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One common theory of how online poker could be rigged is the **'Bad
Beat Theory'**. In 'bad beat theory' good players receive
more 'bad beats' than they should and poor players 'get
lucky' too often when all the chips go in.

It has been argued that a site rigged in this way could increase its profit because the 'fish' would lose their money less quickly, the 'sharks' would win more slowly (and cash-out less often) and more hands would be played.

In this experiment we put PokerStars to the test to see if good players receive more 'bad beats' than they should...

- Hypotheses
- Explanation
- Dataset
- Method
- Results
- Analysis
- Results of Analysis
- Conclusion
- Limitations & Discussion

**Null Hypothesis:** Pokerstars is fair. The cards are
dealt randomly.

**Alternate Hypothesis:** Pokerstars is rigged. The
community cards are biased to favour losing players in 'all-in'
situations.

In this test we anlaysed the results of 'heads up, pre-flop
all-in' hands, i.e. hands in which exactly 2 players were 'all-in'
before the flop. We **compared** the **'Expected
number of hands won'** with the** 'Actual Number of hands
won'** over a large sample size. We also compared the **
'Expected Equity'** (the percentage a hand should win if played
an infinite number of times) with the **'Actual Equity'**
(the percentage the hands actually won).

**If our null hypothesis was true (i.e. Pokerstars is fair):
the 'Expected Equity' would have been very close to the 'Actual Equity' for a very
large sample of hands. **

Since we wished to compare results for winning and losing players it wa necessary to make some assumptions:

Assumption1: good players more often than not get 'their chips in ahead', i.e. their hands will (on average) have an 'Expected Equity' of greater than 50%. Also, poor players more often than not get 'their chips in bad', i.e. their hands will (on average) have an 'Expected Equity' of less than 50%.

Assumption2: a good player's hand will more often than not 'dominate' their opponent's hand than vice-versa in pre-flop all-ins, i.e. their hands will have an 'Expected Equity' of between 68% and 83% more often than they will have an 'Expected Equity' of between 17% and 32%. For more information on 'dominated hands' click here.

Over a large sample if our alternate hypothesis and assumptions were correct:

**1) Hands with an 'Expected Equity' of greater than 50% would
have an 'Actual Equity' which is less than their 'Expected Equity'.
Also, hands with an 'Expected Equity' of less than 50% would have an
'Actual Equity' which is greater than their 'Expected Equity'. **

**2) Hands with an 'Expected Equity' of 68-83% would have an
'Actual Equity' which is less than their 'Expected Equity'. Also, hands
with an 'Expected Equity' of 17-32% would have an 'Actual Equity' which
is greater than their 'Expected Equity'. **

For this test we used **1 million hands** from $3+0.40,
9 player, standard, 'Sit and Go' tournaments played at Pokerstars. The
hand histories were purchased from
PTR* and should
consist of at least 95% of hands played in these tournaments between 5th
April 2011 and 1st May 2011.

Single table **'Sit and Gos' were chosen because these
tournaments have a high proportion of 'pre-flop all-ins'** and
therefore they provide a large sample of hands for analysis. Also,
unlike 'cash games' or 'deep stack' tournaments the results of these
tournaments tend to be largely determined by 'pre-flop all-ins'.

$3 tournaments were chosen because the hand histories were more reasonably priced than higher buy-in 'Sit and Gos' and an almost complete sample of hand histories could be obtained for hands played over a one month period (this was not the case with $1 'SnGs').

Top of Page ♠The hand histories were imported into Poker Tracker* and we used the custom report feature to filter and sort the data. Many thanks to 'White Rider' and 'Kraada' at Poker Tracker who provided excellent support and invaluable knowledge during this process.

The data was filtered to remove all hands that were not**
'heads up, preflop all-ins'**. The remaining hands were filtered
by 'all-in call', i.e. **the hands would be viewed from the
perspective of of the player that called the 'all-in' **bet and
all duplicate hands from other players' perspectives were removed.

The outputs from the custom report were set as:

**Hand I.D.**: the unique reference number that
Pokerstars gives each hand. These were counted to produce the 'Total
Number of Hands'.

**Date:** the date the hand was played.

**Player:** the screen name of the player that called
the 'all-in' bet.

**Hole Cards: **the pocket cards of the 'all-in caller'.

**Expected All-in Equity:** this is expressed as a
percentage, i.e. it is the probability of the caller winning the hand
(p) multiplied by 100. This value is calculated by Poker Tracker using a
'Monte Carlo' method and therefore there are slight errors associated
with each figure, for more details see the limitations and discussion
section.

**Winner:** The screen name of the player that won
the hand.

**Actual Result:** if the hand was won value of 1 was
produced and if the hand was lost a value of 0 was produced, by default
split pots were recorded as won and therefore received a value of 1.
These were summed to give the **'Total Number of Hands Won
(including split pots)'**.

**Split Pots:** if a hand resulted in a split pot a
value of 1 was produced. These were summed to give the** 'Total
Number of Split Pots'.**

The outputs were set to be ordered by 'Expected All-in Equity' so that the hands with the lowest expected all-in equity would appear at the top of the list running down to the hands with the highest expected all-in equity.

**If you want to run this analysis on your own tournament hand
histories you can download the custom report by clicking on the icon
below and then importing into
Poker Tracker* as normal:**

Upon import it was found there were errors when importing a small number of hand histories and there were also a small amount of duplicate hands. These accounted for at most a few thousand of the original 1 million hands and had a negligible effect on sample size.

The report was run and 69750 'heads-up, pre-flop all-ins' were
produced. 88 of these hands were missing the output for 'Expected All-in
Equity' (probably due to a 'bug' in Poker Tracker's equity calculating
program) so **the total number of usable 'All-in' hands produced
from the original sample of 1 million hands was 69663**.

These hands were exported to 5 spreadsheets for analysis. The first was left unchanged and comprised all 69663 hands output from the report. The rest were divided into hands that were 'ahead' preflop, 'behind' preflop, 'dominating' and 'dominated' hands:

**Number of 'ahead' hands (>50% expected all-in equity) = 36101
**

**Number of 'behind' hands (<50% expected all-in equity) =
33524 **

**Number of 'dominating' hands (68%<expected all-in equity<83%)
= 14766 **

**Number of 'dominated' hands (17%<expected all-in equity<32%)
= 12241 **

To view the full results in PDF format click on an icon below (NOTE: the summary is at the bottom of the last page of each spreadsheet).

It is worth noting that the number of 'ahead' hands is significantly greater than the number of 'behind' hands and also that the number of 'dominating' hands is greater than 'dominated' hands. This can be explained by the facts that:

(a) we originally filtered our hands by 'all-in' **caller**,
(i.e. the hand is viewed from the perspective of the player that called
the 'all-in') and

(b) that a significant proportion of players use the 'gap concept' when making 'all-in' decisions. Put simply, the 'gap concept' is the idea that it requires a stronger hand to call an 'all-in' bet than it does to be the initial bettor because the initial bettor has some 'folding equity' while the caller has none.

When viewing the spreadsheets be aware of the formatting of hole cards. Each hole card has a number which represent the card: 2c = 1, 3c = 2, 4c= 3... Ac = 13, 2d = 14, 3d = 15... Ad = 26, 2h = 27... Ah = 39, 2s = 40... As = 52).

Top of Page ♠On each spreadsheet another column was added:

**p(1-p)** where p is the probability of the caller
winning the hand. This value was calculated from the all-in equity and
was summed to give:**∑[p(1-p)]**.

The following outputs were used in calculations:

**Total number of hands, n**

**Number of hands won (incl. split pots), w **

**Number of split pots, s **

The following calculations were carried out in order to compare the actual number of hands won to the expected number of hands won:

**The mean expected equity, x (%)** was calculated
by summing the value of 'expected all-in equity' for every hand and
dividing the total by the number of hands, n.

**Actual (effective) number of hands won, z = w - (s/2).
**It was necessary to adjust the number of hands won to take into
consideration the number of split pots. Since all the hands were
'heads-up' split-pots were considered to have an 'actual equity' of 0.5
when compared with a value of 1 for a hand that was won and 0 for a hand
that was lost. The number of hands won already contained a value of 1
for every split pot and therefore the 'effective' number of hands won
was calculated using the formula shown.

**Expected number of hands won, e = xn/100 **was calculated
in order that this could then be compared to the actual number of hands
won.

**Actual deviation = z-e.** The deviation of the actual
number of hands won from the expected number was calculated by simply
subtracting one from the other.

**Standard Deviation = √∑[p(1-p)]**. To
see if the actual deviation from the expected results was within
reasonable limits the standard deviation of the population was
calculated. In order to achieve this it was assumed that the population
behaved as a binomial distribution. In reality the population is an
imperfect binomial distribution since the probability of success, p,
varied for each hand. In a perfect binomial distribution the
"probability of success of each event, p must be the same for each
trial". For more on this see this discussion.

**Actual mean equity, a = (z/n)100** was also calculated
so that it could be compared to the expected mean equity.

To view the full results in PDF format click on an icon below (NOTE: the summary is at the bottom of the last page of each spreadsheet):

Below is a table of the results for each of the data groups:

Data Group | Equity Range (%) | Total No. Hands | Expected No. Hands Won | Actual No. Hands Won | Actual Deviation | Standard Deviation | Mean Expected Equity | Actual Mean Equity |
---|---|---|---|---|---|---|---|---|

All Hands | 0-100 | 69663 | 35748 | 35850 | +102 | 122 | 51.32 | 51.46 |

Ahead | 50-100 | 36101 | 24002 | 24093 | +91 | 88 | 66.49 | 66.74 |

Behind | 0-50 | 33524 | 11726 | 11738.5 | +12 | 85 | 34.98 | 35.02 |

Dominating | 68-83 | 14766 | 10967 | 11041 | +74 | 53 | 74.27 | 74.77 |

Dominated | 17-32 | 12241 | 3165 | 3181 | +16 | 48 | 25.86 | 25.99 |

From these results we can see that the actual number of hands won is
very close to the expected number of hands won for every data group.
**In all cases the actual deviation is well within 2 standard
deviations** and there are no discrepancies between the 'ahead'
and 'behind' samples or the 'dominating' and 'dominated' samples.

Also, since hands were filtered by 'all-in' caller, the 'ALL HANDS' data group has tested for biases between players that originally made an all-in bet and players that 'called' an all-in bet. The results show no bias in this respect either.

Top of Page ♠If we return to our hypotheses: The evidence shows that our alternate hypothesis is incorrect for this sample of hands, i.e. the community cards are not biased to favour losing players in all-in situations. Also, it is a fair assumption that this sample of hands is representitive of the general population of hands played at $3 'Sit and Go' tournaments at Pokerstars at the present time.

**We can therefore conclude that for $3 'Sit and Go's' our null
hypothesis is correct and that Pokerstars is fair (with respect to 'bad
beats') at the present time. **

This test was performed on a specific game type ($3 SnGs) during a
specific period (April 2011) at a specific poker site (PokerStars) and
the results can be considered true for these conditions only.
Although these results are relevant to online poker in general they
**do not test other circumstances**. Other poker sites may
use different methods for the distribution of cards and other game types
(e.g. cash hold'em) or levels (e.g. $10 SnG) at PokerStars could also use
different programs for the deal. It is also true that the method of
dealing at a given site could change in the future as the site updates.
**Online Poker Watchdog intends to perform this test on other
poker sites for a variety of games and levels. **

Also, this analysis only tests one aspect of potential rigging, i.e.
the 'Bad Beat' theory. In theory there are many other ways that a poker
site could be rigged that this test doesn't examine, for example it
doesn't test the distribution of hole cards between players in any way.
**Online Poker Watchdog intends to perform further analyses,
designed to test other theories of potential rigging of online poker.**

One area of potential debate in this analysis is the assumptions
about 'winning' and 'losing' players, i.e.: 1) good players more often
than not get 'their chips in ahead' and 2) a good player's hand will
more often than not 'dominate' their opponent's hand than vice-versa in
pre-flop all-ins. Since we have filtered hands on 'all-in caller' we are
effectively comparing 'all-in' **calls**, for example: when
we compare the data-groups 'ahead' and 'behind' we are comparing 'good
calls' with 'bad calls'. Therefore, **we are effectively comparing
players that are 'good callers' with players that are 'bad callers'.**

In this regard the main point to be aware of is that this is
just one aspect that makes a winning 'Sit and Go' player. There are
other skills that make a 'winning' player, such as: the ability to steal
binds, good defence from the blinds and good 'long stack' play in the
early rounds. However, although making 'good all-in calls' does not
completely define a winning poker player it is a **major factor**
that contributes to being a winning player. This is especially true for
single table 'Sit and Gos' as the blind structure dictates that much or
the game involves pre-flop all-in play.

Another area that has raised questions is the completeness and legitimacy of the original dataset. Hand histories were purchased from an independent 3rd party and although it is never possible to be 100% sure of the reliability of a 3rd party source there is very little reason to believe that the hand histories are anything but legitimate. The dataset is not complete but is alleged to be at least 95% of all the hands played in the tournaments described above.

Questions have been raised as to the effect of missing hand histories on the analysis. Obviously, if the hands were missed randomly then we simply have a slightly smaller sample size and this wouldn't change our results at all. However, it is possible that the missed hands were due to a software glitch when they were 'datamined' and that there is a pattern behind why they were missed. For this to adversly effect the analysis the missed hands would have to share some characteristic which meant that their actual equities and expected equities were related differently from the rest of the sample. This is highly unlikely.

Top of Page ♠