Paradox or no?
 

Let's say we are playing against a very tight opponent. He plays AA-QQ and AKs-AQs. If you play only hands in his range, you break even (equity-wise). You want to do a little better than that. Which one hand in your range of AA-QQ, AKs-AQs should you remove in order to improve your equity the most?

-pix

Re: Paradox or no?
 

AA

Re: Paradox or no?
 

dumb question

Re: Paradox or no?
 

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dumb question

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You should realize that now you are either way ahead and question is really dumb or way behind and you're just trolling. By raising your voice, you took the wrong line.

-pix

Re: Paradox or no?
 

AQs does slightly worse against that range than QQ, but it would be better to remove the 6 combinations of QQ than the 4 combinations of AQs.

This is interesting, but I wouldn't call it a paradox.

Re: Paradox or no?
 

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AQs does slightly worse against that range than QQ, but it would be better to remove the 6 combinations of QQ than the 4 combinations of AQs.

This is interesting, but I wouldn't call it a paradox.

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I agree, it really is not. It's just that the straightforward answer (AQs) is wrong. Same goes for range of AA,KK,AKs. It's better to remove KK than AKs.

-pix

Re: Paradox or no?
 

I am merely saying that this is question may be paradoxical but is inevetably inpractical and thus i call it stupid

Re: Paradox or no?
 

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I am merely saying that this is question may be paradoxical but is inevetably inpractical and thus i call it stupid

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If you feel inclined to comment in such manner, you have a surprisingly low post count.

Anyway, this is just a quirk I ran across when doing some more or less practical stuff. When analysing hand ranges in certain manner, this phenomenon is quite prominent and you better be aware of it.

-pix

Re: Paradox or no?
 

Another non-paradoxical and useless quirk for all interested.

Suppose opponent's range is AA and KK. Our best two-hand range vs. him is AA and A5s. Now let's say you want to play best 5 unique hands vs. his range. So you play AA and Ad5d? You can do better - play AA and AdKd instead.

EDIT: Here is a practical consideration. You give your opponent top x% of hands and want to know what is your top y% of hands. Your range will be different from the range you would get by simply taking top y% of all hands because your hands will be weighted. I haven't bothered to figure out what is the actual difference yet. Maybe someone of you has insight in this already?

-pix

Re: Paradox or no?
 

If you knew your opponent hand range was that tight wouldnt the most EV thing to do be to expand your hand range to take advantage of all the blinds he is going to be giving up, or am I missing the point here?

Re: Paradox or no?
 

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If you knew your opponent hand range was that tight wouldnt the most EV thing to do be to expand your hand range to take advantage of all the blinds he is going to be giving up, or am I missing the point here?

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This has nothing to do with EV (yet), only equities. Just as if you are in a game with no blinds and your opponent raises those hands anyway. This is purely theoretic, watch out for all 'ifs'.

Ok, continuation from my edit. Let's say opponent opens AA-KK, AKs. What's our equity if we get involved with him with the same range of hands? 50? It is 48,7% because our distribution of hands has changed. I do not know if this was originially considered when defining the 'gap concept', but it definitely supports it. If the opponent takes the best cards and we play looser than him, our disadvantage will be additionally increased by the fact that our hand will include worse cards more often, even if they are evenly distributed in range.

-pix

Re: Paradox or no?
 

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If you knew your opponent hand range was that tight wouldnt the most EV thing to do be to expand your hand range to take advantage of all the blinds he is going to be giving up, or am I missing the point here?

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I don't think you're missing anything. The original question asked which hand to drop to improve your equity, but the discussion seems to assume you're going all-in preflop. The real advantage to playing someone like this is you have so much information about his hand that you will often have a huge edge after the flop, and you'll know when he has you beat.

This may be a theoretical discussion about all-in preflop heads-up play, but if not, if the topic is poker, then your approach makes perfect sense.

Re: Paradox or no?
 

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If you knew your opponent hand range was that tight wouldnt the most EV thing to do be to expand your hand range to take advantage of all the blinds he is going to be giving up, or am I missing the point here?

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This has nothing to do with EV (yet), only equities. Just as if you are in a game with no blinds and your opponent raises those hands anyway. This is purely theoretic, watch out for all 'ifs'.

Ok, continuation from my edit. Let's say opponent opens AA-KK, AKs. What's our equity if we get involved with him with the same range of hands? 50? It is 48,7% because our distribution of hands has changed. I do not know if this was originially considered when defining the 'gap concept', but it definitely supports it. If the opponent takes the best cards and we play looser than him, our disadvantage will be additionally increased by the fact that our hand will include worse cards more often, even if they are evenly distributed in range.

-pix

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How about making some sence of all this for me?? I have absolutely no idea what your talking about at this point. Are the opponents playing the hand agaisnt each other properly?? Are we assumming you are a better player in the hand? Or is this just about Equities?? WHich I really never learned about so I would be intrested to learn.

Thanks

Re: Paradox or no?
 

To make this clear for all: I am only considering equity as % of times your hand or range will win against another hand or range of hands on showdown. You can calculate this using PokerStove, which you can google up.

-pix

Re: Paradox or no?
 

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Ok, continuation from my edit. Let's say opponent opens AA-KK, AKs. What's our equity if we get involved with him with the same range of hands? 50? It is 48,7% because our distribution of hands has changed.

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Our distribution of hands may have changed, but because of our cards, so has our opponent's. If we both play a range of AA,KK, and AKs indiscriminately our equity should still be 50%. PokerStove has it as 50% as well. What am I missing here?

Re: Paradox or no?
 

piki : Interesting OP.

Re: Paradox or no?
 

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Ok, continuation from my edit. Let's say opponent opens AA-KK, AKs. What's our equity if we get involved with him with the same range of hands? 50? It is 48,7% because our distribution of hands has changed.

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Our distribution of hands may have changed, but because of our cards, so has our opponent's. If we both play a range of AA,KK, and AKs indiscriminately our equity should still be 50%. PokerStove has it as 50% as well. What am I missing here?

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I am pretty positive I am not mistaken. I am sorry but I do not know a simple explanation for why it is so, though I am sure it exists. I will try with an example:

Let's say only opponent gets his cards first. He takes a look at his cards and acts accordingly. Then the deck is shuffled and you get your cards. I believe you would agree that in that case, our distribution changes and opponent's is independent of our hand. Is it any different if we're dealt cards at the same time?

EDIT: This is so mind boggling. You take a look at the problem from here, it seems logical it changes, you take another look, it seems logical it does not. Thinking about it again, I confused my self and now I do not know right from left. [img]/images/graemlins/smile.gif[/img]

-pix

Re: Paradox or no?
 

Nash_clown, I stand corrected. Thank you for your input.

I was mislead from a standpoint of an observer of action in our game. Let's say first opponent raises. Since as observer I know his range, I know that second player's distribution has now changed. This is true and obvious. If tight UTG raises, there are bound to be less Aces in hands of opponents. From here I made my conclusion. But when second player does actually get involved, distributions in both players' hands are equal.

-pix

Re: Paradox or no?
 

You have to be careful about your conditioning (not physical shape, logical conditioning).

If both players play the same set of hands, their equity has to be equal. However, suppose the other player plays only AA, KK and AKs, and she opens before I look at my cards. Out of the 1,326 possible combinations of pocket cards, she plays only 16.

I also play only 16 combinations of pocket cards, but since she opened I know that 7 of them are unavailable to me. So out of the 1,225 remaining sets of cards, I will play 9.

Still not looking at my cards, I know if she has AA she wins 80% of the time if I call with the same hand range, KK wins 36% and AKs 35% (counting ties as half wins). Since there are 6 AA's, 6 KK's and 4AKs's, her chance of winning is (6*80% + 6*36% + 4*35%)/16 = 52.2%. Since I call the same fraction of the time when she holds any of these three hands, the computation should be good. I think you are doing something like this to find that I have a slight disadvantage.

We know that logic must be incorrect, because it would say I have the advantage if I look at my cards first. That obviously cannot matter.

The explanation is a bit complicated. She doesn't have the 80% edge every time she holds AA, 1 time in 9 it's 50% (I also have AA), 6 times it's 81% (I have KK) and 2 times it's 88% (I have AKs). If you consider all the possibilities:

1 time out of 6 we have the same cards, 50%
3 time it's AA versus KK, 81%
1 time it's AA versus AKs, 88%
1 time it's AK versus AKs, 66%

each of the last three cases are equally likely to happen with either one of us having the advantage.

Re: Paradox or no?
 

so we are making the assumption that we know his range as a fact? Then, given his known range what the perfect hands to play against him are?

Re: Paradox or no?
 

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so we are making the assumption that we know his range as a fact? Then, given his known range what the perfect hands to play against him are?

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I think the question assumes that we know his range as a fact, but is not asking the perfect hands to play against him, just what is the equity (the profit if both players go all-in heads up) from either playing the same range or subtracting hands from it.