More fun with the HUSNG bankroll simulator

[TNixon]

After the recent bankroll discussion, I went ahead and modified my simulator to try to take some sort of learning period into account. The results are interesting, but pretty much what I expected to see.

Just as a refresher, previous simulator runs worked with the assumptions that you start out with $100, are a 58% winner at every level up to 50, never play higher than $50 games no matter how big your bankroll gets, and play 1000 games at the highest level you currently have the required number of buyins for. Here's the value comparison between working with 10 buyins and 30 buyins under those assumptions:

Trying 10000 times over 1000 games with $100.00. SU: 10.00 SD: 10.00
Survived 9942 times with 4058.91 avg remaining
4035.365807 equity avg exit: 439 latest: 925

Trying 10000 times over 1000 games with $100.00. SU: 30.00 SD: 30.00
Survived 9999 times with 810.26 avg remaining
810.181242 equity avg exit: 333 latest: 333

So by waiting for 30 buyins before moving up, you basically have cost yourself $3200, for a half of a percent better chance of making it through 1k games.

I modified the algorithm so that it uses the following assumptions:

#1: You start out as a 50% winner at $2 and $5 games, a 45% winner at $10 and $20 games, and a 40% winner at $30 and $50 games.
#2: Each game you play at a particular level will slowly increase your winrate, so that after 100 games at a level, you will be a 58% winner at that level, at which point your winrate for the level caps.
#3: You can start at a level appropriate to your bankroll, but you are not allowed to advance to the next level until your winrate caps (after 100 games).

Please note that I feel all of these assumptions are *extremely* conservative, but using these assumptions, here are the numbers using the same bankroll rules:

Trying 10000 times over 1000 games with $100.00. SU: 10.00 SD: 10.00
Survived 9592 times with 530.49 avg remaining
508.845355 equity avg exit: 409 latest: 975
Trying 10000 times over 1000 games with $100.00. SU: 30.00 SD: 30.00
Survived 9993 times with 340.45 avg remaining
340.214472 equity avg exit: 419 latest: 547

The penalty is still there, but it's much less dramatic. You're trading 33% of your potential ending bankroll for a 4% better chance of not going bust. Small changes to either the capped winrate or the learning rate have very dramatic effects though. For example, using 60% as the cap instead of 58%:

Trying 10000 times over 1000 games with $100.00. SU: 10.00 SD: 10.00
Survived 9886 times with 1181.01 avg remaining
1167.544565 equity avg exit: 292 latest: 963
Trying 10000 times over 1000 games with $100.00. SU: 30.00 SD: 30.00
Survived 9999 times with 541.97 avg remaining
541.919008 equity avg exit: 317 latest: 317

In this case, you're trading over half of your potential bankroll for an extra 1% chance of survival.

Leaving the win cap at 58%, but cutting the learning rate in half (meaning you can figure out how to be a 58% winner in 50 games rather than 100) has a similar impact:

Trying 10000 times over 1000 games with $100.00. SU: 10.00 SD: 10.00
Survived 9804 times with 1569.99 avg remaining
1539.219503 equity avg exit: 449 latest: 990
Trying 10000 times over 1000 games with $100.00. SU: 30.00 SD: 30.00
Survived 9994 times with 447.09 avg remaining
446.825677 equity avg exit: 472 latest: 629

Trading 70% of your bankroll for about a 2% better survival rate.

Making both changes, meaning you can figure out how to be a 60% winner in 50 games (and he're we're probably straying into the optimistic, but I think it's still useful as a point of comparison):

Trying 10000 times over 1000 games with $100.00. SU: 10.00 SD: 10.00
Survived 9965 times with 3241.79 avg remaining
3230.445720 equity avg exit: 273 latest: 910
Trying 10000 times over 1000 games with $100.00. SU: 30.00 SD: 30.00
Survived 9999 times with 804.97 avg remaining
804.890084 equity avg exit: 276 latest: 276

Which is very close to the original value penalty with an assumed flat winrate of 58%.



Another bit of interest, using a 58% winrate and a 100 game learning period:

Trying 10000 times over 1000 games with $100.00. SU: 30.00 SD: 20.00
Survived 9995 times with 374.96 avg remaining
374.773244 equity avg exit: 511 latest: 909

The SU and SD numbers are step-up and step-down buyin rules. In this case, the numbers mean step up to a level you have 30 buyins for, and step back down if you fall to 20 buyins for the level. The end result here is higher than the flat "play the highest level you have 30 buyins for" strategy (which gave a result of $340), meaning that even if you ignore the psychology of the situation, there is some statistical truth to the conventional wisdom that says you shouldn't step up until you have enough buyins to be able to play a few games at the level before stepping back down. So with $1500, you'd step up to $50 games, and then step back down if you drop to $1k. But I'm not trying to buck *all* conventional wisdom about bankrolls, after all. Just the parts that don't make sense. [img]/images/graemlins/smile.gif[/img]

If you're playing very agressively with your bankroll, however, you have to discard that bit of conventional wisdom, and be ready to step back down after just one game if you lose, because your chances of busting increase enough to offset the gain from being able to play more games at a level after stepping up:

Trying 10000 times over 1000 games with $100.00. SU: 10.00 SD: 7.00
Survived 8797 times with 560.47 avg remaining
493.049621 equity avg exit: 397 latest: 995

The average finishing bankroll is higher, but equity drops, because of the added chance of failure. And to be honest, my aces get cracked often enough that even though I'm willing to play my bankroll pretty agressively, even I want to have a better than 87% chance of surviving over 1000 games. [img]/images/graemlins/smile.gif[/img]

VERY EXTREMELY SUPER-IMPORTANT NOTE: The agressive bankroll strategy that I'm implicitly recommending by posting these statistics ONLY applies if you're looking for a strategy to build your bankroll. If you're paying bills out of your bankroll, and need to keep it at a consistent level or lose your house, you absolutely MUST play more conservatively than 10 or 20 buyins. You simply cannot afford to spend a significant amount of time playing at a lower level. In that case, 75 or 100 might not be completely out of line if all you play is HUSNGs, and that's a significant and important part of your income.

Of course, at the end of the day, your comfort level has a far greater impact than any statistic I could possibly generate, because statistics won't affect your play, but your comfort level absolutely will. I just want to try to make sure anybody who cares, knows just how much value they're passing up by waiting for 30 buyins to step-up.

[MasterLJ]

Not that it's very scientific, but I can corroborate that there isn't much difference between a 10 BI and 30 BI roll.

I started playing online poker close to 5 years ago... I'm not going to lie and I've freely admitted it to SSNL and others, I probably put $2k+ into the poker economy before I really learned how to be a winning player. I was a winning player, but had bad tilt and bankroll management.

That being said, I rebuilt rolls from $50 to $100 into several grand more times than I care to admit. Never once did I have a BR of more than 20 BIs, and I almost always played HU SNGs.

[BennyMac]

I don't play HUSNGs, but this is an excellent post. You might consider other tournament types if you haven't already, particularly full table sit and go's.

[PrimordialAA]

TRNixon, your my hero ;p, after our little discussion / debate before I converted to your strat and my BR skyrocketed ;p, just from $6 games to $220s in 7 days, am currently at $4k for and from HU only [img]/images/graemlins/smile.gif[/img], I mean i've logged like 4-5k HUs, so I knew the game and was always a winner so the strat fits me, but you def. made me realize how much potential profit I was losing by being overly-nitty and i've shown some of my HU friends that same wisdom, so thx, great post [img]/images/graemlins/smile.gif[/img]

[omgwtfnoway]

there's an article about kelly criterion in the 2p2 magazine this month. kelly criterion is a way to invest your bankroll for maximum roi.
f*=(bp-q)/b
b = odds paid on the bet (0.913 to 1 for a turbo nltrn on stars)
p = winrate
q = 1 - winrate
f* = the proportion of your bankroll each buyin should represent

unless i screwed the math up (more than likely), a 60% winner has f*=16.2% or 6.2 buyins.

[ChicagoRy]

Time to play the 1100s then.

[MasterLJ]

I wish we could get some simulators for cash games =).

There was a post in SSNL where someone used R to create a winrate model, which was cool. Only thing I remember is that at 1 million hands, your win rate is still +/- 2 PTBB/100 from your "true" winrate.