All-In Equity Analysis

Mar 27th, 2010 | Posted by spadebidder

Here we’ll look at preflop all-in hands with a single caller, such that no subsequent decisions are possible in the hand.   These tests will analyze the distribution of the community cards by comparing the preflop equity of the hole cards to the actual outcomes when the community cards have been dealt.  This test is sometimes called an “All-In Luck Analysis” when used for the purposes of comparing a particular player’s expected win rate when all-in preflop, to his actual outcome. In our test we’ll be making this comparison for a sample that includes hands from many players aggregated, and breaking down the distribution of the hands and the outcomes.

The expectation or equity for each player is determined by summing up the number of times that player would win if all 1,712,304 possible boards are dealt out, adding half the number of ties to the wins and then dividing that sum by 1,712,304. This is the calculation used by all Hold’em equity calculators, such as Poker Stove, and it represents the  expected win rate if an infinite number of all-in hands were played with those particular hole cards.  We know that at multi-player tables the community cards will have small biases due to card removal effects as discussed in the Flop Analysis section.  But in a 2-player game there can never be dead cards and so an all-in analysis should be pure and without bias.  We’ll examine both game types here.

In the first set of tests our “players” will be composite players that we’ll call Favorite and Underdog in one test and Big Stack and Small Stack in another test. Hands will be put into those categories just the way the names would imply. We’ll also be dividing up the hands by stakes played, as well as showing the combined result for all stakes.

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This is just a test run on a sample of ~575K hands (stopped when we hit 10K preflop all-ins), which will be replaced with a much larger one and more analysis. We’ll also be comparing full ring to heads-up, as heads-up has zero card removal effects, and full ring should expose card removal effects.

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Site A, 9 player table with 7 minimum active, NL Hold'em, all stakes combined
Using 25 equity bins.
Number of hands analysed: 574,556
Preflop heads-up all-ins: 10000 (Once per 57 hands).

FAVORITE'S
Preflop equity  ExpW% ExpT%/2  TotExp    Hands    ActW% ActT%/2  TotAct  #StDev

[0.98 - 1.00]    .       .       .           0     .       .       .
[0.96 - 0.98]    .       .       .           0     .       .       .
[0.94 - 0.96]  93.37%   0.71%  94.08%        4   75.00%   0.00%  75.00%   -1.67
[0.92 - 0.94]  92.40%   0.66%  93.05%      261   92.34%   0.38%  92.72%   -0.22
[0.90 - 0.92]  90.76%   0.62%  91.38%       47   97.87%   1.06%  98.94%    1.88
[0.88 - 0.90]  88.00%   0.66%  88.65%       34   82.35%   1.47%  83.82%   -0.90
[0.86 - 0.88]  86.76%   0.50%  87.26%      214   84.58%   1.40%  85.98%   -0.57
[0.84 - 0.86]  84.82%   0.31%  85.13%       79   87.34%   0.00%  87.34%    0.56
[0.82 - 0.84]  82.25%   0.29%  82.54%      330   82.73%   0.30%  83.03%    0.24
[0.80 - 0.82]  81.06%   0.23%  81.29%     2013   80.97%   0.20%  81.17%   -0.14
[0.78 - 0.80]  78.95%   0.23%  79.18%       81   80.25%   0.62%  80.86%    0.37
[0.76 - 0.78]  76.29%   0.61%  76.90%       34   70.59%   0.00%  70.59%   -0.88
[0.74 - 0.76]  72.86%   1.88%  74.74%      550   71.64%   2.36%  74.00%   -0.41
[0.72 - 0.74]  71.08%   1.98%  73.06%      446   69.28%   1.68%  70.96%   -1.02
[0.70 - 0.72]  70.20%   0.93%  71.12%      740   70.14%   0.88%  71.01%   -0.07
[0.68 - 0.70]  67.93%   1.25%  69.18%      671   67.66%   1.42%  69.08%   -0.06
[0.66 - 0.68]  66.51%   0.73%  67.24%      332   68.07%   0.75%  68.83%    0.62
[0.64 - 0.66]  64.54%   0.62%  65.16%      334   63.77%   0.60%  64.37%   -0.31
[0.62 - 0.64]  62.03%   0.84%  62.87%      201   65.67%   0.25%  65.92%    0.90
[0.60 - 0.62]  60.39%   0.52%  60.91%      223   63.23%   0.45%  63.68%    0.85
[0.58 - 0.60]  58.35%   0.77%  59.13%      220   53.18%   0.68%  53.86%   -1.60
[0.56 - 0.58]  56.53%   0.32%  56.85%      990   56.06%   0.25%  56.31%   -0.34
[0.54 - 0.56]  54.66%   0.23%  54.89%      870   53.79%   0.17%  53.97%   -0.55
[0.52 - 0.54]  46.14%   6.93%  53.07%      787   44.60%   6.86%  51.46%   -0.98
[0.50 - 0.52]  37.14%  13.76%  50.90%      382   36.91%  13.87%  50.79%   -0.05

[    ALL    ]  67.26%   1.68%  68.94%     9843   66.85%   1.68%  68.53%   -0.90

* With 23 valid bins, expected deviations > 5% signif = 1.2.  Observed = 0
Chi-Sq for Fav =  3.69 w/ 22 DF. Crit nums (.10 .05 .01) 30.81   33.92   40.29

UNDERDOG'S
Preflop equity  ExpW% ExpT%/2  TotExp    Hands    ActW% ActT%/2  TotAct  #StDev

[0.48 - 0.50]  35.35%  13.76%  49.10%      382   35.34%  13.87%  49.21%    0.05
[0.46 - 0.48]  40.00%   6.93%  46.93%      787   41.68%   6.86%  48.54%    0.98
[0.44 - 0.46]  44.88%   0.23%  45.11%      870   45.86%   0.17%  46.03%    0.55
[0.42 - 0.44]  42.83%   0.32%  43.15%      990   43.43%   0.25%  43.69%    0.34
[0.40 - 0.42]  40.10%   0.77%  40.87%      220   45.45%   0.68%  46.14%    1.60
[0.38 - 0.40]  38.57%   0.52%  39.09%      223   35.87%   0.45%  36.32%   -0.85
[0.36 - 0.38]  36.29%   0.84%  37.13%      201   33.83%   0.25%  34.08%   -0.90
[0.34 - 0.36]  34.22%   0.62%  34.84%      334   35.03%   0.60%  35.63%    0.31
[0.32 - 0.34]  32.03%   0.73%  32.76%      332   30.42%   0.75%  31.17%   -0.62
[0.30 - 0.32]  29.57%   1.25%  30.82%      671   29.51%   1.42%  30.92%    0.06
[0.28 - 0.30]  27.95%   0.93%  28.88%      740   28.11%   0.88%  28.99%    0.07
[0.26 - 0.28]  24.97%   1.98%  26.94%      446   27.35%   1.68%  29.04%    1.02
[0.24 - 0.26]  23.39%   1.88%  25.26%      550   23.64%   2.36%  26.00%    0.41
[0.22 - 0.24]  22.49%   0.61%  23.10%       34   29.41%   0.00%  29.41%    0.88
[0.20 - 0.22]  20.59%   0.23%  20.82%       81   18.52%   0.62%  19.14%   -0.37
[0.18 - 0.20]  18.47%   0.23%  18.71%     2013   18.63%   0.20%  18.83%    0.14
[0.16 - 0.18]  17.17%   0.29%  17.46%      330   16.67%   0.30%  16.97%   -0.24
[0.14 - 0.16]  14.55%   0.31%  14.87%       79   12.66%   0.00%  12.66%   -0.56
[0.12 - 0.14]  12.25%   0.50%  12.74%      214   12.62%   1.40%  14.02%    0.57
[0.10 - 0.12]  10.69%   0.66%  11.35%       34   14.71%   1.47%  16.18%    0.90
[0.08 - 0.10]   8.00%   0.62%   8.62%       47    0.00%   1.06%   1.06%   -1.88
[0.06 - 0.08]   6.29%   0.66%   6.95%      261    6.90%   0.38%   7.28%    0.22
[0.04 - 0.06]   5.21%   0.71%   5.92%        4   25.00%   0.00%  25.00%    1.67
[0.02 - 0.04]    .       .       .           0     .       .       .
[0.00 - 0.02]    .       .       .           0     .       .       .   

[    ALL    ]  29.38%   1.68%  31.06%     9843   29.79%   1.68%  31.47%    0.90

* With 23 valid bins, expected deviations > 5% signif = 1.2.  Observed = 0
Chi-Sq for Dog = 10.08 w/ 21 DF. Crit nums (.10 .05 .01) 29.62   32.67   38.93

NO FAVORITE HANDS
Preflop equity  ExpW% ExpT%/2  TotExp    Hands    ActW% ActT%/2  TotAct  #StDev

[exact 50/50]   2.93%  47.07%  50.00%      157    3.82%  46.18%  50.00%

The 50/50 line assumes an even split of the wins between 2 hypothetical opponents
as we have no way to split these results between two specific players or between
the favorite and underdog counts, since this is an aggregate of many players.

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So this sample of 10,000 all-in hands is just to show one of the approaches being used to analyze all-in hands. This post will be updated with more data soon.

 

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