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 Maurer is no longer current.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

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)
--
______________________________________________________________________
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

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