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Is Poker Predominantly Skill or Luck?

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

This Wall Street Journal article describes a one-day meeting in the Harvard 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:

\begin{quotation}The skill debate has been a preoccupation in poker circles since September (2006), when Congress barred the use of credit cards for online wagers. Horse racing and stock trading were exempt, but otherwise the new law hit any game ``predominantly subject to chance. Included among such games was poker, which is increasingly played on Internet sites hosting players from all over the world.\end{quotation}

This, of course, is not a new issue. For example it is the subject of the Mark Twain's short story \emph{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 a Santa 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:

\begin{quotation} 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! \end{quotation}

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:

\begin {table} [ht] %\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

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: \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 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

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