Note: The contact information shown for Dr. Maurer is no longer current.
If necessary, he may be reached at: mjmaurer@yahoo.com .
The player "rayzor" is Darse Billings.
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Subject: Re: Variance for 3-6 hold'em -- IRC statistics
From: Michael Maurer
Date: 13 Jul 94 04:22:27 GMT
Newsgroups: rec.gambling
In sakthi@netcom.com (Sakthi Nayakkar) writes:
>Can some kind soul explain me what is variance.
>And what it is for a typical 3-6 hold'em.
A look at the IRC poker database might hint at the answer to the second
question. I'll stick to the lowest limit IRC game since that is the
loosest and most like 3-6 cardroom holdem.
First, some boring math.
Although everybody loves to shout the word "variance" in mixed company,
it's not a very intuitive quantity because it measures the square of
your bankroll's fluctuations. Also, it's only meaningful when compared
to the average increase (or decrease, heaven forfend) in your bankroll
over the same period of time. Better quantities to look at are the
square root of the variance (the "standard deviation") and the
relationship of this number to your "expectation" or average bankroll
increase. I'm sure that statisticians have a fancy name for this ratio,
but as a typical ignorant engineer I'll just use it cause it works.
For example, if you average a profit of 0.2 small bets per hand in the
long run, and your standard deviation is 6 small bets per hand, it's
going to take a lot of hands for your small edge to show through. [Geek
mode on.] In fact, for you to be 90% sure to walk away a winner you'll
have to play approximately 1500 hands, or 500 hours! In general, you'll
need to play about
N = (1.3 * S / E)^2
hands, where S and E are the standard deviation and expectation per
hand, respectively. If you want 99% good feeling, change 1.3 to 2.3,
and play over three times as many hands.
Now to the good stuff. What is a realistic value for this ratio?
Without poker log book reports from real life players, the best source
of information we have is the IRC poker database. First let's look at
the results achieved by the players who make money the fastest, that is
those players with the highest expectation. I'll use their real
nicknames since one of them is mine :). The fact that I appear on this
list is our first clue that it probably has no relevance for real world
play!
These statistics are compiled from all those who have played at least
1000 hands with 7 or more players each on #holdem (a total of 58
qualifying players dealt 106,611 sets of pocket cards).
E S S/E
Player N hands N win Action profit return SB/hand std/hand std/mean
maurer 1358 136 2330.2 688.2 1.295 0.507 6.216 12.266
rayzor 1907 231 3740.0 956.2 1.256 0.501 6.752 13.467
andrea 3517 543 10648.7 1703.9 1.160 0.484 8.291 17.114
jims 2398 254 4022.4 1129.9 1.281 0.471 5.685 12.066
walter 1255 145 2216.0 558.0 1.252 0.445 5.932 13.341
why2 1046 158 3036.0 408.4 1.135 0.390 8.385 21.475
spiney 1986 262 4879.9 710.8 1.146 0.358 7.006 19.576
rodney 1206 90 1657.5 422.5 1.255 0.350 5.940 16.956
baguette 1435 186 3295.7 495.6 1.150 0.345 7.072 20.477
Nater 3048 291 5015.7 969.3 1.193 0.318 5.788 18.201
The quantities S and E are in here, as is the ratio S/E. (See end of
post for short description of other columns.) Note that a typical value
for S/E is about 15. This means that these players will have to play
about 400 hands to be 90% sure of a win. A single 4-hour session might
have 120 hands, and about 75% of such sessions will be profitable.
(This actually sounds better than real life to me.)
To translate into dollar terms, a player with E=0.4 should average
0.4*120 = 48 small bets per session = $144 profit. Wow! However, if
S/E=15 there is still a 25% chance of walking away a loser. Of course,
in real life we have to pay time charges and dealer tokes of, say, 1/6
small bets per hand. Since this comes right off the top of E while S is
unaffected, one's chances of a losing session are increased. In this
example, E = 0.40-0.17 = 0.23, and S/E nearly doubles to 6/0.23 = 26.
Our more realistic hero has a 33% chance of losing each session.
Now let's look at the ten least successful players. I've changed their
names to prevent embarrassment of their friends and family.
E S |S/E|
Player N hands N win Action profit return SB/hand std/hand std/mean
loser10 1254 130 2669.3 -132.2 0.950 -0.105 5.897 55.933
loser9 1044 174 3443.7 -116.5 0.966 -0.112 7.172 64.270
loser8 1079 141 2994.7 -127.0 0.958 -0.118 6.817 57.918
loser7 1564 170 3317.6 -186.4 0.944 -0.119 5.861 49.174
loser6 1340 139 3224.7 -230.6 0.928 -0.172 6.533 37.963
loser5 2090 319 7895.6 -511.7 0.935 -0.245 9.031 36.886
loser4 1144 170 3878.2 -310.7 0.920 -0.272 8.194 30.170
loser3 1718 118 2669.5 -479.0 0.821 -0.279 4.852 17.402
loser2 3348 414 8651.2 -1099.8 0.873 -0.328 6.292 19.153
loser1 1084 159 3627.5 -381.3 0.895 -0.352 7.363 20.933
Although the values of S/E appear larger, note that the magnitude of E
is generally smaller (they aren't _that_ bad!) and S is typically the
same. The larger value of S/E means it will take more than 400 hands
for them to be sure of sustaining a loss. For example, "loser7" with
S/E=49 can play 100 hands and still win 42% of the time.
What does it all mean? I'd say it just confirms what we all know
already: good players can play a long time and still lose, and bad
players can do the same and win. And that's why we all love the game!
-Michael
P.S. Here is a short description of the columns I neglected to mention.
N hands number of hands in which you were dealt cards
N win number of hands in which you won some or all of the pot
Action total # of small bets made (put into the pot)
profit # of small bets taken out of the pot in excess of your Action
return return on your investment = (action+profit)/action
SB/hand average small bets/hand profit = profit / N hands
std/hand bankroll standard deviation per hand (small = rock,
large = maniac)
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______________________________________________________________________
Michael Maurer maurer@nova.stanford.edu (415) 723-1024
Subject: Re: Variance for 3-6 hold'em -- IRC statistics
From: Michael Maurer
Date: 14 Jul 94 01:01:43 GMT
Newsgroups: rec.gambling
In I wrote
>In fact, for you to be 90% sure to walk away a winner you'll
>have to play approximately 1500 hands, or 500 hours!
As I'm sure you all noticed but were too polite to point out, 1500 hands
takes only about 50 hours, not 500! Duh.
--
______________________________________________________________________
Michael Maurer maurer@nova.stanford.edu (415) 723-1024