# thoughts on risk reward concept, can variance be profitable?

• Bronze
Joined: 26.09.2014
I am referring to this strategy article:
http://www.pokerstrategy.com/strategy/others/1649/1/

So the article states the following equation:

profit = EV – a * variance

And that "a" is determined by the player's psychological robustness and his bankroll management.
And first I wondered why variance would be a factor in expected profits at all, since variance can mean being lucky as well as being unlucky. So it can augment or diminish your profits equally and these effects should cancel each other out over the long run. Except of course that you have to pay rake when reaping the positive side of variance, but not getting rake back when losing (as opposed to trading in the stock market where you can harvest your tax losses and use them to reduce your tax burden on your other profitable investments).
And of course variance might give you a false idea of your strength as a player, making you think you're good, when you just might have gotten lucky or making you think, you're winning strategy is flawed, when you just hit a downswing. Or making you think you've just gotten lucky, when you've been playing really well and so on.
So in that way variance works against everybody and against everyone's profits by tilting people and feeding them false information.
But doesn't this in turn give you an opportunity to do something better than your opponent? In the article they say that a good bankroll management lowers your "a"-value and so does your attitude.
And further that LAG's accept the higher variance for an increased EV (if they are skillful enough to play LAG with +EV or rather with a higher EV than they could play TAG).

So my thinking goes, that if you were to be a LAG like that and would never tilt (or at least less and less severely than your average opponent) and you would do your stat homework after every session, then wouldn't your "a"-value be negative and variance be a proftible element for you. Because then: profits= EV - (negative a)*variance ----> profits = EV + a * variance

So shouldn't you as an aspiring poker player embrace variance, embrace the bad beat, because dealing with it successfully can give you an edge and increase your profits? Thoughts? Did I get one of the premises wrong? Am I crazy, here?
• 10 replies
• Bronze
Joined: 13.08.2009
That's a really interesting perspective. I think that your reasoning that "opponents will make costly mistakes against you due to variance affecting their mindset, but you will not" is correct. But in the model it is explained by a = 0 (ie your mindset is neutral to variance), but our EV is higher.

I think the conclusion should be not that we should intentionally seek higher variance for variance sake, but that we can play profitably in more marginal situations (that inherently have higher variance) because we have EV > 0 purely because our opponent is more prone to making mistakes. If our "a" was greater than zero, there would be fewer situations like this where we could play profitably because we would also be making mistakes.

Does that make sense?
• Super Moderator
Super Moderator
Joined: 02.09.2010
Originally posted by JonDavid
So it can augment or diminish your profits equally and these effects should cancel each other out over the long run. Except of course that you have to pay rake when reaping the positive side of variance, but not getting rake back when losing (as opposed to trading in the stock market where you can harvest your tax losses and use them to reduce your tax burden on your other profitable investments).
Hi, JonDavid,
Welcome to PokerStrategy.com!

The quoted portion is quite accurate, except that all poker rooms have some sort of rewards scheme whereby you can earn back a portion of rake paid.

Have a look at this video:
http://www.pokerstrategy.com/video/25155/

To boil it down:
If you have a positive win rate due to your skill, there will still be runs (sometimes long ones) where you win far less than expected-- the "downswings". There will be stretches when you win more. Over a vast number of hands, these even out, leaving you with your winnings.

Sadly the side on which the video was based is now gone, but this one looks familiar.
[Edited by VorpalF2F: I had to remove the link as the target might be construed as a competing site. If I do this again, I'll have to give myself a stern warning.]
(just google "poker variance simulator" -- there are a few of 'em.)

Note that the general trend line us up, but take a moment to look at the worst case scenario. Over 100K hands the poor bugger got slaughtered.

The danger in knowing all of this is the tendency to blame "variance" when you as a player have unplugged leaks on your game.

There is a tendency to bleed cash while blaming "bad beats", "variance" and a host of other excuses. The only cure is to increase your winrate through study, and to keep your analytical dial set to "self-critical"

Originally posted by JonDavid
So shouldn't you as an aspiring poker player embrace variance, embrace the bad beat, because dealing with it successfully can give you an edge and increase your profits? Thoughts? Did I get one of the premises wrong? Am I crazy, here?
Yes indeed. If you see someone do something outright stupid, make a note.
The next time you play that person, you can use it against him. Note however that the better players occasionally get "caught" doing "something stupid" on purpose.

Let the (Mind) Games begin!

Cheers,
--VS
• Bronze
Joined: 19.04.2010
Welcome to the forum JonDavid

Good thinking there and I think you are correct: we should try to embrace the variance and let it have it's negative effect on our opponents.

I think it was Phil Ivey who told he would often quit opponents pretty quickly if they managed to win a bunch of money early in the match. The reason for that is that many players tend to play better when they're winning (maybe they're more relaxed and can think more straight) and worse when they're losing. Therefore he tried to find a new opponent if the first one was winning and was then ready to play long sessions if he himself won early in the match. I think this is legitimate reasoning and I can relate to that in both ways: I do see my opponents play worse if the lose a lot at first and I guess I don't play my best either if I'm the one losing early.

Not sure if it was Ivey but I think it was. Anyways, if you're better coping with variance, the short term results affect your game less and thus you gain an edge against your opponents.
• Super Moderator
Super Moderator
Joined: 02.09.2010
Originally posted by HuhtalaJ
The reason for that is that many players tend to play better when they're winning (maybe they're more relaxed and can think more straight) and worse when they're losing.
Interestingly, I tend to the opposite.
I have two typical session scenarios:
1  I build up a decent stack with small pots, some non-showdown wins or maybe even I stack somebody. I then piss it away over the next hour or two. I will quit before I drop below 0.
2  I drop a stack early, then over the next hour or so I gain it all back (and more sometimes).

I rarely have a purely losing or winning session.

Phil Ivey can play me any time under scenario #1

Best of luck,
--VS
• Super Moderator
Super Moderator
Joined: 02.09.2010
Here is a poker variance simulator that doesn't contravene the rules:
http://pokerdope.com/poker-variance-calculator/

Enjoy,
--VS
• Bronze
Joined: 19.04.2010
Originally posted by VorpalF2F
Interestingly, I tend to the opposite.
I have two typical session scenarios:
1  I build up a decent stack with small pots, some non-showdown wins or maybe even I stack somebody. I then piss it away over the next hour or two. I will quit before I drop below 0.
2  I drop a stack early, then over the next hour or so I gain it all back (and more sometimes).
I think you're not the only one, winner's tilt is common and some players need to lose first to get the urge to focus and win it back. Now that I think more about it, maybe the concept that I described applies better to heads-up games because they're more personal. When losing there, you lose to one opponent per table and by experience I see a lot more loser's than winner's "tilt" there. In ring games of multiple opponents losing or winning isn't as personal as it is in HU.

Although, what you described is again a negative effect of variance that gives an edge to the one who is less affected
• Bronze
Joined: 26.09.2014
@Zukes Yes, I think we shouldn't seek higher variance intentionally for variance's sake. But that according to Harrington we should seek higher variance/marginal hands, because that's where more non-obvious +EV potential lies for a skillful player, who does his math really well.
That an equally skilled opponent is fazed by variance, when our example player is not would be an advantage for him. We could could of course say that further increases the EV of our ideal player. Or we could say, that increases our profits, because the direct reason is not the technical analysis of our player in one situation, but the factor of variance itself. So we could continue to use the formula profits = EV - a * variance
In a way that is just semantics I suppose, but if we have a negative a value, we should theoretically be able to play some hands profitably that have negative EV (profitable in the long run against a player, not directly profitable in the same hand).
That seems paradoxical, but I think it aligns very well with the -EV-deception play mentioned by VorpalF2F "advanced players getting caught doing something 'stupid' " bringing you profit by masking your true ability or feeding the opponent false information or giving him potential "implied tilt odds" if you win in an obvious -EV-situation.

Mindgames indeed.

@HuhtalaJ
So even opponents at Phil Ivey's level have to win first before they play their best game?
Well, that is encouraging

@VorpalF2F
thanks for the link, definitely using this as a tool to determine my bankroll for my next attempt at profitable poker
• Silver
Joined: 11.01.2009
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.
• Bronze
Joined: 26.09.2014
I think variance here is not defined as ((p[outcome1]*outcome1)²+...+(p[outcome n)* outcome n)² but rather as some kind of constant. Maybe the factor should rather be called "monetary damage of variance" constant. And the whole right term then would be the monetary damage of variance. And the cofactor "a" scales negatively with brm and mental fortitude and positively with range. That was my premise, maybe I was misrepresenting it. Maybe I'll read "Harrington on Cashgames" someday and it's explained in more detail there.

The formula does not take into account the enjoyment derived from gambling for the fishes. Profit is defined here in purely monetary terms. In your example the "rec-reg" relationship is indeed mutually profitable, but profit is equated with utility (or happiness ).

I'm afraid I'll have to have my brother explain the logarhythmic utility and the isoelastic utility function to me tomorrow, before I can comprehend more of your post. But please keep the posts coming. I can't wait to understand them, though it might take me a while.