is a high-school problem, the answer being

k_optimal = (1/rho) * networth * (EV / variance)
(or k_optimal=1 if the right-hand side is bigger than 1, as obviously hardly anyone will be interested in you selling yourself short; in this case, you should consider moving up).

The classical Kelly formula corresponds to rho=1. The famous 'half-Kelly' approach assumes that rho=2 (hence the 1/rho=1/2 multiplier).

If we define the optimal bankroll as the minimum net worth for which k_optimal=1, then we get

bankroll_optimal = rho * variance / EV
Example:

100 hands of microstakes NLHE; estimated EV = 0.02*BI [2 bb/100 winrate

plus cashback minus life expenses - please be cautious in winrate assumptions, you might be just running good; rather, look at how good your skill is in comparison with other players]; standard devation (trackers have such a stat, it measures the sq. root of the variance) std_dev = 80 bb = 0.8*BI; variance = std_dev * std_dev = (0.8*BI)*(0.8*BI). Assume rho=1.

bankroll = variance/EV = (0.8*BI)*(0.8*BI) / (0.02*BI) = 32 BI

So if you have, say, 16 BIs for the stake you wish to grind (e.g. it gives you additional VIP benefits as opposed to grinding a limit lower 100% on your won), then sell 50% of the action (* i.e. with stakers giving you only 50% of BIs in SnGs or covering only 50% of your losses in cash games *) or arrange another kind of staking deal.

*** Please note that Kelly BRM assumes that you immediately move down or up stakes when your bankroll / net worth changes, so it's advised to have a few more BIs because you can't track bankroll changes during a session ***

Let me stop here for now as posting is addictive I need to grind at least a bit

I'll expand in a while by an explanation of the Chen-Ankenman 'cutoff bankroll formula' for moving up/down in the poker environment, where there's a discrete ladder of stakes and we have to make decisions like 'either NL50 or NL100' when we'd rather play NL75, and also the Sharpe ratio and its application to staking.