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Message from discussion PokerStars tournament all-in pot equities v. pots won

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More options May 18 2003, 6:05 pm
Newsgroups: rec.gambling.poker
From: "Steve Brecher" <s...@my.signature.at.end>
Date: Sun, 18 May 2003 14:37:46 -0700
Local: Sun, May 18 2003 5:37 pm
Subject: PokerStars tournament all-in pot equities v. pots won
Using emailed hand histories, I tabulated the results of 1675 pots in
PokerStars tournaments in which betting ended prior to the river because
players in the pot were all-in, and in which none of the all-in players was
drawing dead.  The great majority of the tournaments were multi-table; a few
were two-table, and fewer still single-table.  There were a total of 3420
all-in hands (usually two per pot, but in a few instances three or even
four).

For each all-in, I collected the player's equity in the pot at the time of
the all-in and the proportion of the pot the player won (1.0 for a win, 0.0
for a loss, 0.5 for a two-way split, etc.)  The pot equity of a hand is the
average proportion of the pot that a player would win over the long run.
For example, in the following sample Hold 'Em Showdown output,

990 pots with board cards: Tc 9d 7s

9h9s         8d8s
% chance of outright win     74.040404    24.343434
% chance of win or split     75.656566    25.959596
expected return, % of pot    74.848485    25.151515
fair pot odds:1               0.336032     2.975904
pots won:                       741.00       249.00

--the "expected return, % of pot" result shows the pot equity of each hand.

The chi square statistic does not indicate significant variance of actual
from expected (3392.7 on 3421 hands).  A plot of rolling 30-"bin" averages
of sorted equity vs. pots won looks linear; the linear regression is
y = 0.9908x + 0.0045
which is not significantly different from the theoretical y = 1.0x + 0 that
describes the "no dealing bias" case.

Looking at the "big" favorites/underdogs, and expressing the favorite's
chances as odds against the underdog:

Odds at least      # of hands     Avg equity    Avg pot won
2:1                1121           0.8123        0.8134
3:1                 753           0.8617        0.8597
4:1                 623           0.8806        0.8777
9:1                 260           0.9383        0.9321

In sum, I see no evidence that "bad beats" are more frequent than expected.

(I am grateful to Paul Pudaite for analytical guidance, but any errors are
my own.)

--
For mail, please use my surname where indicated:
st...@surname.reno.nv.us (Steve Brecher)