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Math Help - Model for No-limit Texas Hold'em

  1. #1
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    Model for No-limit Texas Hold'em

    Greetings. This is my first post.
    Here comes a rather long article of my dig into the analogy between NL Hold'em & coin flip and other models. I hope you enjoy reading it especially if you know how to play it. Comments & corrections are most welcome.

    ***

    I've heard that people using the analogy that Poker is like a coin flip. For an average player the coin is exactly 50-50. If you are highly skilled, your coin is slightly favored to your side. A 60-40 advantage would be huge. At that time I thought "60-40 is not huge. I can still lose 40% of the time. A 90-10 would be huge". Yeah in my dreams.

    For a few days during my walk to work I've been thinking of a model to describe this coin flip analogy. Let's say each hand is a coin flip. How big an advantage does a good player have on his coin? How good is your coin?

    First, let's define the problem a little. We probably agree that a good player earns, in average, about 10 x Big Blinds (BB) per 100 hand.

    Let's use 10XBB / 100 hands as an example.

    Next we need to know the average pot size. I went to PokerStars.com and took a look at all available NL Hold'em tables at the time:

    Table (BB=) Average Pot (XBB)
    $6 --- 11.4
    $4 --- 11.9
    $2 --- 12.6
    $1 --- 12.4
    $0.5 --- 12.2
    $0.25 --- 15.1
    $0.1 --- 12.3
    $0.05 --- 20.7
    $0.02 --- 25.5

    You can see that at higher limit, the average pot size is around 12 X BB, while this ratio grows at lower micro limits. Let's use 12 times big blind as the magic number for now.

    So how does 10XBB / 100 hands look like in a coin flip?

    In 100 hands, there will be a total pot of 100 x 12BB = 1200 BB.
    You get a profit of 10 BB of it.

    Say in this coin flip, you bet $1, and your opponents (in the poker sense, you are playing against all of your opponents as one entity) bet $1.
    The coin is flipped. If you win, you get $2 (pot size = $2). Your net win is $1.
    If you lose, you get nothing. Your net loss is $1.

    Next we can look at a situation of 60-40.
    When you invest $1 for 100 times, in average --
    Your result is 60 x 2 + 40 x 0 = 120.
    You win $20 per 100 hands, with an average pot size of $2.

    We know the average pot size is 12 times of BB. BB in this case = 2/12 = 1/6.

    Therefore, your winning per 100 hand of a 60-40 coin is:
    20 / (1/6) = 120 (BB). You are expected to win 120BB per 100 hands.

    Now this is 12 times better than your average earning of 10BB / 100 hands!
    In reality, whoever wins 10 BB / 100 hand has a coin 12 times weaker than a 60-40 coin.
    The coin is actually a 50.83 vs. 49.17 coin.

    That's not really an excellent coin flip you might say! But keep in mind that you are also fighting against the casino rake. From my experience, the online casino rakes about 6 x BB / 100 hands from each player at the table. Therefore, you are actually flipping something close to 51.2 vs. 48.8.

    ***

    Another interesting question is to see how good the professionals are doing. Last time I read an article saying that a pro is earning an hourly rate of $100 (just as an example) but his standard deviation is $600. Now how good is his coin flip?

    First I checked Pokerstars and worked out the average hands / hour. It is about 90. For simplicity we will say 100 hands / hour.

    Let's first simulate what is going to happen with a 60-40 coin, flipping 100 times.

    If everything goes as expected, the player wins 60 pots and loses 40 pots, netting a 20 pots average (per hour).

    His standard deviation would be 0.98 pots (thanks to MS Excel).
    That's better than 20:1 ratio between profit and standard deviation -- too good to be true. He is actually getting 1 profit : 6 standard deviation!

    The pro is actually getting a coin that is 120 times worse than the 60:40 coin. It is 50.083 vs. 49.917.

    ***

    But, are we really 10 times better than a professional? There might be something wrong in the model. When we use our own examples in the coin model, we start from our BB/100 hand. However, when we apply the coin model to the pro, we start from his standard deviation. Now, if our coin model underestimates the variance, we are going to give the pro a poorer coin than he deserves.

    Actually, it is very likely that the coin model is underestimating the variance of NL hold'em. This is because:

    - People don't play every hand. By playing fewer hands there are more variance. Our coin model assumes that you do play every hand.

    - The frequency of all-ins (either doubling up or lose everything) is not negligible, and when that happens the pots are usually HUGE.

    The next closest thing that comes up in my mind is a DICE. A standard dice has 6 faces, numbering from 1 to 6. When you roll the dice you can expect 1-6 to show up with equal chances.

    A standard dice would have a higher variance than the coin model. By rolling the dice 100 times, you are expecting an average of (1+6)/2 = 3.5 (or, 0 gain). Your standard deviation would be 1.71. That STDEV is 70% higher than flipping a coin.

    Yet the good thing about dice is that we can change its numbers on the faces. Let's now exaggerate the dice a bit to reflect:

    (1) Most of the times you don't play a hand, hence the gain/loss is minimal
    (2) Sometimes when you do get heavily involved, your either win a lot or lose a lot

    Let's paint the dice with:

    5
    1
    0
    0
    -1
    -5

    This is the amount you are going to win/lose.

    The next interesting concept is the average pot size.
    For one approach we can average the dice numbers. (5+1+1+5)/6=2
    For another approach, in reality both you and your opponents (as one entity) still each contribute $1 to roll, so the pot is still $2.

    Actually, I purposely designed the dice to be this way, so the two approaches are equivalent. Now we don't have to worry about which approach is correct. (In fact I have no idea which one is correct. )

    Now let's roll this 5,1,0,0,-1,-5 dice for 100 times. For an average player:

    Expected win: $0
    Standard deviation: $2.94

    Now we have successfully elevated the variance of our model, from $1 (coin) to $1.71 (standard dice) to $2.94 (our customized poker dice).

    That's for an average player. How about a good player? The dice is now biased towards the positive side. We can imagine the positive side of the dice (0, 1, 5) as one side of the coin, and the negative side (0, -1, -5) as the other side of the coin. Therefore, a 60-40 dice has 20% (60/3) chance to hit 0, 1 or 5, and 13.33% (40/3) chance to hit 0, -1, -5.

    Now let's revisit our examples:

    [Case 1] Professional with his hourly rate:standard deviation = 1:6

    Let's still assume he plays 100 hands per hour.
    It turns out that the Pro is using a poker dice of 50.122 vs. 49.878.
    (Compare to our previous coin model of 50.083 vs. 49.917.)

    You might think there isn't a lot of difference at all - indeed there isn't! Our pro, using our new poker dice model, is just having a coin (dice) that is 50% better.

    [Case 2] You earn 10BB / 100 hands.

    Everything remains the same. Your poker dice is 50.83 vs. 49.17.

    Your poker dice is still much better than the Pro! But if you pay a closer attention, the difference between the Pro and you have reduced a bit, due to the increased variance of our new model.

    ***

    It seems to me the poker dice is doing ok but not that great. I further improved the model with an imaginery dice of 20 faces.

    Face 1 -- 15
    Face 2 -- 4
    Face 3 -- 1
    Face 4-17 -- 0
    Face 18 -- -1
    Face 19 -- -4
    Face 20 -- -15

    Now that sounds like poker even more! When I run the same calculation, the standard deviation becomes 4.92. This is the turbo version of our poker dice (stdev = 2.94).

    So what kind of Turbo dice is our pro using? I will save you the trouble of reading and just tell you the result: 50.205 vs. 49.795. Our pro's true edge is getting more obvious now.

    Yes, this is still weak compared to your dice (10BB/100 hands). You are doing better probably because our opponents are much weaker at low/micro limits.

    ***

    I still remember the first night I learned poker as a loose passive donk. After watching people bad beat each other like drinking water, I thought "poker is not really all about skill. I might say it is 3 skill -- 7 luck in any given night".

    For this 3 skill (PROFIT) - 7 luck (STDEV) or better to happen, say 100 hands are dealt during the night, you would need an edge better than 50.5 vs. 49.5 per hand. Otherwise you are more helpelessly manipulated by the poker gods.

    ***

    By the way, Casino games usually feature odds much worse (for you) than that. For example, a single 0 Roulette gives the house an edge of 1/37. This is like a 0.514 vs. 0.486 coin. Roulette with 0 and 00 would further double the house's edge.

    Yet once I read that a good pro's edge at a casino NL hold'em table is as high as 5% (52.5 vs. 47.5 coin). I guess at that time most people didn't really know what they were doing.

    ***

    To show that you didn't waste your past 10 minutes, here comes the summary:

    (1) A good player's edge is tiny (usually less than 51:49 per hand) in poker. Casino games usually give the house a greater edge.

    (2) Small advantages build up to give you a profit, in the long run.

    Thanks!
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  2. #2
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    I am an avid poker player and a big studier of poker theory for NL Hold Em. You have constructed an interesting analogy of poker being a weighted coin, but unfortunately you have made some assumptions that very much restrict the ability of a poker player to win the most money playing ideally.

    I want for now just to say that the fundamental money making concept in poker is the expected value of betting.

    E.V. = (% of winning the total pot) * (total pot size - your bets)

    You can see that we want both our percentage of winning the pot to be very large and the total pot size as well to be as large as possible.

    Thank your very much for your post. I don't know if you play poker but the fact that you are able to look at it mathematically is a great indication that you can be a winning player.
    Last edited by Jameson; November 8th 2007 at 12:06 PM.
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  3. #3
    Super Member angel.white's Avatar
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    I don't know what you suckers are talking about, I just get a feeling about whether or not my card is coming...
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  4. #4
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    I want to start by saying that I think doing analysis of this particular form of poker can be more difficult than other casino games. Take blackjack for instance. In BJ the betting is placed at the beginning of the hand and will not increase unless you double down. This means the most one can lose is twice the original bet placed.

    In no limit hold em the amount won or lost has a much greater range. The most one can lose in the whole amount in front of the player and the most one can win is that amount times the number of players at the table.

    Let's talk about how optimal play can affect the edge one has in winning for either game. In blackjack, there is a well known optimal strategy that is the mathematically best choice of action in each hand depending on the cards one holds and the shown dealer card. If one places the same bet every time AND plays with this optimal strategy, the house has an edge of less than .5% I believe. The way blackjack is beaten is by counting cards and knowing when the few times occur that the player will have an advantage over the house. At these times, the player should place a larger bet the correlates with the predicted edge at the time, and in the long run this can lead to the player winning.

    I want to take a look at something you said.

    Yet the good thing about dice is that we can change its numbers on the faces. Let's now exaggerate the dice a bit to reflect:

    (1) Most of the times you don't play a hand, hence the gain/loss is minimal
    (2) Sometimes when you do get heavily involved, your either win a lot or lose a lot
    I hope you don't take offense, but I feel both of these assumptions are't 100% correct and will lead to an incorrect analysis.

    (1)If one doesn't play a hand, meaning that player folds before the flop, then the gain/loss is 0 unless that player was in either blind. If in the small blind the amount lost is one half of the big blind for that table and if in the big blind that player would lose the amount of that big blind. You weren't incorrect in calling this minimal, but we can be more exact.

    (2)I don't understand what you mean by "heavily involved" and "a lot", but I would think that if one doesn't play very far into the hand then one would not have put a large portion of their stack into the pot. I would like it if you could elaborate on this please.

    I have to go for now, but I'll post more. I hope you return to continue this discussion. I would love for more of your thoughts.
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  5. #5
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    Yep, the models are never perfect. I was just trying to describe poker in some simple terms. Maybe the dice can be refined a bit to stress the nature of blinds (thus you get a -1 and -2 every X times). The very high end of the dice should be of larger value because you may win a lot more because there were multiple players going all-in with you with inferior hands.

    The ideal model of any game (say, a Saturday night home game that you paricipate) would be running a simulations with algorithms matching each player's personality, skills even positions, through thousands, even millions of hands.

    Even if the current coin/dice analogies are not so close to reality, I hope the main idea holds true - luck is still very important for any given night. Just like Casino is bound to win money in the long run, poker pros make a living by playing a lot more hands!

    And... if you have a good feeling about a good card is coming next, you are not spending the thinking time in the right place! You should consider your odds and your opponents cards, what they think you have, what they think you think they have, and what they think you think they think you think they have... that's probably a little too much.
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  6. #6
    Super Member angel.white's Avatar
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    Quote Originally Posted by maltz View Post
    And... if you have a good feeling about a good card is coming next, you are not spending the thinking time in the right place! You should consider your odds and your opponents cards, what they think you have, what they think you think they have, and what they think you think they think you think they have... that's probably a little too much.
    No, sometimes I just feel it.




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  7. #7
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    Quote Originally Posted by maltz View Post
    The ideal model of any game (say, a Saturday night home game that you paricipate) would be running a simulations with algorithms matching each player's personality, skills even positions, through thousands, even millions of hands.
    Very good point. My sentiments exactly.
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  8. #8
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    I don't know what you suckers are talking about, I just get a feeling about whether or not my card is coming...
    Maybe that is why You are loosing so much money at poker....


    OK, one basic mistake in Your models I found is that You are assuming to play against an entity of opponents.

    If I make a $1 Bet in Texas Holdem then the pot isn`t $2 unless I play against only one opponent. If I play against 4 people then the pot is $5.
    In No Limit Holdem good players fold most of their hands and win HUGE pots in the few once they are playing.

    Another thing is that in Poker You don`t have to hold the best hand to win. You can force Your opponents to fold by making HUGE bets. The ability to bluff is a major difference to all other casino games.

    Think of a roulette game in which no one sees where You put Your chips.You tell the croupier that You have chosen the number that has shown up. You offer him to pay You 10 times Your bet amount without showing Your bet. Or You show and he has to pay 37 times Your bet.

    Another major difference is that You have a chance of changing Your odds in poker. If You look at it Your way ( Playing against an entity ) then Your cahnces of winning with AA ( two aces being dealt to You ) are only 28% , assuming there are 10 players at the table. If You force some people to fold by making a huge bet You can end up as a 70 to 30 favorite.
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