Is Poker predominately a game of skill or luck?

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

Poker a big deal
The Economist, Dec 19, 2001.

The WSJ 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 predominantly a game of skill.

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 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.

The question of weather pa game is predominatly a game of skilll or luck applies to numerous state laws. A good discussion of the cases that involve poker is given in the article Is Poker in the U..S. a Game of Skill? by Chuck Humphrey, a lawyer who has specialized in gambling laws. He writes:

I have not been able to find any case law that has ever squarely held poker to be a game of skill free from illegality under applicable state anti-gambling laws. There have been some passing references to poker as a game of skill in a few cases.[1] But these are only references that go to whether any skill is involved in the game, not to the level of that skill as compared with the element of chance in the game. The actual decisions did not involve poker, let alone the more relevant question of the legality of offering poker games in a setting where the house directly or indirectly makes money by raking the game, charging an entry fee or selling food, beverage or merchandise to players. The decided cases hold that in order to be a “game of skill” the elements of skill must predominate over those of chance in determining the outcome.

I suggest that those interested in improving the law on skill v. chance work on expanding that definition to better specify the principal elements that constitute skill and chance. A weighing mechanism that could be considered by a judge or jury should also be set forth.

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 topic: A Conceptualization and Quantification of Skill 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 understand Sherman's method we have to understand two concepts that mathematician use for chance events. These are expected value and standard deviation. We will illustrate these in terms of a poll do decide how Omama and Hillary will do in the Washington contest. Let p be the proportion who plan to vote for Omma and q the proportion who plan to vote for Hillary. We choose a sample of n democrats and ask them how they plan to vote. Let p' be the proportion of the sample who say Omama and q' the proportion who say Hillary. Then the expected value for the number who say Omama is p and the standard deviation is sqr(pq/n). Then it is possible to show that the probability that p' differs from p by more than 2 standard deviations is .954

The New York Times often puts at the end of an article about a poll, an explanation of how their poll was carried out. In a recent poll of 1,166 people, the article stated that the margin of error was 3%. In their explanation of how the poll was carried out they explained the margin of error by the statement "In theory, in 19 cases out of 20 the results based on such samples will differ by no more than three percentage points in either direction from what would have been obtained by seeking out all American adults."

There are those who think the gambling markets give better predictions that the pollsters. This started with an education project [1]The Iowa Electronic Market] in which you trade with real money but there is a limit to how much you can bet. A purley gambling example is Tradesports

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:

Month
Placebo
Vitamin ME
1
4
7
2
6
5
3
8
8
4
4
9
5
5
7
6
3
9
Mean
5
7.5

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. There are two group means:

Mean1 = [math]\frac{(4+6+8+4+5+3)}{6}=\frac{30}{6}= 5.0 [/math]

Mean2 = [math]\frac{(7+5+8+9+7+9)}{6}= \frac{45}{6}=7.5 [/math]

Then a grand mean over all observations:

Mean = [math]\frac{(30+45)}{(6+6)} = 6.25[/math]

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] [math][(4-5.0)^2+...+(9 - 7.5)^2][/math]

+ [math][(5.0 - 6.25)^2+...+(7.5 - 6.25)^2][/math]

Check: 46.25 = 27.5 + 18.75

In the usual abbreviations:

SST = SSE + 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.

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.

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.

Sally
Laurie
John
Mary
Sarge
Dick
Glenn
Game 1
-6.75
-10.10
-5.75
10.35
9.7
4.43
-1.95
Game 2
4.35
-4.25
.40
-.35
-8.8
-.15
5.8
Game 3
6.95
-4.35
.18
-7.75
7.65
-5.9
3.9
Game 4
-1.23
-11.55
4.35
2.9
4.85
-3.9
3.25
Game 5
6.35
-1.5
-.45
-.65
-.25
-4.9
1.42
Mean
1.934
-6.35
-.254
.9
2.63
-2.084
2.484

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 example. The players are now seen in the role of treatments. Each player has a mean net gain over the set of games. For each outcome in the table we write the difference between this outcome and the overall mean as the sum of two terms: the difference between the outcome and the player's mean plus the difference between the player's mean and the overall mean. Sherman suggests that the difference between the outcome and the players mean is due primarily to luck while the difference between the players mean and the overall mean is due primarily to skill. This leads him to define the skill % as the ratio of the of the group sums of squares to the total sums of squares and the luck % as the ratio of the within group sums of squares to the total sums of squares.

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:

[math]{\rm Game's\ Skill\ Percentage} = \frac{\rm Systematic\ Variance}{\rm Systematic\ Variance + Random\ Variance}[/math]

and similarly,

[math]{\rm Game's\ Luck\ Percentage} = \frac{\rm Random\ Variance} {\rm Systematic\ Variance + Random\ Variance}[/math]

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%.

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%.

http://www.dartmouth.edu/~chance/forwiki/skillvsluck.jpg

Sherman concludes his articles with the remarks:

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.

Discussion questions:

(1) Do you think that Sherman's measure of skill and luck in a game is reasonable? If not, why not?

(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?