Poker Mathematics & Arithmetic

Mathematics, Strategy

September 4, 2007

I’d like to say a little bit about my philosophy on mathematics in poker. Generally speaking, there seem to be two vocal camps on this topic. One camp is full of math-phobic players and writers (often forum posters) who will try to convince you that poker is a game of psychology, not numbers. The other group is the ever-growing number of mathematician players and writers who seem to talk about equilibrium solutions and similar high math incessantly.

By inclination, I side with the mathematicians since my academic background is heavy in applied mathematics. However, I’ve come to believe that fundamentally, both positions are wrong. Poker is indeed first and foremost a game of math. A firm grasp of the mathematical and technical aspects of poker is sufficient to defeat or break even against all but a small cadre of the most skilled opponents. Online a technically solid game is arguably all you need.

I believe the psychological elements of poker are dependent on and subservient to the mathematical ones. Specifically, it’s necessary to use mathematics to integrate the information gleaned from tells and psychological observations into your overall play. If this integration is done incorrectly, you may actually get worse results despite highly accurate observations about your opponents. As such, I find the anti-math camp laughable. That said, I believe there is more money to be won from psychology than technical play at the highest levels.

Much as I might like to, I can’t bring myself to side with the mathematicians either. Specifically, I think they’re doing the wrong math. There are a number of errors being made:

  • An unreasonable focus on precision – I always find it a little laughable when I read poker writing that states mathematical results out to four significant figures. While additional significant figures are usually a good thing for scientific applications, in poker they’re useless. Poker reasoning simply isn’t that precise – in most decisions one card being present of absent from the remaining deck tilts the probabilities 2% or more. Given that, reporting results at the tenth of a percent level is just silly as even a minor change in the assumptions causes a much bigger change than that.
  • Use of esoteric disciplines – The next time I see someone try to apply chaos theory to poker, I’m going to puke. Even game theory, which initially seemed to be highly applicable to poker appears to have been somewhat of a dud. I’ll have more to say about why game theory hasn’t panned out in another post.
  • Excessive computation – I’m always suspicious of poker math that has to be done on a computer because you don’t get to bring a computer with you to the table. Poker math should be simple.
  • Unwarranted averaging – you should always be suspicious whenever anyone averages anything. I’ll explain why in detail in a future post, but suffice to say that averaging often gives you a seemingly very accurate answer that’s in fact totally useless.
  • Silly assumptions, especially for simuations – There’s a classic computer programming expression: Gargabe In, Garbage Out or GIGO for short. What it means is that if you do math with the wrong inputs, even if it’s the right math, you get the wrong answer. So often, poker math is incorrect in its assumptions, and then even if you do the most correct and elegant math, you get a garbage answer.

Now, I’ve told you what I see wrong with poker math. Since I’ve already hinted that I’m none the less a big fan of math at the table, I owe it to you to tell you what good poker math looks like. Here goes:

  • Logical – The key to all good poker math is to adopt the right logical way of thinking about the current scenario. If the logic piece is wrong, you can do all the math you want but it’s the wrong math and you’ll come to the wrong conclusion.
  • Approximate – accuracy is not king in poker arithmetic. This stems from that fact that a small error in arithmetic can only produce a small error in play. It’s better to get almost the right answer in a means that’s easy enough to do at the table than it is to figure out the exact answer after you lose your money. Sadly, approximate arithmetic has been missing from the school curriculum since the demise of the slide rule, and almost no one knows how to do it any more. I’m here to help – stay tuned.
  • Turns complex concepts into simple math – Often times when mathamatical concepts are stated, they’re made more complex than necisary in terms of the amount and difficult of computation involved. I believe most if not all poker math can be done using the 4 basic arithmetic operations on integers under 100. In other words, 3rd grade math. Math that’s simple enough to do in your head at the table. And I believe this can be done without sacrificing any of the power derived from mathematical reasoning.
  • Integrated – Poker math needs to be capable of taking observations about psychology and player behavior and seamlessly meld them with knowledge about the card odds, pot size, and other factors in order to make decisions that are relevant in light of all the available information, not just the stuff that appears math-friendly on the surface. This is a tall order, and something nearly absent from the poker literature. I’ll have a lot to say on it in the future.
  • Precomputed – When computation-heavy math comes up, it should be done ahead of time and the results boiled down into a form that’s simple and easy to apply. A blackjack basic strategy card is a perfect example of precomputation working. The simpler, more generic, and more insightful the “condensed” form, the better.

Anyways, I’ve made a lot of promises of future material here. Bear with me as I get this stuff posted.

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