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Best Percentage Of Hands To Be Played


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Before I played much or read much, I thought a lot and came up with a theory on what is the optimal number of hands to play, and came up with a theory. Now, for the first time, revealed to the world, is the Hblask Optimal Play Selection (HOPS) theory, described in detail.Whenever I want to think about a general poker problem, the first thing to do is simplify the situation. So for this problem, the simplification is: everyone plays exactly the same, based on how you play, but it takes them a while to react depending on what changes you make. So, how many hands should you play?If you play the top 10% in a 10-handed game, on average 1 person will see the flop each hand and will win it outright. If you play 20%, 2 people will, on average, see the flop, and you will win half the hands you are involved in. All else being equal, if you win half the hands, you will be a break-even player. If you play more than 20%, there will be more than 2 people per flop, and you will be a losing player. So what does this converge to? If you play 10%, you are leaving money on the table compared to the people who play 11%, because they win the ones you do, plus a certain percentage of those extra 1%. So basically, all else being equal, you should play just less than 20% of hands at a 10 person table. More generally, you should play just enough hands so that there will be slighlty fewer than 2 people seeing the flop if each person plays that percentage, so at a 9 person table, that's 2/9, or 22%, at a 5 person table, that's 2/5 or 40%.Now, everyone doesn't play equally, so you have to adjust your own percentage up and down by the following factors: quality of pre-flop play of opponents; percent of hands opponents are seeing; quality of post-flop play by opponents; your ability to put opponents on a hand so that marginal hands can be played profitably, quality of your pre- and post-flop play, and margins of error in all of these things, differing betting patternsl, and a bunch of other things. I will leave it as an exercise to the reader to figure out how to adjust based on each of these factors.Usually, for an average player, that makes the formula for profitability in hand selection to be approximately (2 / number of players) - 2%. Excellent players should play more, weak players should play fewer. (In it's extreme, this becomes the Smasharoo Theory).I'm convinced that anyone who understands and implements this theory alone will be profitable. I've never seen any detailed discussion of this and don't even know if anything has ever been published on it. And of course it all depends on everyone knowing which hands are good and which aren't, but that's sort of figured into my minus 2% factor above.OK, tear it apart. On your mark, get set, go....

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Okay, I first want to clarify our assumptions. Are we assuming that everybody plays the same percentage of hands that we do? And are we assuming that everybody has the same level of skill that we do? Assuming the last is exactly equivalent to assuming that with N players seeing the flop, any one of the players will win the hand 1/N times (in our simplified model where players play only the top percentage of hands in accordance with what percentage we give them). How do we feel about these assumptions?Okay, let's dig in:

If you play more than 20%, there will be more than 2 people per flop, and you will be a losing player.
I don't understand this part. Why will having three players to a flop make us a losing player? There will be three units of money in the pot and we should win it 1/3 of the time, so we should be break even.In fact, under your assumptions, it seems to me that everyone will just break under any percentage that we give them for hand ranges. The only thing that will make a person not break even in the long run is if we give them a skill advantage. We would have to assign a certain number of times they win the hand on average based on a number of players on the flop. For instance, if we state that they win 55% of all two person flops and 40% of all three person flops, or something, we can come up with how much they win and what range they should play.I may be wrong about our assumptions and therefore my assessment of your theory could be off. Let me know.
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Okay, I first want to clarify our assumptions. Are we assuming that everybody plays the same percentage of hands that we do? And are we assuming that everybody has the same level of skill that we do? Assuming the last is exactly equivalent to assuming that with N players seeing the flop, any one of the players will win the hand 1/N times (in our simplified model where players play only the top percentage of hands in accordance with what percentage we give them). How do we feel about these assumptions?Okay, let's dig in:I don't understand this part. Why will having three players to a flop make us a losing player? There will be three units of money in the pot and we should win it 1/3 of the time, so we should be break even.In fact, under your assumptions, it seems to me that everyone will just break under any percentage that we give them for hand ranges. The only thing that will make a person not break even in the long run is if we give them a skill advantage. We would have to assign a certain number of times they win the hand on average based on a number of players on the flop. For instance, if we state that they win 55% of all two person flops and 40% of all three person flops, or something, we can come up with how much they win and what range they should play.I may be wrong about our assumptions and therefore my assessment of your theory could be off. Let me know.
You are right, I explained it fairly badly. My simplifying assumption is that everyone is playing identically, and you are allowed to change only the % of hands you play, which the other players do not react to, at least not initially. So now what percentage do you want to play? If everyone plays an equal number of hands, it will always, on average, be a tie, only losing to the rake.But if it is currently 20%, and you bump yours to 21%, your average hand will be worse than the other players' average hands, so you will win slightly fewer showdowns than anyone else, because of the weaker hands. It is only when you are heads up with another player with equal hands that you are 50/50. If you play fewer and they don't change, your average hand goes up and you win more than 50/50. If you play more, your average hand goes down and you win fewer than 50/50. If you are each playing 15% of hands, that will be 1.5 people per flop, and a third (?) of the time a hand will be won uncontested. Therefore, by moving your percentage up, you will contest hands that you can expect to have the best of a certain amount of the time because you are still better than the break even 50/50 point.Hope that makes more sense, although you have me doubting everything again.
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It makes more sense now. Give me a minute to come up with my solution.
Bzzzt, time's up. The theory stands. Bye.By the way, I should add, this is for cash games. I have been unable to develop a theory for tournament play yet, although it is an interesting problem.
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Hblask,does your theory account for the fact we can actualy fold/bet/raise/call/check post flop?Does it account for the fact we can play a wider range of hands after several others have limped? or from Late Position ? or against players who take marginal hands too far, or are passive post flop ?this theory is worthless.there's no complex problem behind it.It's just completely oversimplifiedYour theory might address the "If we played a game of push/fold preflop with equal stacks, how many hands should we play, assuming everyone else plays the Top 20% " So ya got that going for you.

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Money's already in the pot from blinds. Being a 49/51 dog is still profitable.
I think I threw that into the slop factor, but there could certainly be a mathematical derivation of this particular bit of info.
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Hblask,does your theory account for the fact we can actualy fold/bet/raise/call/check post flop?Does it account for the fact we can play a wider range of hands after several others have limped? or from Late Position ? or against players who take marginal hands too far, or are passive post flop ?this theory is worthless.there's no complex problem behind it.It's just completely oversimplifiedYour theory might address the "If we played a game of push/fold preflop with equal stacks, how many hands should we play, assuming everyone else plays the Top 20% " So ya got that going for you.
As for position and limping, etc, I'm assuming that evens out over the long run, since there's no reason for any particular person to get a certain position or a certain hand in a certain position more than others.My theory accounts for the fact that people playing at a particular level tend to have similar skills, although that is certainly not true all the time. But certain tricks and techniques either work or don't, so they will be adapted by all players over time based on their success. Continuation bets are a good example. When they were relatively unheard of they worked most of the time. Then everyone learned about them, and "picking off continuation bets" became a good theory. Now they've sort of reached an equilibrium. And that is my point in all of this. As people incorporate techniques, and the level of play equalizes, the optimal percent of hands to play will settle in a very narrow range, because if someone is that much better based on using some particular betting pattern or psychological technique, others will pick up that pattern and negate it. That leaves some un-describable characterisic called "inate ability" to differentiate the good and bad players, and for most games in the middle range, there's not much difference. So what does that leave? Hand selection.Now, good players recognize the limitations of this theory and quickly adapt. DN in the Main Event is a good example. He said he was raising around 90%. Clearly this is theoretically insane. But if your post-flop play exceeds those of the other people at the table by vast amounts, it can be profitable. It'll work if the players are very timid and fold to most any bet pre-flop and/or post-flop. Then it's just you winning by stealing and 1/3 of the time with random junk hitting.Unless you believe that you are much much better than the other players in the room, then my theory is a useful guideline. I've been in rooms where I play almost 40% of the hands 10 player because I know everyone sucks so bad post-flop. I suspect the further you get from the middle range of play, the less valuable my theory is. The 0.10 rooms and High Stakes Poker probably can ignore it safely. Everyone else should at least think about how it applies. What are your strengths? How many extra percentage of hands does this buy you? Is this offset by your weaknesses? The truth is the rule is only marginally useful, but my best description of how to use it is to use it as a starting point and adjust up or down based on your level of play and of your opponents at any given time. So the next time someone asks "what percentage of hands should I be playing?", you can answer: Start with 2/number of players - 2% and adjust for your skill level.
I'm slowly coming to the conclusion that this problem is extremely difficult.
Yes, it is. I started thinking about it because I was thinking of writing a Rational Player simulation on my computer, and needed a starting place. The difficulties just in this simple part of the problem convinced me to not attempt the program.
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So, the amount that we earn from a given playing style is related to the following:i is the amount of players in the handM is the percentage of hands we playN is the percentage of hands that our opponents playM*Sum(from i=0 to 9) of [(average pot size for i players)(percentage that there are i players in the pot given that they all play N percent of pots)(percentage that we win the pot against i players playing N percent of pots when we play M percent of pots)]All we have to do is maximize this equation in terms of M. The most difficult part of all of this is the last part, since one basically needs to use a poker calculator program to get good numbers for pitting one range against another. If one could come up with an equation (in terms of N, M, and I) for the percentage of the time that M wins the pot, that would totally solve the equation and we would have an analytical solution for the problem.The problem can be solved given fixed numbers for N and M, but I'm pretty sure it's impossible to exactly form a general equation.

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So, the amount that we earn from a given playing style is related to the following:i is the amount of players in the handM is the percentage of hands we playN is the percentage of hands that our opponents playM*Sum(from i=0 to 9) of [(average pot size for i players)(percentage that there are i players in the pot given that they all play N percent of pots)(percentage that we win the pot against i players playing N percent of pots when we play M percent of pots)]All we have to do is maximize this equation in terms of M. The most difficult part of all of this is the last part, since one basically needs to use a poker calculator program to get good numbers for pitting one range against another. If one could come up with an equation (in terms of N, M, and I) for the percentage of the time that M wins the pot, that would totally solve the equation and we would have an analytical solution for the problem.The problem can be solved given fixed numbers for N and M, but I'm pretty sure it's impossible to exactly form a general equation.
Very interesting... I think you are correct on everything here. If M=N it seems 20% is the breakeven point, right? So to do better than that, you'd have to have a smaller number for M to make your average hand quality higher. And if M and N = 10%, it seems moving M up has got to help a lot.All this is complicated by the effects of position, as Actuary discussed. I thought about that some more, and wonder if the number of hands to play may be more accuartely described as the percentage of hands that should be played based on number of people who left to act. So at a 10 person table, if the first two fold, should player #3 play hands in the top 2/8 = 25% - 2% = 23%? I'm thinking that's not correct, because that would say that from first position you should play 18% of the hands, which seems a bit high.I may be back to writing a simulator after all.
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Oh, how I wish I was more intelligent in math. Can we say, "way over my head".This is how I'm understanding this:Play less hands, but not to little. Sometimes if someone plays with more money in the pot you should up your hands or lower your hands played?I'm so dumb, I wish I had worked more on my math than my science skills.Congratulations on having a higher IQ than myself.Great read btw. I love theory even if I can't comprehend the theory at hand.Good day.

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Oh, how I wish I was more intelligent in math. Can we say, "way over my head".This is how I'm understanding this:Play less hands, but not to little. Sometimes if someone plays with more money in the pot you should up your hands or lower your hands played?I'm so dumb, I wish I had worked more on my math than my science skills.Congratulations on having a higher IQ than myself.Great read btw. I love theory even if I can't comprehend the theory at hand.Good day.
This doesn't account for people changing the amount they bet, that is blatantly assumed to equal out across players -- an obvious falsehood, but I don't believe a fatal one.I hope people think about this, because I think it goes to the heart of advantage in poker. If I had time to work on this I could have great fun with it, slowly getting rid of the simplifying assumptions.
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you play $5/10?then you should know there is no "optimal".what is optimal for one fellow would not be for another.what is optimal on one table would not be for another.I'll ball park a range: 18 - 31 %
Hblask,ok, it's not "useless and worthless" but it's irrelevant.Skill differences will always be a greater impact on the bottom line than trying to affix a percentage of hands opponents play and acting accordingly. Your theory is not practical in application.And how do you arive at 2/n - 2% ?I'll reread that part...But does it assume opponets play 2/n ?LLY formula seems to be better at finding M..although no more practical
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If you play the top 10% in a 10-handed game, on average 1 person will see the flop each hand and will win it outright. If you play 20%, 2 people will, on average, see the flop, and you will win half the hands you are involved in. All else being equal, if you win half the hands, you will be a break-even player. If you play more than 20%, there will be more than 2 people per flop, and you will be a losing player. So what does this converge to? If you play 10%, you are leaving money on the table compared to the people who play 11%, because they win the ones you do, plus a certain percentage of those extra 1%. So basically, all else being equal, you should play just less than 20% of hands at a 10 person table. More generally, you should play just enough hands so that there will be slighlty fewer than 2 people seeing the flop if each person plays that percentage, so at a 9 person table, that's 2/9, or 22%, at a 5 person table, that's 2/5 or 40%.
I'm not getting what is wrong with more than 2 to a hand?What's wrong with losing 66% of hands you see preflop?
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I'm not getting what is wrong with more than 2 to a hand?What's wrong with losing 66% of hands you see preflop?
Here's the basis for it: that the level of play in any given room is pretty consistent. People who crush a room move up to higher levels; people who lose disappear or learn. When I think about it there may be inflection points around 33% for three people. But probably not. Again, if you assume you can crush the room, you can ignore the guideline completely.
seems so to meHblask appears hung on winning 50% of time at showdown.He's shown bad math skills in the past and stubborness, so we'll see where this goes.
Huh? Me? I persist getting questions answered, but I'm not stubborn. Bad math skills? I think I'm better than 98% of the people on here. I don't understand this.At any rate, I think this has gone far enough. If you were looking to change your game, this was the wrong thread to pursue. It was more a curiousity than anything, and I think I'm correct. But you are right, the practical value is somewhat limited by real world concerns.
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Here's the basis for it: that the level of play in any given room is pretty consistent. People who crush a room move up to higher levels; people who lose disappear or learn. When I think about it there may be inflection points around 33% for three people . But probably not. Again, if you assume you can crush the room, you can ignore the guideline completely.Huh? Me? I persist getting questions answered, but I'm not stubborn. Bad math skills? I think I'm better than 98% of the people on here. I don't understand this.
What do you mean by the bolded part?You have me confused on whether 2/n - ~2% is always going to be the optimum amount to play over the long run, or just when your opponents play 2/n ? You do seem to think losing more than 50% is bad. You also don't understand that you win more money in pots you win than you lose in pots you lose, assuming you actually fold sometimes before showdown.Were you not the guy that screwed up the Greenstein math and could not understand how the implication that Ax and Kx hands are not folded as much would lead to AK in LP being a favorite over 88 if lots of hands are folded in EP/MP ( not that it's useful, but the concept) Maybe I got the wrong guy. But this alone is crazy.************Abbaddabba,you got something there.
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