sahaguje
08-06-2004, 07:54 AM
Hi,
This post is originally part of a discussion about hand selection at PLO. I stated that the structure of a full poker game had an effect of the proportion of hands one should play, and that it did not really depended on the game. My intuition was that you should play around half of the average numbers of players per flop, more if you are better or if there is any dead money (blinds, ante...) preflop. I repost here, cause I really would like some comments by people aware of game theory, cause I dont know it at all.
OK, after a good sleep night, I think I was right, and I am going to defend my position...
Let s redefine it : in each game, considering your position, the stacks and the level of other players, you can define an optimal number of hands to play. This number is closely related to two things :
- The numer of players in the game, because it defines your probability to have the first, second, third etc. best hand.
- The distribution of probabilities of winning among the hands : The best hand has a probability of p(1), to win, the second hand p(2) etc, in each poker game, but p(i)-p(i-1) is different in every form of poker.
In a future post, I am gonna try to prove this mathematically. Here I just want to explain why I think this is theoretically right to play only a certain proportion of hands, and this proportion is linked with the distribution of probabilities of winning the hands between players. First I will answer to pete's post, "fraught with logical errors"
Quote:
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This little thought-experiment is deeply misguided. The notion that you can guarantee e.v. merely by cutting the “losing” 20% of hands is ridiculous. Imagine for one second that the table contains 10 clones of yourself, and you play perfectly. On average, you play 30% of hands. Do you think one of your clones could actually improve his E.V. by decided to play only 15% of hands? What hands would he be cutting? Well, by the stipulation that you play perfectly, he would be dumping only profitable situations. Thus, the clone who only played 15% of his hands would be the fish! He would be failing to extract equity when it exists! Suddenly everyone at the table would have a + E.V. except for him.
And just to make an obvious point, let’s say that clone actually could make money by playing half as many hands as you. What would the game-theoretical response to that be? Well, the other clones should start playing 7.5%, 3.75%, and so on, until all clones play 0 hands, and you can just go home.
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Thanks for giving me such an example. The problem in what you say is that you dont consider the other clones' hands, but only mines. It is faulty to say that if I miss occasions to make +EV play, that means I am gonna be a fish. If there were no blinds, I could wait for AA ds and only play them, this would not be the most profitable play, but that does not mean at all I would have become a fish, and people now all win money just because I am more selective...
Anyway, if the table is full of clones (so no one is a better player), and if we consider there is no dead money in the pot, then when the pot is 4-handed, in the long run my probability of winning the hand has to be more than 25%, or else I would lose money. Remember I cannot make more money than the other players can when I have a hand, since we are all clones. But my probability to win the pot postflop when we are 4 is of course related with my probability to win the pot preflop when there are ten players, at least in the long run. The reason of this is obvious : in order to calculate the probability of a hand to win the pot, you must deal a great number of flops, and count each time this hand wins. That is what twodimes do, for example.
So, lets get back to the problem : each hand has p(i) chances to win the pot, i being the rank of the hand if we compare the probabilities ( ie : p(i-1)<p(i)<p(i+1) ). If the game is 10-handed, you should only play your p(1) hands, cause now you will be a favorite on every hand.
Notice two things : 1) that it is only true in the long run, and for a particular hand the p1 hand wont always be a favorite post flop. But if you play a great number of flops, it will be.
2) Since all players are clones, in the long run you cannot count on implied odds etc, because the other players will gain the same implied odds when they will be in your situation. So if we consider there is no dead money, 4 handed post flop, every hand that has more than 25% to win is a winner, and every hand that has less is a loser. Now you can tell there is money in the pot from preflop bets. That is true, and when I will calculate the "good" proportion of hands to play, I will try to take it into account, but here I just want to explain my point theoretically.
So you want to play your p1 hands only if you want to be a favorite each hand you play, or any hand that has more than 25% odds to win the pot if you want to take every opportunity to make a +EV play. Now sometimes, only 1 hand will have for than 1 to 3 odds, sometimes 3, but most of the times two hands will have odds sufficient enough, and 2 wont. If everybody does good hand selection, only p1, p2, p3 and p4 hands will be played each time ; so in the long run, p1 and p2 hands will show a profit, and p3 and p4 hands will lose money. So do maximize your winnings, you should only play 20% of your hands, if it is 4 handed post flop on average.
It is the same for every number of players : if it is 6 handed, you have to get 1-to-5 odds preflop to show a profit. In the long run, 3 hands will have more than these odds postflop, and in the long run again, it will be p1, p2 and p3 hands, so you should play 30% of your hands. etc etc etc.
That is what I meant when I said you should play half hands the other does ; I should have said you should play half the number of players in the pot, but of course it is the same thing. Now maybe it will not seem intuitive multiway, but it is obvious heads up : if there was no dead money and if you had the same level than your opponents, obviously the best strategy is to play only the hands that beat him, ie that has more than 50% odds to win the pot.
Now you see that all we said was true for every form of game. But since there are the blinds, and players bet money preflop, you are not forced to have more than 25% odds to win the pot if it is 4 handed post flop to make money, even if all the players are equal. It depends on the average pot size, and it can be easily calculated, I ll do that in a next post. So for example we can imagine if it is 4 handed post flop on average, you just have to have 20% (and not 25%) odds to win to make it profitable. The type of game (HE, stud, Omaha) determines the distribution of probabilities to win, so it now affects how many hands are above the 20% line. For example, p(i)-p(i-1) is usally really superior at hold em than at omaha, so there is a good chance that in the long run more hands at omaha can have a probability to win superior to 20% post flop when 4 handed. So you should play more hands at omaha than at HE. But notice that it only comes from the fact that there is money in the pot when flop betting round begins. And I guess it wont make an enormous difference, but once again, it is easy to calculate, and I will try to do it.
So the good proportion of hands to play should theoretically be slighty superior (5-10% max, I guess) to half the average number of players per flop, all other conditions being equal.
Now pete made a good point : if everyone knows that, they should reduce the numer of hands they play until 0, and there will be no game. It is totally true. It is the same at NLHE : theoretically, if there was no dead money, everyone should wait for AA, as long as everyone else does it ; to put it differently, if 9 players only play AA, and one player plays AA and KK, if there is no dead money this player will be a big loser in the long run.
So why are there games ?
1) Well, in the games I know, there are some blinds to fight for.
2) Players dont make the proper adjustment.
3) Hands you evaluate as p1 are always playable if they really are the best hands. That is 10% of hands. But since you cannot know for sure which is the best hand, you will play more or less hands.
4) There is a virtuous circle : if 2 players enter the pot, you can play at least your 15% best hands, so the player after you can play his 20% best hands, etc. That is one of the reasons you can play more hands when you have position, i.e. more informations on the future conditions of the hand played.
5) Most players think they are on average able to make money when they win, more than the addition of all the money they lose when they lose their hands. Most of them are wrong, but this is that bad evaluation that makes them play. Otherwise, there will be no poker games.
I wont comment the rest of pete's post. It is interesting, but not appropriate. Every example in it supposes there is blind money, and/or that he is better than the other players. For example, he says he should not reduce the number of hands he plays when they are only rocks at the table, but on the contrary raise this number. Major mistake, cause he can play as many hands he wants, if the other players just play the best 10% of their hands, then pete will be an outdog everytime he is paid preflop. To make money, he has to steal the blinds, or play better than his opponents post flop. If in addition to playing only their best hands, the rocks play them heavily and with great agressivity, both pre and post flop, pete will lost incredible amounts of money very quickly.
I hope this post was interesting, and not too boring. I ll try to mathematically apply that toughts to omaha, but only if my reasonig is good, so I will wait a little for that. Please, comment and criticize what I wrote.
See you
sahaguje
This post is originally part of a discussion about hand selection at PLO. I stated that the structure of a full poker game had an effect of the proportion of hands one should play, and that it did not really depended on the game. My intuition was that you should play around half of the average numbers of players per flop, more if you are better or if there is any dead money (blinds, ante...) preflop. I repost here, cause I really would like some comments by people aware of game theory, cause I dont know it at all.
OK, after a good sleep night, I think I was right, and I am going to defend my position...
Let s redefine it : in each game, considering your position, the stacks and the level of other players, you can define an optimal number of hands to play. This number is closely related to two things :
- The numer of players in the game, because it defines your probability to have the first, second, third etc. best hand.
- The distribution of probabilities of winning among the hands : The best hand has a probability of p(1), to win, the second hand p(2) etc, in each poker game, but p(i)-p(i-1) is different in every form of poker.
In a future post, I am gonna try to prove this mathematically. Here I just want to explain why I think this is theoretically right to play only a certain proportion of hands, and this proportion is linked with the distribution of probabilities of winning the hands between players. First I will answer to pete's post, "fraught with logical errors"
Quote:
--------------------------------------------------------------------------------
This little thought-experiment is deeply misguided. The notion that you can guarantee e.v. merely by cutting the “losing” 20% of hands is ridiculous. Imagine for one second that the table contains 10 clones of yourself, and you play perfectly. On average, you play 30% of hands. Do you think one of your clones could actually improve his E.V. by decided to play only 15% of hands? What hands would he be cutting? Well, by the stipulation that you play perfectly, he would be dumping only profitable situations. Thus, the clone who only played 15% of his hands would be the fish! He would be failing to extract equity when it exists! Suddenly everyone at the table would have a + E.V. except for him.
And just to make an obvious point, let’s say that clone actually could make money by playing half as many hands as you. What would the game-theoretical response to that be? Well, the other clones should start playing 7.5%, 3.75%, and so on, until all clones play 0 hands, and you can just go home.
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Thanks for giving me such an example. The problem in what you say is that you dont consider the other clones' hands, but only mines. It is faulty to say that if I miss occasions to make +EV play, that means I am gonna be a fish. If there were no blinds, I could wait for AA ds and only play them, this would not be the most profitable play, but that does not mean at all I would have become a fish, and people now all win money just because I am more selective...
Anyway, if the table is full of clones (so no one is a better player), and if we consider there is no dead money in the pot, then when the pot is 4-handed, in the long run my probability of winning the hand has to be more than 25%, or else I would lose money. Remember I cannot make more money than the other players can when I have a hand, since we are all clones. But my probability to win the pot postflop when we are 4 is of course related with my probability to win the pot preflop when there are ten players, at least in the long run. The reason of this is obvious : in order to calculate the probability of a hand to win the pot, you must deal a great number of flops, and count each time this hand wins. That is what twodimes do, for example.
So, lets get back to the problem : each hand has p(i) chances to win the pot, i being the rank of the hand if we compare the probabilities ( ie : p(i-1)<p(i)<p(i+1) ). If the game is 10-handed, you should only play your p(1) hands, cause now you will be a favorite on every hand.
Notice two things : 1) that it is only true in the long run, and for a particular hand the p1 hand wont always be a favorite post flop. But if you play a great number of flops, it will be.
2) Since all players are clones, in the long run you cannot count on implied odds etc, because the other players will gain the same implied odds when they will be in your situation. So if we consider there is no dead money, 4 handed post flop, every hand that has more than 25% to win is a winner, and every hand that has less is a loser. Now you can tell there is money in the pot from preflop bets. That is true, and when I will calculate the "good" proportion of hands to play, I will try to take it into account, but here I just want to explain my point theoretically.
So you want to play your p1 hands only if you want to be a favorite each hand you play, or any hand that has more than 25% odds to win the pot if you want to take every opportunity to make a +EV play. Now sometimes, only 1 hand will have for than 1 to 3 odds, sometimes 3, but most of the times two hands will have odds sufficient enough, and 2 wont. If everybody does good hand selection, only p1, p2, p3 and p4 hands will be played each time ; so in the long run, p1 and p2 hands will show a profit, and p3 and p4 hands will lose money. So do maximize your winnings, you should only play 20% of your hands, if it is 4 handed post flop on average.
It is the same for every number of players : if it is 6 handed, you have to get 1-to-5 odds preflop to show a profit. In the long run, 3 hands will have more than these odds postflop, and in the long run again, it will be p1, p2 and p3 hands, so you should play 30% of your hands. etc etc etc.
That is what I meant when I said you should play half hands the other does ; I should have said you should play half the number of players in the pot, but of course it is the same thing. Now maybe it will not seem intuitive multiway, but it is obvious heads up : if there was no dead money and if you had the same level than your opponents, obviously the best strategy is to play only the hands that beat him, ie that has more than 50% odds to win the pot.
Now you see that all we said was true for every form of game. But since there are the blinds, and players bet money preflop, you are not forced to have more than 25% odds to win the pot if it is 4 handed post flop to make money, even if all the players are equal. It depends on the average pot size, and it can be easily calculated, I ll do that in a next post. So for example we can imagine if it is 4 handed post flop on average, you just have to have 20% (and not 25%) odds to win to make it profitable. The type of game (HE, stud, Omaha) determines the distribution of probabilities to win, so it now affects how many hands are above the 20% line. For example, p(i)-p(i-1) is usally really superior at hold em than at omaha, so there is a good chance that in the long run more hands at omaha can have a probability to win superior to 20% post flop when 4 handed. So you should play more hands at omaha than at HE. But notice that it only comes from the fact that there is money in the pot when flop betting round begins. And I guess it wont make an enormous difference, but once again, it is easy to calculate, and I will try to do it.
So the good proportion of hands to play should theoretically be slighty superior (5-10% max, I guess) to half the average number of players per flop, all other conditions being equal.
Now pete made a good point : if everyone knows that, they should reduce the numer of hands they play until 0, and there will be no game. It is totally true. It is the same at NLHE : theoretically, if there was no dead money, everyone should wait for AA, as long as everyone else does it ; to put it differently, if 9 players only play AA, and one player plays AA and KK, if there is no dead money this player will be a big loser in the long run.
So why are there games ?
1) Well, in the games I know, there are some blinds to fight for.
2) Players dont make the proper adjustment.
3) Hands you evaluate as p1 are always playable if they really are the best hands. That is 10% of hands. But since you cannot know for sure which is the best hand, you will play more or less hands.
4) There is a virtuous circle : if 2 players enter the pot, you can play at least your 15% best hands, so the player after you can play his 20% best hands, etc. That is one of the reasons you can play more hands when you have position, i.e. more informations on the future conditions of the hand played.
5) Most players think they are on average able to make money when they win, more than the addition of all the money they lose when they lose their hands. Most of them are wrong, but this is that bad evaluation that makes them play. Otherwise, there will be no poker games.
I wont comment the rest of pete's post. It is interesting, but not appropriate. Every example in it supposes there is blind money, and/or that he is better than the other players. For example, he says he should not reduce the number of hands he plays when they are only rocks at the table, but on the contrary raise this number. Major mistake, cause he can play as many hands he wants, if the other players just play the best 10% of their hands, then pete will be an outdog everytime he is paid preflop. To make money, he has to steal the blinds, or play better than his opponents post flop. If in addition to playing only their best hands, the rocks play them heavily and with great agressivity, both pre and post flop, pete will lost incredible amounts of money very quickly.
I hope this post was interesting, and not too boring. I ll try to mathematically apply that toughts to omaha, but only if my reasonig is good, so I will wait a little for that. Please, comment and criticize what I wrote.
See you
sahaguje