# The Importance of Mathematics in Poker

Poker is an awesome and very difficult game. But as a probability/strategy game is a subject of mathematics. However, I found a lot a rounders, even pros, that are completely blind to the power of mathematics in this game. The most evident example is in heads up play, that is completely resolved from the point of view of Nash equilibria.

Looking closer at heads up strategy of the majority of pro players, it is possible to find a lot of trivial strategic errors. And yet, even on this useful website, it is still not possible to use latex to explain probabilistic formula! Why are people so reluctant to use mathematics in poker?

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Your premise is off here. It isn't that high level players ignore math, it's that a lot of other concepts start to become more important. The incomplete nature of the information that we are given in poker means that it will never be a purely mathematical challenge. Figuring out what flaws exist in your opponent's game and then exploiting those flaws will result in better results than an approach based purely on the math of the game can provide. As you face more advanced opponents, the importance of these types of concepts only increases.

Further, as IHars mentioned, even in the heads-up arena, math doesn't solve the game, because neither player ultimately plays according to the mathematical optimum. And for the one who is best at adjusting, they will end up achieving better results than a Nash-based approach would yield.

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To add to this very good answer, it can also be a mathematically sound decision to make a sub-optimal play (one not supported by the mathematics of the hand) if it will allow you to achieve higher value in future hands because of it. (ie, if the other players perceive you as a "bluffer" because of how you played a few hands early, when you stop bluffing and start value betting in a way that looks like a bluff, it can be very profitable.) –  lnafziger Feb 3 '13 at 22:32
I'm not sure the idea that because we have incomplete information that means that it will never be a purely mathematical challenge. I'm not saying that poker is a purely mathematical challenge but there is a lot (most of applied maths) maths dedicated to treating situations where we have incomplete information. –  hmmmm Mar 9 '13 at 12:56

Heads Up is very special type of game... Adjustment your style to the opponent, bluffing is more important than following to some mathematically optimal strategy.

Look at Viktor Blom (Isildur1). His hyper-aggressive and "non-optimal" style baffles most opponents on high stakes.

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Bluffing correctly would actually be part of a mathematically optimal strategy, so although it might be more frequent, without extra information about profitability, it's problematic to assume that it's more or less important than any single part of a total strategy. –  Toby Booth Jan 30 '13 at 20:02

The point of winning in poker is to maximize your EV. Math is great tool for analyzing multiple situations, and so part of your decision process. Many successful players, not knowing math, are really good at estimating their optimal play without using it. That is all about it, deciding your optimal moves better then your opponent. If you can do it w/o math does not matter.

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Well my answer for you question comes in a few parts.

First off I'm not too sure if you understand what you mean when you say that "heads up play is completely solved from the point of view of nash equilibria". Yes you can google for a couple of charts and get a good idea of some basic heads up strategy but saying "completely solved" is a bit misleading. Being in a Nash equilibria for a two player game means that player A cannot change his strategy unilaterally, in the knowledge of players B's strategy to make more profit and vice versa. One problem here is that player A needs to know player B's strategy and vice versa, in poker this is simply not the case. If both players are playing heads up play completely according to Nash equilibria then great but that isn't really the case and if it was then one player would surely adjust to make their play more explotive.

The thing to notice about the Nash equilibria is that it give exploitative play but not optimal play. So saying that heads up play is completely solved by Nash equilibria is maybe a bit misleading (there is a Nash solution but it's not optimal).

So I would say that my first point is that some people do seem to overestimate the importance of mathematics in poker. You can't just look up a few charts and have the game solved and maybe more importantly people sometimes fail to notice when mathematics is giving an explotive solution not an optimal one.

Bringing me onto my second point which is maybe a bit more in response to some of the answers here is that people also seem to underestimate the importance of mathematics in poker. In the above answers I can see maybe see some misconceptions already (maybe not, I could just be reading them wrong). In reference to Isuldur1's hyper agressive play being "mathematically poor" and non-optimal, I would say is maybe a bit naive. Looking at that sort of strategy with some pretty basic mathematical tools we can see some advantages of it.

(from phil gordon's little green book of poker) If you consider a heads up match with a hyper aggresive player (HAP) and player A playing regular poker, consider the following stacks are $5000, preflop pot$500:

Player A (A, k)

HAP (7,7) (Ts, 8s) ( 8s,5h) (5h,9s)

Flop (Ah,5s,6s)

HAP moves all in. What does player A do? Well, if you are in the knowledge that you are playing a hyper aggresive player with top pair on a flop such as this you may be thinking great, easy call but lets look at a rough table:

Hand A chances A equitiy 1 (Ts, 8s) 52 5,481 2 (8s,5h) 63 6,615 3 (7,7) 1.6 168 4.(5h,9s) 76.3 8012

Player A total equitiy vs HAP: -\$1448

Now that is interesting. Playing very aggressively with these hand can actually give you a positive equity against a strong made hand.

Now I am not attempting to argue that this in any way clears the matter up or that this is an in depth analysis- it clearly isn't but what it does show is a particular example of where some (very basic) math can give us a good strategy and some idea of where Isuldur1 and other players hyper aggresive strategy comes from.

Of course if you want some more in depth analysis then you need to use some more advanced mathematics (it's easy to come up with a strategy that counters the above type of play) but as you start to look in more generality the math becomes more vague. This does not make it less useful and (I would argue contrary to some of the above answers) it does not make it less applicable in higher level games it just means you need some higher level maths.

For anyone that is interested in maths and poker I would reccomend Bill Chen's Mathematics of Poker as the best book by far.

I may need to come back and edit this post as it is a bit long. Also I would also like to say that we really should get latex (mathjax or something) for this forum as it would be extremely useful! (I might bring it up on meta as a suggestion)

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Nice post, but I think the specific example you give is incorrect. maybe a typo? The matchup you state give's the Hero ~82% equity at this point. Otherwise, the post brings up some interesting points. –  Toby Booth Jan 30 '13 at 19:56

Up to a point, mathematics is important. A good grasp of it will allow to beat average players.

Beyond that point, psychology is key. "Good" players have a grasp of the basics. Then it resolves into, "he knows that I know that he knows," and then things resolve themselves into how deep the chain of thinking goes (how many "ply") in mathematics. That is something that say, a computer cannot be taught.

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