Poker is totally mathematical

I have just read a thread in which people said "poker is not all about maths," "luck is a factor in poker," "it is not always right to take +$EV decisions."

I am now going to show that everything in poker can be expressed mathematically, and prove that those people who say poker is not all about maths are incorrect

Luck does not exist. Each poker "event" has a likelihood based on some measure of prevision or "probability" which is in turn based on your own judgements about the likely ranges of your opponents hands (from reads and bet sizing) and more deeply about your own judgements of the mechanics of the game.

If you believe in this mysterious "luck" idea then in order to be coherent you must alter your probability of winning a showdown. To make this concrete ill use an example from another thread:

I have AK and I believe my opponent has a pocket pair. He has gone all in after people limped and I had raised. I believe that the deck has been shuffled well so that as far as I can tell any card which I have not seen is equally likely to come down on the board. I have made a coherent probability statement so now I can work out my EV for the decision to call and the decision to fold, and take the decision with the maximum EV. In the thread the call was the right choice.

However believing in some mystical force called luck changes my statement that each card is equally likely to come out, and that in some way my aces and kings are not as likely/more likely to appear than the other cards. I must now respecify my probabilities and use those to work out my equity for the hand. If that turns out to be -$EV then I should fold. But I fold ONLY because by believing in the mystical force luck I have changed the nature of the game in some sense.

But what of the argument that $EV is not the only factor?

Well it is not! But all decision making is mathematical. You have a reward, which in the case of poker is money. You specify your utility for that reward, i.e. a measure of how much you prefer one amount over another amount. Then you make the decision that maximises your expected utility. This completely specifies the problem because not choosing this optimal decision means you didnt really believe your utilities in the first place and is therefore incoherent.

Clearly then $EV is just a specific case of expected utility based on the assumption that we prefer, for example, $2 twice as much as $1. You can't just use $EV all the time and think you are making the right decision though!! You have to be careful:

e.g. I have only $100 in the world. Without it I can't eat and will starve for a whole week. Someone offers me to toss a coin for my $100 and says he will give me $210 if it is heads and moreover that I can use any fair coin and toss it myself (so that he has no influence on the game.)
My $EV is clearly $155 for playing and $100 for not playing so based on that I should play. But would you knowing you'd starve if you lost????? In this case my Utility for money is not linear. I need my $100 so badly that it dominates the function. Assymptotically, as my wealth increases and losing $100 does not matter, then I always take the bet, as my utility tends to $EV.

This has deep meaning for poker. If you are not properly rolled, then many decisions that are +$EV are incorrect for you. Which is the mathematical reasoning behind playing with a proper bankroll, because if you do then your utility for money (in your roll) is proportional to the money itself and so $EV decisions make sense.

So if you ever worry about the amount of money you lose if you make the call, (and this includes thinking that "later ill win more if i save this"), that is the same as playing scared and means that for you maximising $EV is not a coherent way of making your decisions.

If maximising $EV is not optimal for you, (and mathematically it really doesn't have to be and in many cases shouldn't be), then the decision making advice on this forum is incorrect for you. Only with a proper bankroll which must be treated as if infinite (in comparison to the stakes you play) so that your utility for money is completely linear, can the advice on here (and in nearly any book) make any sense.

If you are arguing for not taking an edge therefore, you are making a subjective statement equivalent to, "the money i'm risking here is too much for me" and thus posters on this forum can only tell you that you should stop playing the stakes you are if you want the mathematics of poker to make sense.

We would be wrong to say that you should always maximise youe $EV, because as I have shown this may not make sense for you and your preferences about money. Clearly however we cannot give any advice to you either if your preferences are not completely specified to us. Furthermore if you do not subscribe to a linear utility function for money in your bankroll then you will not make as much money long term as someone who does.

If there is any decision in poker that is not mathematical, I challenge you to find it!

Great job Gingerwizard.

*Last edited by HalvSame on Wed, 21 Feb 2007, 3:32pm; edited 1 time in total*

Great post. Now dissect these statements mathematically, they drowned in the Folding AK preflop thread when it descended into luck hell:

Great post,
I spend a chunk of time every home game trying to explain to people that luck has nothing to do with poker with very little success. I'll have to point them to this post, very well put Gingerwizard.

Depends on your definition of mates. guys I know, work with, or just play poker with, I tell them nothing. My Freinds, real freinds, they know my commitment, and think I'm full of shit, with how much I win and never deposit. They think that there is nothing to learn about poker, get your cards, bet or fold. That's what they play.

Meh, I tell anyone who asks how dedicated I am. I also point them to this forum. I don't really play my home game for profit only, it's more a social/recreational thing. I don't think any of them have taken my advice to go to the forums and study the game. Hell, my roommates play like a couple of hours online every day, and I've only just convinced them to get PT and PAHUD, they don't even bother posting hands in the forums. Bottom line is, to make money in this game takes dedication, and your average fish isn't going to have that dedication.

That was excactly what I was looking for. Thank you.

Thx Halv.

dont say much coz i want the games to be serious in nature. im sure they also wont bother with any advise etc as you say.

I said this in the other thread but for those who believe in luck plz know that it's going to stunt your growth in poker.

This is a great post. I'm gonna add it to the beginners digest in the BC. Thanks for writing it, gingerwizard.

"Gingerwizard's post"

QFT QFT QFT

Superb post and very well explained and reasoned. I agree 100% - it's a great explanation of why playing with a sufficient BR is critical to this game.

BTW nice post ginger

Gingerwizard wrote:

Yes I remember this argument from the thread: The mathematical answer goes like this:
You see your hand and have beliefs about how good it is. Then information is provided in the form of the betting activities of the other players and you need to use this information to update your beliefs about the likelihood that you hand wins in a showdown. So e.g. when 3 players fold your prevision for the event that you win the hand will increase almost surely.

So if you believe that players limping means Ax a certain % of the time then you must update your probabilty of winning a showdown with the PP hand given that some players limped. Once you have done that you are free to work out your $EV. This makes it purely a math call, to use the language of the guy who mentioned it.

Now whether it is valid to assign a certain probability to a limped hand having been Ax is a subjective matter of choice (as with all meaningful probabalistic statements). Of course the limped hands already have their prior probabilities of having been Ax which of course are already in the standard $EV calculation (all cards equally likely), and if we wished we could attempt to update this probability conditioned on the event that they limped the hand based on our own judgements of what range they would limp.

However in my own subjective view, to do so would be folly unless playing a bot with a known limping range. As a limp can be based on card strength, position, the fact that there is a loose player they have position on, feeling lucky, being on tilt, being tired, bored, "mixing it up", or even a randomly timed event based on some players decision to, say, limp/shove from EP if he ever got [6d3s], one can, and in my view should, treat a limped holding that has been mucked after a raise as any two unkown cards. In that sense all cards you have not seen are therefore possible values for the board.

Gingerwizard,

A more practical way of deciding the likelyhood that the six callers held an ace or king would be to compare the activity in this hand to the activity in prior hands. If the button is often facing six callers then my notion that aces and kings are likely to be held by the callers is questionable. However, if this is far from the norm then I don't think it unreasonable to think a high probability exists that one or two of your outs are gone. Think about it. Six callers have paid money to play their cards. That's twelve cards out of the thirteen available. Does it seem unreasonable to you that the two highest cards would have each been played?

I don't mean to detract from your thread but that other thread got out of hand. I'm still curious about the calculation of the AK call given X outs are gone. Anyone?

Aww. One of our posters is growing up. *sniffle*

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Poker is totally mathematical