Math and Poker

Brain Damage

2010-10-11T23:39:00.000-07:00

Are winning poker players brain damaged?

From Cheap Talk

Each brain-damaged person got a wad of play money, and instructions to gamble on 20 rounds of coin tossing (heads-you-win/tails-you-lose, with some added twists). Other people who had no such brain lesions got the same money and the same gambling instructions.

The brain-damaged gamblers pretty consistently ended up with more money than their healthier-brained competitors. The researchers speculate that when “normal” gamblers encounter a run of unhappy coin-toss results, they get discouraged and become cautious – perhaps too cautious. Not so the people with brain-lesion-induced emotional disfunction. Encountering a run of bad luck, they plough on, undaunted. And then enjoy a relatively handsome payoff. At least sometimes.

A course in multi level models

2010-06-17T20:32:00.000-07:00

I don't know much about multilevel modeling but I tend to think about opponent modeling in a non-hierarchical multilevel way. Opponents are multi-dimensional, tight, aggressive, tricky, weak, etc.

Here's an online course on the topic that I think I should take. It's completely free, lectures and a textbook online. It just requires registration.

Twitter

2010-03-20T16:44:00.000-07:00

I broke down an opened a twitter account --- @garycarson

I'll be twitting about some ncaa bets I'm making based on the LRMC model.

Decision Analysis

2010-02-10T16:46:00.000-08:00

An intoductory lecture to decision analysis

Calculating variance

2010-02-09T21:03:00.000-08:00

A simple optimization problem

2010-02-07T11:23:00.000-08:00

Here's a simple optimization problem that doesn't require anything other than high school algebra and maybe some basic calculus to solve.

The technique it illustrates is just one of basic substitution when you have a problem that can easily be reduced to a problem with a single variable.

What is optimization?

2010-02-06T10:08:00.000-08:00

Here's a video of Leon Lasdon giving a talk with some examples of successful applications of operations research.

Lasdon is one of the old guys in operations research, and he says something at the start of his talk that struck me. "Everybody knows what optimization is", he says.

I used to think that was true also. But it's really not. In the poker world many people confuse the idea of optimal with that of equilibrium. In game theory you can find the equilibrium solution as an optimal solution of a min/max problem. So among poker math dweebs it's become common to term a solution of a game theory model as optimization and to describe exploitave.

In the poker world optimization has become exploitive and equilibrium has become optimization.

Outside the poker world optimization just means doing the best you can within whatever operating constraints you might have, just like Lasdon says in the video.

Sometimes I feel like Don Quixote tilting at windmills to try to use optimization to mean optimization when talking to poker dweebs.

What is operations research?

2010-02-05T15:21:00.000-08:00

A 6 minute overview of uses of operations research.

Orthogonal

2010-01-15T06:21:00.000-08:00

In recent arguments before the Supreme Court some lawyer used the word "orthogonal" in a way that I thought was a pretty natural meaning. (see Simple Justice, Washington Post, or Althouse.

MR. FRIEDMAN: I think that issue is entirely orthogonal to the issue here because the Commonwealth is acknowledging -
CHIEF JUSTICE ROBERTS: I’m sorry. Entirely what?
MR. FRIEDMAN: Orthogonal. Right angle. Unrelated. Irrelevant.
CHIEF JUSTICE ROBERTS: Oh.
JUSTICE SCALIA: What was that adjective? I liked that.
MR. FRIEDMAN: Orthogonal.
CHIEF JUSTICE ROBERTS: Orthogonal.
MR. FRIEDMAN: Right, right.
JUSTICE SCALIA: Orthogonal, ooh.

I was going to make a post about what dummies we have on the Supreme Court until I mentioned it to my girlfriend (whose degree is in psychology but she's still not a dummy) and she said, "Huh, what does that mean?".

So I guess the dummy is me, the one that just assumed that if it's a word I've commonly used it must be a word that everyone knows.

Card shuffling

2009-12-18T12:07:00.000-08:00

How many dovetail shuffles suffice?

Anchors

2009-12-06T15:12:00.000-08:00

Poker is partly a game of card distributions and partly a game of psychology. The card distribution aspects of the game can easily be straightforwardly modeled via rather simple mathematics (even if if might be computationally complex). The psychological aspects are often not so straightforward.

One of the psychological complications is the concept of anchoring.

Anchoring and adjustment is a psychological heuristic that influences the way people intuitively assess probabilities. According to this heuristic, people start with an implicitly suggested reference point (the "anchor") and make adjustments to it to reach their estimate. A person begins with a first approximation (anchor) and then makes adjustments to that number based on additional information.

Anchoring can manifest itself in many ways in poker, the most prevelant is the tendency to use pot size as an anchor in determining bet size in no limit poker. Bhy manipulating the pot size we can effect the tendency of our opponents to pick a bet size on future betting rounds. Anchoring our bets on pot size has become so common in poker that it seems natural.

It's a tendency that can be manipulated -- not just by manipulating the pot size, but by manipulating the physical size displayed by the number of chips. By using larger denomination chips we might make the pot seem smaller to the opponent. Using a green chip ($25) in a red chip ($5) game might do that. But using a black chip ($100) in a red chip ($5) game might have the opposite effect, the unusually large denomination might have the effect of making the pot seem huge.

One thing you can be sure of -- your bet denomination choices will effect your opponents future behavior, even if you can't be sure what that effect is.

A Computer Scientist in a Business School has some examples of anchoring.

Informs Blog

2009-10-12T06:51:00.000-07:00

INFORMS is an academic society for Operations Research and Management Science. It was formed a few years ago by merging two groups (Operations Research Society of America and The Institute of Mangement Science) that were formed separately in the 1950's. They used to be separate organizations that had joint meetings -- twice a year. One called ORSA/TIMS and one called TIMS/ORSA. I used to go to almost every meeting but since nobody is paying my travel and registration expenses these days I don't go anymore.

I did go to the one in Pittsburgh a couple of years ago to give a short talk on Poker Bankrolls and was going to go to a Regional Meeting in College Station last year to give a similar talk, but I had a heart attack and went to the hospital instead.

The merged group doesn't have two meetings a year, just one. And this year it's in San Diego and it's going on right now. What's cool about this years meeting is that it's being blogged in a group blog.

The blog is worth checking one, it's giving a pretty good superficial introduction to what Operations Research is all about.

I went to an ORSA/TIMS (or was it TIMS/ORSA?) meeting in San Diego back in the 1970's and gave a talk on a flaw in the way a game theory model of poker that had been published in the flagship journal (Operations Research) treated hand values in draw poker. Back then there were entire all-day sessions devoted to gambling models. Primarily because blackjack research was in it's heyday and modern mathematical finance was just starting out and often using gambling models to study financial markets.
INFORMS doesn't cover gambling so well anymore. The topic has kind of played itself out among math dweebs. They do cover applications in sports, but not so much sports betting as in sports tactical decisions.

Probability estimation in poker: A qualified success for unaided judgment

2009-09-27T14:15:00.000-07:00

In Journal of Behavioral Decision Making.

by
James Liley, Tim Rakow *
University of Essex, Colchester, UK
email: Tim Rakow (timrakow@essex.ac.uk)

Abstract
Poker players make strategic decisions on the basis of imperfect information, which are informed by their assessment of the probability they will hold the best set of cards among all players at the conclusion of the hand. Exact mental calculations of this probability are impossible - therefore, players must use judgment to estimate their chances. In three studies, 69 moderately experienced poker players estimated the probability of obtaining the best cards among all players, based on the limited information that is known in the early stages of a hand. Although several of the conditions typically associated with well-calibrated judgment did not apply, players judgments were generally accurate. The correlation between judged and true probabilities was r > .8 for over five-sixths of the participants, and when judgments were averaged across players and within hands this correlation was .96. Players slightly overestimated their chance of obtaining the best cards, mainly where this probability was low to moderate (<.7). Probability estimates were slightly too strongly related to the strength of the two cards that a player holds (known only to themselves), and insufficiently influenced by the number of opponents. Seemingly, players show somewhat insufficient regard for the cards that other players could be holding and the potential for opponents to acquire a strong hand. The results show that even when judgment heuristics are used to good effect in a complex probability estimation task, predictable errors can still be observed at the margins of performance.

Shifting Gears

2009-09-13T13:25:00.000-07:00

The current issue of Management Science has a cool article entitled "Poker Player Behavior after Big Wins and Big Losses".

The actual journal article is firewalled for subscribers only, the link takes you to an earlier version on one of the author's website.

Looking at playing histories collected from the 25/50 blind NLHE tables at FullTilt they found that after a big loss players tend to loosen up and after a big win players tend to play less aggressively.

It's a long article that goes into some of the behavioral theories about how wins and losses might effect gamblers.

Data Mining

2009-09-12T07:22:00.001-07:00

When I was a student, a long, long time ago, the term data mining described a frowned upon practice. Only bad people did data mining. It was a term used to describe the practice of doing a series of tests of significance on a set of data until a statistically significant effect was found. It was a form of sequential testing but it was a very unscientific, atheoretical form. Bad practice.

All that's changed. Data Mining today is a generally accepted practice, a tool that any good data analyst should have in their tool box. But today's version of data analysis isn't really the same as yesterday's version. There's some subtle but important distinctions to be made.

Wikipedia gives a simple definition for the term data mining

Data mining is the process of extracting patterns from data.

That's as good as any place else as a place to start defining the concept, but it's just a start. When we use the term data mining we're talking about more than just data, we're talking about a lot of data, a whole lot.

That's one of the ways that data mining differs from statistics. The field of statistics was developed as a tool to extract information from small amounts of data. Small sample statistics is the backbone of statistical theory and practice. We don't do that with data mining -- we're looking at extraction of information from huge collections of data.

Statistics is based on using probability models to fit data or to test hypotheses. Data mining is a process of uncovering models, not fitting models. A process of forming hypotheses, not testing them.

Here's a video of the first lecture a course on data mining given for first year grad students at Stanford and given simultaneously at the Google headquarters.

In this lecture he's mostly just defining the concept. I'll follow along in subsequent posts.

Having some problems

2009-08-25T18:40:00.001-07:00

I'm having some problems with my host site. The blog will return shortly.