Pokerscott
01-20-2005, 05:59 PM
Many times in the SnG tournament, you will find yourself in a situation where you have to decide if you should push all in with this hand or wait for a better spot.
Lots of factors go into this decision (blinds, position, etc). One factor that I haven't seen much information on is what type of hand can I expect if I wait? It clearly depends on the number of hands you can afford to wait, so I wanted to address this question.
Methodology:
In order to address this question you have to make two decisions up front. First, how are you going to measure hand strength? Second, how are you going to summarize the possible strength of the hands you may get in the future?
In terms of evaluating hand strength, I used the data from here (http://www.wizardofodds.com/games/holdem/) to rate the strength of the various hands. I did the analysis using the 2-handed strength ratings and the 4-handed strength ratings. I'm not sure which is best to apply to the push or wait question. Likely there are 4+ hands left, which argues for the 4-way. However, you are only going to be generally facing one opponent who calls, argueing for the 2-way. I personally think the 2-way is more correct, but the information is there if you feel otherwise...
In terms of summarizing the future hands, I decided to report the median best hand you could expect given N hands remaining.
How can you calculate median best hand given N hands remaining?
Imagine the all the possible hands ordered by hand strength. For each hand you also know the probability of getting that type of hand (e.g. AJs is 0.3%). If you draw a random number from a uniform [0,1] distribution, you can then map it to your ordered list of hands to determine a particular hand associated with that random number. If you want to figure out the best hand given two remaining hands, choose two uniform [0,1] random numbers and take the minimum. For N hands, choose N random numbers and take the minimum. (I choose the minimum instead of the maximum simply because the hands are ordered starting with the best first. Low percentiles correspond to better and better hands)
At this point, you basically have one possibility in terms of best hand given N future hands. If you want to know the median best hand given N future hands, you simply repeat the process thousands of times and take the median result. I did it 100k times to generate the table below (and yes I know the answer for 1 hand should be 50% /images/graemlins/smile.gif )
Here is the table that summarizes the results:
http://img3.imagevenue.com/loc260/f6e_MedBestHand.GIF
How to read this table:
# hands left:
-This is the number of hands you are going to get to look at if you fold your current hand (e.g. if the big blind crushes you in 3 hands, you might want to look at the 3 or maybe the 2 row...)
Median 'Best-Hand' Strength Percentile:
-This is the result of the analysis reported in percentile. For example, if you have seven hands to go, the median 'best-hand' you can expect over those seven hands will be a hand strength percentile of 9.4%. Half the time your best hand will be better than the 9.4 percentile and half the time it will be worse.
Implied Hand (2-way rank)
Now that you know the hand strength percentile you can expect with 7 hands left, all you have to do is figure out the right ordering of hand strengths. If you think 2-way rank is right, then the hand that corresponds to the 9.4 percentile is an ATo.
Implied Hand (4-way rank)
Same information as the 2-way ranking except hands are ranked assuming 4 hands start and go all the way to the river. For the 4-way rank, the 9.4 percentile hand is pocket 7s.
Let me know if you have questions or think a tweak to this analysis would be a lot more helpful.
Cheers,
Pokerscott
Lots of factors go into this decision (blinds, position, etc). One factor that I haven't seen much information on is what type of hand can I expect if I wait? It clearly depends on the number of hands you can afford to wait, so I wanted to address this question.
Methodology:
In order to address this question you have to make two decisions up front. First, how are you going to measure hand strength? Second, how are you going to summarize the possible strength of the hands you may get in the future?
In terms of evaluating hand strength, I used the data from here (http://www.wizardofodds.com/games/holdem/) to rate the strength of the various hands. I did the analysis using the 2-handed strength ratings and the 4-handed strength ratings. I'm not sure which is best to apply to the push or wait question. Likely there are 4+ hands left, which argues for the 4-way. However, you are only going to be generally facing one opponent who calls, argueing for the 2-way. I personally think the 2-way is more correct, but the information is there if you feel otherwise...
In terms of summarizing the future hands, I decided to report the median best hand you could expect given N hands remaining.
How can you calculate median best hand given N hands remaining?
Imagine the all the possible hands ordered by hand strength. For each hand you also know the probability of getting that type of hand (e.g. AJs is 0.3%). If you draw a random number from a uniform [0,1] distribution, you can then map it to your ordered list of hands to determine a particular hand associated with that random number. If you want to figure out the best hand given two remaining hands, choose two uniform [0,1] random numbers and take the minimum. For N hands, choose N random numbers and take the minimum. (I choose the minimum instead of the maximum simply because the hands are ordered starting with the best first. Low percentiles correspond to better and better hands)
At this point, you basically have one possibility in terms of best hand given N future hands. If you want to know the median best hand given N future hands, you simply repeat the process thousands of times and take the median result. I did it 100k times to generate the table below (and yes I know the answer for 1 hand should be 50% /images/graemlins/smile.gif )
Here is the table that summarizes the results:
http://img3.imagevenue.com/loc260/f6e_MedBestHand.GIF
How to read this table:
# hands left:
-This is the number of hands you are going to get to look at if you fold your current hand (e.g. if the big blind crushes you in 3 hands, you might want to look at the 3 or maybe the 2 row...)
Median 'Best-Hand' Strength Percentile:
-This is the result of the analysis reported in percentile. For example, if you have seven hands to go, the median 'best-hand' you can expect over those seven hands will be a hand strength percentile of 9.4%. Half the time your best hand will be better than the 9.4 percentile and half the time it will be worse.
Implied Hand (2-way rank)
Now that you know the hand strength percentile you can expect with 7 hands left, all you have to do is figure out the right ordering of hand strengths. If you think 2-way rank is right, then the hand that corresponds to the 9.4 percentile is an ATo.
Implied Hand (4-way rank)
Same information as the 2-way ranking except hands are ranked assuming 4 hands start and go all the way to the river. For the 4-way rank, the 9.4 percentile hand is pocket 7s.
Let me know if you have questions or think a tweak to this analysis would be a lot more helpful.
Cheers,
Pokerscott