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==Is Poker Predominantly Skill or Luck?==
<math>Mean1 = (4+6+8+4+5+3)/6 = 30/6 = 5.0<br>
Mean2 = (7+5+8+9+7+9)/6 = 45/6 = 7.5</math><br>


Harvard ponders just what it takes to excel at poker<br>
Then a grand mean over all observations:<br>
''Wall Street Journal'', May 3, 2007, A1<br>
Mean = (30+45)/(6+6) = 6.25<br>
King, Neil, Jr.


The ''Wall Street Journal'' article describes a one-day meeting in the Harvard
Variance is always a sum of square deviations divided by degree of freedom:
Faculty Club of poker pros, game theorists, statisticians, law students and
SS/df.  This is also called a mean squared deviation MS.
gambling lobbyists to develop a strategy to show that poker is not predominantly
a game of chance. In the article we read:


<blockquote>he skill debate has been a preoccupation in poker circles
ANOVA begins by expressing the deviation of each observation from the grand mean as a sum of two terms: the difference of the observation from its group mean, plus the difference of the group mean from the grand mean.   Writing this out explicitly for the example, we have, for the placebo group:<br><br>
since September (2006), when Congress barred the use of credit cards for online
(4 - 6.25) = (4 - 5.0) + (5.0 - 6.25)<br>
wagers. Horse racing and stock trading were exempt, but otherwise the new law
(6 - 6.25) = (6 - 5.0) + (5.0 - 6.25)<br>
hit any game ``predominantly subject to chance''. Included among such games was
...<br>
poker, which is increasingly played on Internet sites hosting players from all
(3 - 6.25) = (3 - 5.0) + (5.0 - 6.25)<br>
over the world.</blockquote>
 
This, of course, is not a new issue. For example it is the subject of the Mark
Twain's short story ''Science vs. Luck'' published in the October 1870 issue
of The Galaxy. The Galaxy no longer exists but co-founder Francis Church became
an editor for the ''New York Sun'' and will always be remembered for his reply to Virginia:  ``Yes, Virginia, there is aSanta Claus''.
 
In Mark Twain's story, a number of boys were arrested for playing ``old sledge''
for money.  Old sledge was a popular card game in those days and often played
for money.  In the trial the judge finds that  half the experts  say that old
sledge is a game of science and half that it is a game of skill. The boys'
lawyer for suggests:
 
<blockquote>Impanel a jury of six of each, Luck versus Science -- give
them candles and a couple of decks of cards, send them into the jury room, and
just abide by the result! </blockquoe>
 
The Judge agrees to do this and so four deacons and the two dominies (Clergymen)
were sworn in as the ``chance'' jurymen, and six inveterate old seven-up
professors were chosen to represent the``science'' side of the issue.  
 
They retired to the jury room. When they came out, the professors had ended up
with all the money. So the Judge ruled that the boys were innocent.  
 
Today more sophisticated ways to determine if a gambling game is predominantly
skill or luck are being studied. Ryne Sherman has written two articles on this
``Towards a Skill Ratio'' and ``More on Skill and Individual Differences'' in
which he proposes a way to estimate luck and skill in poker and other games.
These articles occurred in the Internet magazine Two + Two Vol. 3, No. 5 and 6
but are not available since the journal only keeps their articles for three
months.
 
To estimate skill and luck percentages Sherman uses a statistical procedure
called analysis of variance (ANOVA). 
To understand Sherman's method of comparing luck and skill we need to understand
how ANOVA works so we will do this using a simple example.  
 
Assume that a clinical trial is carried out to determine if vitamin ME improves
memory. In the study, two groups are formed from 12 participants. Six were
given a placebo and six were given vitamin ME. The study is carried out for a
period of six months.  
At the end of each month the two groups are given a memory test. Here are the
results:


and for the vitamin ME group:
   
(7 - 6.25) = (7 - 7.5) + (7.5 - 6.25)<br>


<math>\begin {table} [ht]
(5 - 6.25) = (5 - 7.5) + (7.5 - 6.25)<br>
%\caption{Results of memory tests}
...<br>
\centering
(9 - 6.25) = (9 - 7.5) + (7.5 - 6.25)<br>
\begin{tabular}{|c| |c| |c|}
\hline\hline
Month & Placebo & Vitamin ME \\
\hline\hline
1 & 4 & 7 \\
2 & 6 & 5 \\
3 & 8 & 8 \\
4 & 4 & 9 \\
5 &5 & 7 \\
6 & 3 & 9 \\
\hline\hline
Mean & 5 & 7.5\\
\hline\hline
\end{tabular}
\label {table:nonlin}
<\math>


The numbers in the  second  column are the average number of correct answers
for the
placebo group and those in the third column are the average number of correct
answers
for the Vitamin ME group.  ANOVA can be used to see if there is significant
difference between the groups. Here is Bill Peterson's explanation for how this
works.
\noindent
There are two group means:\\ \\
$${\rm Mean}1 = \frac{(4+6+8+4+5+3)}6 = \frac{30}6 = 5.0$$
$${\rm Mean}2 = \frac{(7+5+8+9+7+9)}6 = \frac{45}6 = 7.5\,.$$
Then a grand mean over all observations:
$${\rm Mean} = \frac{(30+45)}{(6+6)} = 6.25\,.$$
Variance is always a sum of squared deviations divided by degrees of freedom:
$SS/df$.  This is also called a mean squared deviation $MS$.
\par
ANOVA begins by expressing the deviation of each observation from the grand mean
as a sum of two terms:  the difference of the scores from their group mean
and the difference of the group means from the grand mean.  Writing this out
explicitly for the example, we have, for the placebo group:
<math>\begin{eqnarray*}
(4 - 6.25) &=& (4 - 5.0) + (5.0 - 6.25)\cr
(6 - 6.25) &=& (6 - 5.0) + (5.0 - 6.25)\cr
&\ldots& \cr
(3 - 6.25) &=& (3 - 5.0) + (5.0 - 6.25)\cr 
\end{eqnarray*}</math>
and for the vitamin ME group:
\begin{eqnarray*}
(7 - 6.25) &=& (7 - 7.5) + (7.5 - 6.25)\cr
(5 - 6.25) &=& (5 - 7.5) + (7.5 - 6.25)\cr
&\cdots& \cr
(9 - 6.25) &=& (9 - 7.5) + (7.5 - 6.25)\,.\cr
\end{eqnarray*}
The magic (actually the Pythagorean Theorem in an appropriate dimensional space)
The magic (actually the Pythagorean Theorem in an appropriate dimensional space)
is that the sums of squares decompose in this way:
is that the sums of squares decompose in this way.<br>
\begin{eqnarray*}
(4-6.25)^2 +\cdots+(9-6.25)^2 & = &
\bigl((4-5.0)^2+\cdots+(9 - 7.5)^2\bigr)+\cr
& &\bigl((5.0 - 6.25)^2+\cdots+(7.5 - 6.25)^2\bigr)\,.\cr
\end{eqnarray*}
\noindent
Check:  46.25 = 27.5 + 18.
\par
\noindent
In the usual abbreviations,
 
<center> SST= rm{SSE} + SSG.</center>
 
where these three quantities are the total sum of squares, the error sum of
squares, and the group sum of squares. In ANOVA, scaled versions of SSE and SSG
are compared to determine if there is evidence that there is a significant
difference among the different groups.
\par
The SSE is a measure of the variations within each group and so should not tell
us much about the effectiveness of the treatments and is often called the
nuisance variation.  On the other hand the SSG is a measure of the variation
between the groups and would be expected to give information about the
effectiveness of the treatment.
\par
Sherman uses this same kind of decomposition for his measure of skill and
chance for a game.  We illustrate how he does this using data from
five weeks of our low-key Monday night poker games.  In the table below, we show
how much each player lost in five games and their mean winnings.
\begin {table} [ht]
\centering
\begin{tabular}{|c| |c| |c| |c| |c| |c| |c| }
\hline\hline
& Game 1 & Game 2 & Game 3 & Game 4 & Game 5&Mean  \\
\hline
Sally  & -6.75 & 4.35  & 6.95 &-1.23 & 6.35 &1.934 \\
Laurie  & -10.10 & -4.25 & -4.35 & -11.55 & -1.5 &-6.35 \\
John  &-5.75 & .40 & .18 & 4.35 &-.45 &-2.54 \\
Mary & 10.35 & -.35 & -7.75 & 2.9 &-.65 & .9 \\
Sarge & 9.7 & -8.8 & 7.65 & 4.85 & -.25 & 2.63 \\
Dick&4.43 &-15 &-5.9 &-3.9 &-4.9 & -2.084\\
Glenn&-1.95 & 5.8 & 3.9 & 3.25 &1.42 & 2.489\\
\hline\hline
\end{tabular}
\label {table:nonlin}
\end{table}
 
To compare the amount of skill and luck in these games Sherman would have us
carry out an analysis of variance in the same way we did for our medical
example. Here the players are the groups and the games are the treatments.  The
group means are the averages of the players winnings and are in the last row of
the data. 
\par
The grand mean (gm) is the sum of all the winnings divided by 35
which gives us a grand mean of $-.105714$. The sums of squares within  groups
is the  sum of  the squares of the differences between the winnings and the
player's mean winnings:
$$(-6.75 -1.934)^2 + \dots + (-1.42-2.489)^2 = 758.499\,.$$
The sums of squares between groups is the sum of the squares of the differences
between the winnings and the grand mean:
$$(-6.75 - \mbox{gm})^2  + \cdots+  (-1.42-\mbox{gm})^2 = 311. 447\,.$$
Thus the total sums of squares is
$$758.499 + 311.447 = 1069.95\,.$$
\par
Sherman assumes that the variation  in the amount won within groups  is
primarily due to luck and  calls this ``Random Variance'' and the variation
between groups is due primarily  to skill and calls this ``Systematic Variance".
He then defines
 
$${\rm Game's\ Skill\ Percentage} = \frac{\rm Systematic\
Variance}{{\rm(Systematic\  Variance} + {\rm Random\  Variance})}\,,$$
and similarly,
$${\rm Game's\ Luck\ Percentage} = \frac{\rm Random\ Variance}{{\rm(Systematic\
Variance} + {\rm Random\  Variance})}\,.$$
So, in our poker game, the Random Variance is 758.499 and the Systematic
variance is 311.477. So the Skill Percentage  is 29.1\% and  the Luck Percentage
is 70.9\%.
\par
In his second article, Sherman reports  the Skill Percentage  he obtained using
data from a number of different types of games.  For example, using data for
Major League Batting, the Skill Percentage for hits was 39\% and for home runs
was 68 \%. For NBA  Basketball it was 75\% for points scored. For poker stars in
weekly tournaments it was 35\%.
\par
Sherman concludes his articles with the remarks:
 
\begin{quotation} If two persons play the same game, why don't both achieve the
same results? The purpose of last month's article and this article was to
address this question.  This article suggests that there are two answers to this
question: Skill (or systematic variance) or Luck (or random variance).  Using
both the correlation approach described last month and the ANOVA approach
described in this article, one can estimate the amount of skill involved in any
game.  Last, and maybe most importantly, Table 4 demonstrated that the skill
estimates involved in playing poker  (or at least tournament poker) are not very
different from other sport outcomes which are widely accepted as
skillful.\end{quotation}
\vskip .2in
\noindent
Discussion questions:
\vskip .1in
\noindent
(1) Do you think that Sherman's measure of skill and luck in a game is
reasonable?  If not, why not?
\vskip .1in
\noindent
(2) There is a form of poker modeled after duplicate bridge.  Do you think that
the congressional decision should apply to this form of gambling?
 
 
\theendnotes
 
 
 
 
 
 
 
 
 
 
 


<math>(4-6.25)^2 +...+(9-6.25)^2 =</math>
[(4-5.0)^2+...+(9 - 7.5)^2] + [(5.0 - 6.25)^2+...+(7.5 - 6.25)^2] <br>
Check:  46.25 = 27.5 + 18.75<br>


From ???@??? Tue Oct 16 08:53:02 2007
In the usual abbreviations:<br>
To: laurie snell


</math>
SST = SSE + SSG<br>

Revision as of 19:49, 17 October 2007

<math>Mean1 = (4+6+8+4+5+3)/6 = 30/6 = 5.0
Mean2 = (7+5+8+9+7+9)/6 = 45/6 = 7.5</math>

Then a grand mean over all observations:
Mean = (30+45)/(6+6) = 6.25

Variance is always a sum of square deviations divided by degree of freedom: SS/df. This is also called a mean squared deviation MS.

ANOVA begins by expressing the deviation of each observation from the grand mean as a sum of two terms: the difference of the observation from its group mean, plus the difference of the group mean from the grand mean. Writing this out explicitly for the example, we have, for the placebo group:

(4 - 6.25) = (4 - 5.0) + (5.0 - 6.25)
(6 - 6.25) = (6 - 5.0) + (5.0 - 6.25)
...
(3 - 6.25) = (3 - 5.0) + (5.0 - 6.25)

and for the vitamin ME group:

(7 - 6.25) = (7 - 7.5) + (7.5 - 6.25)

(5 - 6.25) = (5 - 7.5) + (7.5 - 6.25)
...
(9 - 6.25) = (9 - 7.5) + (7.5 - 6.25)

The magic (actually the Pythagorean Theorem in an appropriate dimensional space) is that the sums of squares decompose in this way.

<math>(4-6.25)^2 +...+(9-6.25)^2 =</math> [(4-5.0)^2+...+(9 - 7.5)^2] + [(5.0 - 6.25)^2+...+(7.5 - 6.25)^2]
Check: 46.25 = 27.5 + 18.75

In the usual abbreviations:

SST = SSE + SSG

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