NLHE bankroll issues (math majors apply within)

Two things:

1) Is the conventional wisdom that one should have 20X the buy-in for a specific NLHE cash game level to feel well staked? (e.g. $20K for $5/$10 blinds, $1000 max buy-in with approx 5% chance of going bust). Is there mathematical evidence that this rule of thumb is roughly equivalent to the recommendation of having 300X the big bet for LHE games? (this all assumes a "winning player")

2) For the sake of argument let's assume that 20X the max. buy-in bankroll for NLHE cash games is sufficient. If one is multi-tabling games at the same level (e.g. 4 tables at $5/$10 blinds/$1000 max buy-in) then how much more of a bankroll would be needed to have the same level of comfort that one only has a 5% chance of going broke. It seems to me that playing multiple tables, again assuming a winning player, would decrease variance because of the evening out of streaks, bad beats, etc. If this is true then obviously a player would not need an $80K bankroll, in this example, to have the same 5% chance of going broke while playing 4 tables. I'm also assuming no degredation of play due to multi-tabling and less attention paid to each table.

Since I am not a math whiz I can only articulate this problem and am hoping that those with good math skills can help with the answer.

[Sponger15SB]

Try posting this in the Probability forum as well.

[TwoNiner]

The only thing you'd have to worry about with multi-tabling is your expected winrate decreasing. If you could play 4 tables individually with the same winrate you would possess playing 1 table at a time then there is no adjustment needed for multi-tabling. If 20k was your bankroll for 1 table, then use it for 4 tabling. In reality you won't win as much per hand multi-tabling, and you will most likely be a little more tilt prone and you'll have to adjust a little bankroll wise, I'd think. As to the specific requirements I'll leave that up to others to answer.

[whitelime]

B = -(sigma^2/2u)ln(r)

r = exp(-2uB/sigma^2)

where u is your BB/100
sigma is your SD/100
r is your desired risk of ruin
B is your bankroll

Here are some approximate numbers:

A solid winrate in a high stakes limit game is about 1.5BB/100

SD is around 15

A solid winrate for a high stakes NL game is 6BB/100

SD is around 50

[whitelime]

ROR BR WR SD
81.87% 50 5 50
67.03% 100 5 50
54.88% 150 5 50
44.93% 200 5 50
36.79% 250 5 50
30.12% 300 5 50
24.66% 350 5 50
20.19% 400 5 50
16.53% 450 5 50
13.53% 500 5 50
11.08% 550 5 50
9.07% 600 5 50
7.43% 650 5 50
6.08% 700 5 50
4.98% 750 5 50
4.08% 800 5 50
3.34% 850 5 50
2.73% 900 5 50
2.24% 950 5 50
1.83% 1000 5 50
1.50% 1050 5 50
1.23% 1100 5 50
1.01% 1150 5 50
0.82% 1200 5 50
0.67% 1250 5 50
0.55% 1300 5 50
0.45% 1350 5 50
0.37% 1400 5 50
0.30% 1450 5 50
0.25% 1500 5 50
0.20% 1550 5 50
0.17% 1600 5 50
0.14% 1650 5 50
0.11% 1700 5 50
0.09% 1750 5 50
0.07% 1800 5 50
0.06% 1850 5 50
0.05% 1900 5 50
0.04% 1950 5 50
0.03% 2000 5 50

[whitelime]

RoR BR WR SD
79.11% 20 1.5 16
62.58% 40 1.5 16
49.50% 60 1.5 16
39.16% 80 1.5 16
30.98% 100 1.5 16
24.51% 120 1.5 16
19.39% 140 1.5 16
15.34% 160 1.5 16
12.13% 180 1.5 16
9.60% 200 1.5 16
7.59% 220 1.5 16
6.01% 240 1.5 16
4.75% 260 1.5 16
3.76% 280 1.5 16
2.97% 300 1.5 16
2.35% 320 1.5 16
1.86% 340 1.5 16
1.47% 360 1.5 16
1.16% 380 1.5 16
0.92% 400 1.5 16
0.73% 420 1.5 16
0.58% 440 1.5 16
0.46% 460 1.5 16
0.36% 480 1.5 16
0.29% 500 1.5 16
0.23% 520 1.5 16
0.18% 540 1.5 16
0.14% 560 1.5 16
0.11% 580 1.5 16
0.09% 600 1.5 16
0.07% 620 1.5 16
0.06% 640 1.5 16
0.04% 660 1.5 16
0.03% 680 1.5 16
0.03% 700 1.5 16
0.02% 720 1.5 16
0.02% 740 1.5 16
0.01% 760 1.5 16
0.01% 780 1.5 16
0.01% 800 1.5 16

[ QUOTE ]
B = -(sigma^2/2u)ln(r)

r = exp(-2uB/sigma^2)

where u is your BB/100
sigma is your SD/100
r is your desired risk of ruin
B is your bankroll

Here are some approximate numbers:

A solid winrate in a high stakes limit game is about 1.5BB/100

SD is around 15

A solid winrate for a high stakes NL game is 6BB/100

SD is around 50

[/ QUOTE ]

The problem with the standard RoR calculations though is that they assume you never cash-out. Hence the more you win the less risk you have. Reality for most people is that you're going to cash out regularly. I haven't worked out how to factor this in with the above formula, but the other way to do this is to simulate your range of results by setting up a macro in excel. You can then factor in the impact of cashing out at a certain level to see how likely it is that you go bust or need to drop down limits.

Upshot is it increases your RoR very significantly if you expect to always cash-out once you get to a certain level.

[Matt Flynn]

You have the cash out problem, which can be mitigated by dropping down in limits when losing.

Multitabling has the problem of magnifying tilt. This is a very serious problem. If you multitable and have a tendency to tilt, you need more than 20 buyins.

I have better stats than posted for NL and have had about an 18 buyin downswing. And I turtle when losing hard, meaning I buyin for less and get very averse to playing in profitable but not good games.

If you cannot replace the money, will take out the money as you win, are averse to or embarrassed by dropping down such that you are likely to push your bankroll when losing, play a loose style, play shorthanded, occasionally have significant tilt events and a few other situations, you need a lot more than 20 buyins. Playing shorthanded and playing a loose style dramatically increase bankroll needs. If you are a pro and bankroll is your only toolkit, the only way 20 is enough is if you drop to half the limit when you hit 10. Better to have 30 and drop at 20, and far better to just have 40 and not worry about it (unless you play HU and three-handed aggressive games - then 40 is not enough).

All these discussions of 20 buyins depend on longhanded poker with tight play and a decent winrate.

Matt

[SunOfBeach]

the math has been well presented to you, as has some great justification by matt flynn on the hows and whys of varying from the 'standard' 20*buyin number. that being said, if/when you develop enough hands of data to have some idea of your own winrate and SD, you can then decide how much risk of ruin youre willing to accept, and then proceed from there. and as matt said, i virtually guarantee that most people are either looser, not as good, play shorthanded more, or are more risk averse than the 'standard' person who would find that the 20*buyin number is sufficient. examine your own winrate and SD and see what works for YOU, and YOUR style of play, in the games that YOU play.

these guidelines can be a disaster - its not a cookie cutter... how many buyins do you think gus hansen would need with his style of play, vis-a-vis david grey... even if they were playing in the same game?

Solid guideline.

[Marlow]

Great post, Matt. I'm in a nasty downswing now and this helps me understand some of the implications for my BR needs.