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==Is Poker Predominantly Skill or Luck?==


Harvard ponders just what it takes to excel at poker<br>
''Wall Street Journal'', May 3, 2007, A1<br>
King, Neil, Jr.


The ''Wall Street Journal'' article describes a one-day meeting in the Harvard
==Forsooth==
Faculty Club of poker pros, game theorists, statisticians, law students and
gambling lobbyists to develop a strategy to show that poker is not predominantly
a game of chance. In the article we read:


<quotation>he skill debate has been a preoccupation in poker circles
==Quotations==
since September (2006), when Congress barred the use of credit cards for online
“We know that people tend to overestimate the frequency of well-publicized, spectacular
wagers. Horse racing and stock trading were exempt, but otherwise the new law
events compared with more commonplace ones; this is a well-understood phenomenon in
hit any game ``predominantly subject to chance''. Included among such games was
the literature of risk assessment and leads to the truism that when statistics plays folklore,
poker, which is increasingly played on Internet sites hosting players from all
folklore always wins in a rout.”
over the world.</quotation>
<div align=right>-- Donald Kennedy (former president of Stanford University), ''Academic Duty'', Harvard University Press, 1997, p.17</div>


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''
"Using scientific language and measurement doesn’t prevent a researcher from conducting flawed experiments and drawing wrong conclusions — especially when they confirm preconceptions."
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:


<blockquotation>Impanel a jury of six of each, Luck versus Science -- give
<div align=right>-- Blaise Agüera y Arcas, Margaret Mitchell and Alexander Todoorov, quoted in: The racist history behind facial recognition, ''New York Times'', 10 July 2019</div>
them candles and a couple of decks of cards, send them into the jury room, and
just abide by the result! </blockquotation}


The Judge agrees to do this and so four deacons and the two dominies (Clergymen)
==In progress==
were sworn in as the ``chance'' jurymen, and six inveterate old seven-up
[https://www.nytimes.com/2018/11/07/magazine/placebo-effect-medicine.html What if the Placebo Effect Isn’t a Trick?]<br>
professors were chosen to represent the``science'' side of the issue.
by Gary Greenberg, ''New York Times Magazine'', 7 November 2018


They retired to the jury room. When they came out, the professors had ended up
[https://www.nytimes.com/2019/07/17/opinion/pretrial-ai.html The Problems With Risk Assessment Tools]<br>
with all the money. So the Judge ruled that the boys were innocent.  
by Chelsea Barabas, Karthik Dinakar and Colin Doyle, ''New York Times'', 17 July 2019


Today more sophisticated ways to determine if a gambling game is predominantly
==Hurricane Maria deaths==
skill or luck are being studied. Ryne Sherman has written two articles on this
Laura Kapitula sent the following to the Isolated Statisticians e-mail list:
``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
:[Why counting casualties after a hurricane is so hard]<br>
called analysis of variance (ANOVA). 
:by Jo Craven McGinty, Wall Street Journal, 7 September 2018
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:


The article is subtitled: Indirect deaths—such as those caused by gaps in medication—can occur months after a storm, complicating tallies
Laura noted that
:[https://www.washingtonpost.com/news/fact-checker/wp/2018/06/02/did-4645-people-die-in-hurricane-maria-nope/?utm_term=.0a5e6e48bf11 Did 4,645 people die in Hurricane Maria? Nope.]<br>
:by Glenn Kessler, ''Washington Post'', 1 June 2018


\begin {table} [ht]
The source of the 4645 figure is a [https://www.nejm.org/doi/full/10.1056/NEJMsa1803972 NEJM article].  Point estimate, the 95% confidence interval ran from 793 to 8498.
%\caption{Results of memory tests}
\centering
\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}
\end{table}


The numbers in the  second  column are the average number of correct answers
President Trump has asserted that the actual number is
for the
[https://twitter.com/realDonaldTrump/status/1040217897703026689 6 to 18].
placebo group and those in the third column are the average number of correct
The ''Post'' article notes that Puerto Rican official had asked researchers at George Washington University to do an estimate of the death toll.  That work is not complete.
answers
[https://prstudy.publichealth.gwu.edu/ George Washington University study]
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:
\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*}
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)
is that the sums of squares decompose in this way:
\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,
$${\rm SST} = \rm{SSE} + {\rm SSG}\,,$$
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
:[https://fivethirtyeight.com/features/we-still-dont-know-how-many-people-died-because-of-katrina/?ex_cid=538twitter We sttill don’t know how many people died because of Katrina]<br>
carry out an analysis of variance in the same way we did for our medical
:by Carl Bialik, FiveThirtyEight, 26 August 2015
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


$${\rm Game's\ Skill\ Percentage} = \frac{\rm Systematic\
----
Variance}{{\rm(Systematic\  Variance} + {\rm Random\  Variance})}\,,$$
[https://www.nytimes.com/2018/09/11/climate/hurricane-evacuation-path-forecasts.html These 3 Hurricane Misconceptions Can Be Dangerous. Scientists Want to Clear Them Up.]<br>
and similarly,
[https://journals.ametsoc.org/doi/abs/10.1175/BAMS-88-5-651 Misinterpretations of the “Cone of Uncertainty” in Florida during the 2004 Hurricane Season]<br>
$${\rm Game's\ Luck\ Percentage} = \frac{\rm Random\ Variance}{{\rm(Systematic\
[https://www.nhc.noaa.gov/aboutcone.shtml Definition of the NHC Track Forecast Cone]
Variance} + {\rm Random\  Variance})}\,.$$
----
So, in our poker game, the Random Variance is 758.499 and the Systematic
[https://www.popsci.com/moderate-drinking-benefits-risks Remember when a glass of wine a day was good for you? Here's why that changed.]
variance is 311.477. So the Skill Percentage  is 29.1\% and  the Luck Percentage
''Popular Science'', 10 September 2018
is 70.9\%.
----
\par
[https://www.economist.com/united-states/2018/08/30/googling-the-news Googling the news]<br>
In his second article, Sherman reports  the Skill Percentage  he obtained using
''Economist'', 1 September 2018
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


[https://www.cnbc.com/2018/09/17/google-tests-changes-to-its-search-algorithm-how-search-works.html We sat in on an internal Google meeting where they talked about changing the search algorithm — here's what we learned]
----
[http://www.wyso.org/post/stats-stories-reading-writing-and-risk-literacy Reading , Writing and Risk Literacy]


[http://www.riskliteracy.org/]
-----
[https://twitter.com/i/moments/1025000711539572737?cn=ZmxleGlibGVfcmVjc18y&refsrc=email Today is the deadliest day of the year for car wrecks in the U.S.]


==Some math doodles==
<math>P \left({A_1 \cup A_2}\right) = P\left({A_1}\right) + P\left({A_2}\right) -P \left({A_1 \cap A_2}\right)</math>


<math>P(E)  = {n \choose k} p^k (1-p)^{ n-k}</math>


<math>\hat{p}(H|H)</math>


<math>\hat{p}(H|HH)</math>


==Accidental insights==


My collective understanding of Power Laws would fit beneath the shallow end of the long tail. Curiosity, however, easily fills the fat end.  I long have been intrigued by the concept and the surprisingly common appearance of power laws in varied natural, social and organizational dynamics.  But, am I just seeing a statistical novelty or is there meaning and utility in Power Law relationships? Here’s a case in point.


While carrying a pair of 10 lb. hand weights one, by chance, slipped from my grasp and fell onto a piece of ceramic tile I had left on the carpeted floor. The fractured tile was inconsequential, meant for the trash.
<center>[[File:BrokenTile.jpg | 400px]]</center>
As I stared, slightly annoyed, at the mess, a favorite maxim of the Greek philosopher, Epictetus, came to mind: “On the occasion of every accident that befalls you, turn to yourself and ask what power you have to put it to use.”  Could this array of large and small polygons form a Power Law? With curiosity piqued, I collected all the fragments and measured the area of each piece.


<center>
{| class="wikitable"
|-
! Piece !! Sq. Inches !! % of Total
|-
| 1 || 43.25 || 31.9%
|-
| 2 || 35.25 ||26.0%
|-
|  3 || 23.25 || 17.2%
|-
| 4 || 14.10 || 10.4%
|-
| 5 || 7.10 || 5.2%
|-
| 6 || 4.70 || 3.5%
|-
| 7 || 3.60 || 2.7%
|-
| 8 || 3.03 || 2.2%
|-
| 9 || 0.66 || 0.5%
|-
| 10 || 0.61 || 0.5%
|}
</center>
<center>[[File:Montante_plot1.png | 500px]]</center>
The data and plot look like a Power Law distribution. The first plot is an exponential fit of percent total area. The second plot is same data on a log normal format. Clue: Ok, data fits a straight line.  I found myself again in the shallow end of the knowledge curve. Does the data reflect a Power Law or something else, and if it does what does it reflect?  What insights can I gain from this accident? Favorite maxims of Epictetus and Pasteur echoed in my head:
“On the occasion of every accident that befalls you, remember to turn to yourself and inquire what power you have to turn it to use” and “Chance favors only the prepared mind.”


<center>[[File:Montante_plot2.png | 500px]]</center>
My “prepared” mind searched for answers, leading me down varied learning paths. Tapping the power of networks, I dropped a note to Chance News editor Bill Peterson. His quick web search surfaced a story from ''Nature News'' on research by Hans Herrmann, et. al. [http://www.nature.com/news/2004/040227/full/news040223-11.html Shattered eggs reveal secrets of explosions].  As described there, researchers have found power-law relationships for the fragments produced by shattering a pane of glass or breaking a solid object, such as a stone. Seems there is a science underpinning how things break and explode; potentially useful in Forensic reconstructions.
Bill also provided a link to [http://cran.r-project.org/web/packages/poweRlaw/vignettes/poweRlaw.pdf a vignette from CRAN] describing a maximum likelihood procedure for fitting a Power Law relationship. I am now learning my way through that.


Submitted by William Montante


From ???@??? Tue Oct 16 08:53:02 2007
----
To: laurie snell

Latest revision as of 20:58, 17 July 2019


Forsooth

Quotations

“We know that people tend to overestimate the frequency of well-publicized, spectacular events compared with more commonplace ones; this is a well-understood phenomenon in the literature of risk assessment and leads to the truism that when statistics plays folklore, folklore always wins in a rout.”

-- Donald Kennedy (former president of Stanford University), Academic Duty, Harvard University Press, 1997, p.17

"Using scientific language and measurement doesn’t prevent a researcher from conducting flawed experiments and drawing wrong conclusions — especially when they confirm preconceptions."

-- Blaise Agüera y Arcas, Margaret Mitchell and Alexander Todoorov, quoted in: The racist history behind facial recognition, New York Times, 10 July 2019

In progress

What if the Placebo Effect Isn’t a Trick?
by Gary Greenberg, New York Times Magazine, 7 November 2018

The Problems With Risk Assessment Tools
by Chelsea Barabas, Karthik Dinakar and Colin Doyle, New York Times, 17 July 2019

Hurricane Maria deaths

Laura Kapitula sent the following to the Isolated Statisticians e-mail list:

[Why counting casualties after a hurricane is so hard]
by Jo Craven McGinty, Wall Street Journal, 7 September 2018

The article is subtitled: Indirect deaths—such as those caused by gaps in medication—can occur months after a storm, complicating tallies

Laura noted that

Did 4,645 people die in Hurricane Maria? Nope.
by Glenn Kessler, Washington Post, 1 June 2018

The source of the 4645 figure is a NEJM article. Point estimate, the 95% confidence interval ran from 793 to 8498.

President Trump has asserted that the actual number is 6 to 18. The Post article notes that Puerto Rican official had asked researchers at George Washington University to do an estimate of the death toll. That work is not complete. George Washington University study

We sttill don’t know how many people died because of Katrina
by Carl Bialik, FiveThirtyEight, 26 August 2015

These 3 Hurricane Misconceptions Can Be Dangerous. Scientists Want to Clear Them Up.
Misinterpretations of the “Cone of Uncertainty” in Florida during the 2004 Hurricane Season
Definition of the NHC Track Forecast Cone

Remember when a glass of wine a day was good for you? Here's why that changed. Popular Science, 10 September 2018

Googling the news
Economist, 1 September 2018

We sat in on an internal Google meeting where they talked about changing the search algorithm — here's what we learned

Reading , Writing and Risk Literacy

[1]

Today is the deadliest day of the year for car wrecks in the U.S.

Some math doodles

<math>P \left({A_1 \cup A_2}\right) = P\left({A_1}\right) + P\left({A_2}\right) -P \left({A_1 \cap A_2}\right)</math>

<math>P(E) = {n \choose k} p^k (1-p)^{ n-k}</math>

<math>\hat{p}(H|H)</math>

<math>\hat{p}(H|HH)</math>

Accidental insights

My collective understanding of Power Laws would fit beneath the shallow end of the long tail. Curiosity, however, easily fills the fat end. I long have been intrigued by the concept and the surprisingly common appearance of power laws in varied natural, social and organizational dynamics. But, am I just seeing a statistical novelty or is there meaning and utility in Power Law relationships? Here’s a case in point.

While carrying a pair of 10 lb. hand weights one, by chance, slipped from my grasp and fell onto a piece of ceramic tile I had left on the carpeted floor. The fractured tile was inconsequential, meant for the trash.

BrokenTile.jpg

As I stared, slightly annoyed, at the mess, a favorite maxim of the Greek philosopher, Epictetus, came to mind: “On the occasion of every accident that befalls you, turn to yourself and ask what power you have to put it to use.” Could this array of large and small polygons form a Power Law? With curiosity piqued, I collected all the fragments and measured the area of each piece.

Piece Sq. Inches % of Total
1 43.25 31.9%
2 35.25 26.0%
3 23.25 17.2%
4 14.10 10.4%
5 7.10 5.2%
6 4.70 3.5%
7 3.60 2.7%
8 3.03 2.2%
9 0.66 0.5%
10 0.61 0.5%
Montante plot1.png

The data and plot look like a Power Law distribution. The first plot is an exponential fit of percent total area. The second plot is same data on a log normal format. Clue: Ok, data fits a straight line. I found myself again in the shallow end of the knowledge curve. Does the data reflect a Power Law or something else, and if it does what does it reflect? What insights can I gain from this accident? Favorite maxims of Epictetus and Pasteur echoed in my head: “On the occasion of every accident that befalls you, remember to turn to yourself and inquire what power you have to turn it to use” and “Chance favors only the prepared mind.”

Montante plot2.png

My “prepared” mind searched for answers, leading me down varied learning paths. Tapping the power of networks, I dropped a note to Chance News editor Bill Peterson. His quick web search surfaced a story from Nature News on research by Hans Herrmann, et. al. Shattered eggs reveal secrets of explosions. As described there, researchers have found power-law relationships for the fragments produced by shattering a pane of glass or breaking a solid object, such as a stone. Seems there is a science underpinning how things break and explode; potentially useful in Forensic reconstructions. Bill also provided a link to a vignette from CRAN describing a maximum likelihood procedure for fitting a Power Law relationship. I am now learning my way through that.

Submitted by William Montante

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