Is Poker predominantly a game of skill or luck?

Is Poker predominantly a game of skill or luck?

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

The WST article reports on 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:

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.

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 will always be remembered for his reply to Virginia's letter to the New York Sun, "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 times 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 lawyer for the boys suggests:

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!

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 o.ld 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.

To estimate skill and luck percentages Sherman uses a statistical procedure called analysis of variance (ANOVA). There many discussion of ANOVA on the web. One of these Variance and the Design of Experimentsl begins with the following hypothetical data.

 

Treatment 1	Treatment 2
4	7
6	5
8	8
4	9
5	7
3	9

These might be the result of a clinical trial to determine if vitamin WR improves your memory. In the study one group is given a placebo and the second group takes vitamin WR for a month. At the end of the month the two groups are given a memory test and the numbers in the columns represent the number of correct answers the participants had. Then an ANOVA test is made to see if there is a signicant difference between the groups.

Here is Bill Peterson's explanation for how this works.

There are two group means:

   mean1 = (4+6+8+4+5+3)/6 = 30/6 = 5.0
   mean2 = (7+5+8+9+7+9)/6 = 45/6 = 7.5

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.

The idea of ANOVA is to compare variation between groups (which measures the treatment effect) to the variation within groups (which is noise or error). To this end, we express the deviation from the grand mean as a sum of two parts: the difference from the group mean ("within") and the difference of the group mean from the grand mean ("between").

Thus:

   (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)
   
   (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)

These are obvious arithmetic.  The magic (actually the Pythagorean Theorem)

is that the sums of squares decompose in this way.

   (4-6.25)^2 +...+(9-6.25)^2 =
       [(4-5.0)^2+...+(9 - 7.5)] + [(5.0 - 6.25)^2+...+(7.5 - 6.25)^2]

Check:

   46.25 = 27.5 + 18.75

In the usual abbreviations:

   SST = SSE + SSG

   (total sum of sqs = error sum of sqs + group sum of sqs)

Fisher's F statistic is F = MSG/MSE. Large values of F are effectively a large signal to noise ratio, and we conclude that there is a real treatment effect.