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PokerStars tournament all-in pot equities v. pots won

Steve Brecher May 18, 2003 2:37 PM
Posted in group:
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

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: (Steve Brecher)