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Subject: Short-Handed Holdem Variance
From: Michael Maurer
Date: 14 Sep 94 21:54:22 GMT
Newsgroups: rec.gambling
In mhall@netcom.com (Michael Hall) responds
to Darse and Winner777:
>It sounds like you are saying that the variance is higher when
>there are more players in the game. I disagree.
>Short-handed hold 'em is a hair-raising high variance game. Imagine
>it heads up. You get the blinds much more often (every hand), and you
>have to play far more loosely and aggressively and frequently than you
>would at a full table, because otherwise your opponent will eat up your
>blinds. At a full table, you can sit around and wait for premium hands.
>Short-handed, you have to raise with that anemic A2 offsuit and worse.
My intuition was the same as yours, but then I analyzed the IRC poker
database and found it to be wrong. Surprisingly, short-handed holdem is
less volatile than a full game.
You actually hint at the reason yourself: in a full game, you must sit
and wait around for premium hands. These hands often win, but not
always, and it takes many hours to reach the long turn. In contrast, a
short-handed game forces you to play a higher fraction of your hands,
and because of the small number of players you are dealt more hands per
hour. Although each hand may be a crapshoot compared to the premium
hands you play in the full game, the law of large numbers soon comes to
your rescue.
To back up my position with figures, assume for the moment that IRC
poker provides a good estimate of bankroll variance in Texas Holdem.
(Personally, I think that despite the unrealistic nature of IRC poker
this is one of the few comparisons to real life that is valid.) IRC
blind structure is 0.5-1 small bets, betting structure 1-1-2-2. The
following table shows the bankroll standard deviation per hand (averaged
over the entire IRC player population) as a function of the number of
players in the game. I estimate standard deviation per hour based on
some reasonable number of hands dealt.
IRC Holdem Results
Players Hands in Action/ Std.Dev./ Hands/ Action/ Std.Dev./
in Game Sample Hand Hand Hour Hour Hour
======= ======== ======= ========= ====== ======= =========
2 87346 2.64 4.24 60 158.8 32.84
3 110966 2.75 4.44 50 137.5 31.40
4 138517 2.60 4.68 45 117.3 31.39
5 146584 2.46 4.97 40 98.4 31.44
6 138741 2.37 5.24 38 90.1 32.30
7 122491 2.34 5.66 35 81.8 33.51
8 98655 2.29 5.92 32 73.5 33.53
9 72414 2.36 6.51 29 68.6 35.05
10 46520 2.38 7.02 26 61.8 35.82
11 28644 2.41 7.44 23 55.5 35.68
12 17808 2.54 8.20 20 50.7 36.69
In the table, action is the average number of small bets one puts in the
pot, and std.dev. is the standard deviation of one's bankroll in units
of small bets. Note that my choice of Hands/Hour causes the
Std.Dev/Hour to be almost flat, meaning that on an hourly basis one's
variance is independent of game size. (The sample size is in number of
pockets dealt, so there were 87346/2 2-handed games and 17808/12
12-handed games. A very small number of hands may be missing.) The
Action/Hand column is also surprisingly flat; apparently at a full table
one puts more bets into the pot on those rare hands one chooses to play.
Just for comparison, here are similar results for Omaha-8-or-better.
Note the increased action and std. dev., making Omaha volatile relative
to Holdem in all game sizes. Again, I taka a guess at Hands/Hour being
2/3 that of Holdem.
IRC Omaha Results
Players Hands in Action/ Std.Dev./ Hands/ Action/ Std.Dev./
in Game Sample Hand Hand Hour Hour Hour
======= ======== ======= ========= ====== ======= =========
2 11672 4.39 5.61 40 175.6 35.48
3 14012 4.54 5.64 33 151.3 32.57
4 16945 4.33 6.29 30 129.9 34.47
5 17730 4.16 6.78 27 111.0 35.01
6 15861 4.20 7.47 25 106.5 37.63
7 13377 4.14 7.79 23 96.6 37.63
8 9760 4.24 8.91 21 90.4 41.17
9 6804 4.28 9.30 19 82.8 40.89
10 3590 4.26 9.49 17 73.8 39.51
11 2409 4.24 10.16 15 65.0 39.79
In both cases, the standard deviation per hour is 30 to 40 small bets,
or $90 to $120 for $3-6 holdem. That means that swings of $200 in a
4-hour session should be commonplace, and indeed they are.
On a hand-by-hand basis, short-handed play is surprisingly less volatile
than full-table play; typically the variance is reduced by a factor of
4. But on an hourly basis, one shouldn't notice much difference between
short-handed and full-table play. Do people's real-life experiences
(not just perceptions) support this prediction?
-Michael
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______________________________________________________________________
Michael Maurer maurer@nova.stanford.edu (415) 723-1024