Chance News 29
Quotations
"There are few things that are so unpardonably neglected in our country as poker. The upper class knows very little about it. Now and then you find ambassadors who have sort of a general knowledge of the game, but the ignorance of the people is fearful. Why, I have known clergymen, good men, kind-hearted, liberal, sincere, and all that, who did not know the meaning of a "flush." It is enough to make one ashamed of the species".
Forsooth
The following Forsooths are from the September 07 issue of the RSS NEWS.
THE BIGGEST KILLER BY FAR
Heart disease claimed the lives of one in five men and about one in six women last year , figures indicate.
The Times
26 May 2006
[Hanson plc is the]] Largest aggregates producer in the world and 3rd largest in the USA
3 March, 2006
This Forsooth was suggested by Jerry Grossman.
In addition, a person's odds of becoming obese increased by 57 percent if he or she had a friend who became obese over a certain time interval. If the two people were mutual friends, the odds increased to 171 percent.
This discussion relates to an article The Spread of Obesity in a Large Social Network over 32 Years that appeared in the July 26, 2007 issue of the New England Journal of Medicine and seems to be freely available. Of course, here the increased to 171 percent is increased by 71%.
Jerry remarks "The NEJM article is interesting to those of us interested in the mathematical aspects of the social network."
This forsooth was suggested by Paul Alper
I've done 120 short-term energy outlooks, and I've probably gotten two of them right.
Mark Rodekohr, a veteran Department of Energy (DOE) economist
Minnesota Star Tribune
August 12, 2007Is Poker predominantly 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 Twains short story Science vs. Luck published in October 1870 issue of The Galaxy. In the story a number of boys were arrested for playing "old sledge" for money. Old sledge was a popular card game in those times 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 that this is a good way to decide if old sledge is a game of science or luck 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 and when they came out the professors had ended up with all the money and 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 "Towards a Skill Ratio" in the Article 1 and Article 2 in which he proposes a way to estimate luck and skill in poker and other games.
Sherman writes:
If two poker payers play in two equally difficult ring games (games played for monny but not in a tournament), why don't both players win the same amount of money? Typically there are two answers to this question. Either one player was more skilled, or one player was luckier. The purpose of this article is to provide a quantitative way to measure both of these differences.
We will illustrate how accomplishes this using data from five weeks of our Retirment Communitie's Monday night poker games.
To estimate skill and luck percentages Sherman uses a statistical procedure called "Analysis of variance" (ANOVA). There is a nice discussion of ANOVA here and the author begins with the following simple example to show how ANOVA works. We recommend that you check his arithmetic to better underand how it works.
Group 1 ˝Group2 Observation 1
Observation 2
Observation 3
2
3
1
6
7
5
Mean
Sums of Squares(SS)
2
2
6
2
Overall Mean
Total Sums of Squares
4
28
To carry out an Analysis of variance, the total sum of squares (28) is partitioned into the within group sum of squares (2+2 = 4) and between group sums of squares 28-(2+2) = 24.
With larger data set this partitioning can be found using statistical packages with programs that carry out analysis of variance.
Note that for our poker data theThe variation between groups corresponds to the variation within games and Sherman assumes that this variation is due primarily to skill. The variation within groups corresponds to the variation within players and Sherman assumes that this is primarily due to luck. This leads him to define the skill factor as the ratio of the between group variation to the total variation and the luck factor as the ratio of the between group variation and the total variation. These are indicated more simply as Skill % and Luck %.
Using our poker data and the SAS ANOVA program we find that the total sums of squares is 1069.95, the between groups the sums of squares is 311.447and the within groups the sums of squares is 758.499. Thus from our poker data we would estimate the skill % to be 311.447/1069.95 = 29.1% and the Luck % to be 758.499/1069.06 = 70.9%.
In his second article , Sherman reports the skill % he obtained using data from a number of different types of games. For example, using data for Major League Batting hits the skill % was 39% and home runs it as 68%. For NBA Basketball for Points Scored the skill % was 75%. For Pokertars in weekly tournaments the skill % was 35%.
Second chance lottery drawing
Ask Marilyn
Parade, 5 August 2007
Marilyn vos SavantA reader poses the following question.
Say that a state runs a lottery with scratch-off tickets and has a second-chance drawing for losing tickets. The latter are sent to a central location, where they are boxed and stored until it’s time for the drawing. An official then chooses one box and draws a ticket from it. All the other boxes are untouched. Is this fair, compared to storing all the tickets in a large container and then drawing a ticket from it?
Marilyn responds that, "The methods are equivalent, and both are perfectly fair: One winner was chosen at random", and suggests that the method is used purely for physical convenience. (In a state lottery, however, we imagine the whole affair would be conducted electronically.)DISCUSSION QUESTIONS:
(1) Marilyn's answer is almost correct. What has been implicitly assumed here?
(2) Here is a related problem (from Grinstead & Snell, Introduction to Probability, p. 152, problem 23).
You are given two urns and fifty balls. Half of the balls are white and half are black. You are asked to distribute the balls in the urns with no restriction placed on the number of either type in an urn. How should you distribute the balls in the urns to maximize the probability of obtaining a white ball if an urn is chosen at random and a ball drawn out at random? Justify your answer.
Submitted by Bill PetersonThe understanding and misunderstanding of Bayesian statistics
Gambling on tomorrow, The Economist, Aug 16th 2007
Scientists try new ways to predict climate risks, Reuters 12 Aug 2007.
Too late to escape climate disaster?, New Scientist, 18 Aug 2007.
Earth Log - Complex lesson, Daily Telegraph, 17 Aug 2007.
The latest edition of one of the Royal Society's journals, Philosophical Transactions is devoted to the science of climate modelling:
predictions from different models are pooled to produces estimates of future climate change, together with their associated uncertainties
the Royal Society said, and it partly focusses on 'the understanding and misunderstanding' of Bayesian statistics. So this Economist article discusses the difference between the frequentist and Bayesian view of statistics, in the context of forecasting the weather.
The article starts by claiming that there were just two main influences on the early development of probability theory and statistics: Bayes and Pascal. It claims that Pascal's ideas are simple and widely understood while Bayes are not. Pascal adopted a frequentist view, which The Economist characterises as the world was that of the gambler: each throw of the dice is independent of the previous one. Bayes promoted what we now call Bayesian probability, which The Economist characterises as incorporating the accumulation of experience into a statistical model in the form of prior assumptions:
A good prior assumption about tomorrow's weather, for example, is that it will be similar to today's. Assumptions about the weather the day after tomorrow, though, will be modified by what actually happens tomorrow.
But prior assumptions can influence model outcomes in subtle ways, The Economist warns:
Since the future is uncertain, (weather) forecasts are run thousands of times, with varying parameters, to produce a range of possible outcomes. The outcomes are assumed to cluster around the most probable version of the future. The particular range of values chosen for a parameter is an example of a Bayesian prior assumption, since it may be modified in the light of experience. But the way you pick the individual values to plug into the model can cause trouble. They might, for example, be assumed to be evenly spaced, say 1,2,3,4. But in the example of snow retention, evenly spacing both rate-of-fall and rate-of-residence-in-the-clouds values will give different distributions of results. That is because the second parameter is actually the reciprocal of the first. To make the two match, value for value, you would need, in the second case, to count 1, ½, ⅓, ¼—which is not evenly spaced. If you use evenly spaced values instead, the two models' outcomes will cluster differently.
It goes on to claim that those who use statistical models often fail to account for the uncertainty associated with such models:
Psychologically, people tend to be Bayesian—to the extent of often making false connections. And that risk of false connection is why scientists like Pascal's version of the world. It appears to be objective. But when models are built, it is almost impossible to avoid including Bayesian-style prior assumptions in them. By failing to acknowledge that, model builders risk making serious mistakes.
One of the Philosophical Transactions papers authors', David Stainforth of Oxford University, says
The answer is more comprehensive assessments of uncertainty, if we are to provide better information for today's policy makers. Such assessments would help steer the development of climate models and focus observational campaigns. Together this would improve our ability to inform decision makers in the future.
Questions
- What influences on the early development of probability theory and statistics can you think of, other than Pascal and Bayes?
- Is the frequentist view of statistics nothing more than each throw of the dice is independent of the previous one. What other characteristics would you associate with this view of statistics? Can you offer a better one-line summary? What about a better descrption of Bayesian statistics than incorporating the accumulation experience into a statistical model in the form of prior assumptions.
- In one of the Royal Society's papers, authors David Stainforth from Oxford University and Leonard Smith from the LSE, advocate making a clearer distinction between the output of model experiments designed for improving the model and those of immediate relevance for decision making. What do you think they meant by that? Can you think of a simple example to illustrate your interpretation?
- The Economist claims that scientists are not easily able to understand Bayes because of their philosophical training in the rigours of Pascal's method. How would you reply to this assertion?
Further reading
- Confidence, uncertainty and decision-support relevance in climate predictions, David Stainforth, Oxford University and Leonard Smith, LSE.
- This paper discusses the sources of uncertainty in the interpretation of climate model simulations as projections of the future.
- See also Climateprediction.net.
Sumbitted by John Gavin.