知其然不知其所以然

听说我们北面那个叫红鹿的小镇，打1-2一群翻牌前推的。好奇心起，我到计算器上算一些翻牌前的百分率。发现一个现象，我想不太明白，希望霍华德等年轻的数学好的同学帮我说一下为什么。

单挑的时候，如果翻牌前推了，不同花KQ不如不同花A5，差挺多呢，57对42.当有第三个人进来的时候，比如第三个人是对8，那结果就变成88-38%，KQ-35%，A5-26%，由于第三个人的加入，KQ比A5厉害了。我想不太好为什么。是不因为他的两个牌都成了8的 over card，而A5只有一个。同时，还有顺子可能。这个结论告诉我，当有3个人翻牌前all的时候，A带小变成啥也不是了，而KQ仍然是很强的牌。

然后今天我就打了这样一把牌，我拿了红桃KQ，一个手里只有80的先推了，由一个鱼靠，这家伙很松，不会是AA，KK，AK这东西。只要不是这三个东西，我们3个一块进来，大大提升了KQ的战斗力。可以肯定，无论第一个80推的是A带小还是一小对，我都有超过1/3的机会赢。外加上我可能把这条鱼打走的可能，所以我推了，果然鱼扔了。这样，我和80推的那哥们每个人都是用80去赢160，很好的买卖。翻牌出K了，可最后来了A，AJ赢了。

假如这牌没有那条鱼进来我是坚决不能靠的。那样我就成了42%的时候用80赢80，是亏钱的。

我觉得许多同学玩锦标赛的时候，都喜欢拿A带小推，如果有两个人靠你，你还不如推KQ或者QJ了。

Howard 发表于 2009-12-16 01:29
墙兄，我可能比你小几岁但是也不再年轻了，数学也不好只是喜欢瞎算一些东东而已

你的例子很有代表性，A5虽 ...

佩服佩服

windstormm说的很有风格和逻辑
我非常同意
英文花花公主windstormm是ftp未来的大白鲨 ！
许多老蕉不是天才的对手
哈哈

HOWARD大侠回答很有风度.  我就不抬杠了.   [s:167]   看得出HOWARD有不错数理背景, 也喜欢钻研. 很好很强大... I only tried to point out another perspective. 希望有些同学不要被误导.
I have seen plenty of pushes on reraise  with small pocket pairs in our weekly freerolls. I hope it is only because it is in freeroll, not how they really play out there.  Not everyone can pull a Cada all the time and have a winning bank roll.

windstormm 写一手漂亮的英文，观点也非常鲜明，是本论坛新发现的人才。
我本来一看英语就躲远远的。如果在英文论坛里，没办法，只好捏着鼻子看了。但在中文里，一般英语就躲开。刚才看了你最后的回帖，觉得你的英文真是很棒，文章内容也相当好。欢迎来坛里，可否自我介绍一下。

墙兄太客气了.  讲得我不好意思. 其实我写的有好几处语法错误. 打英文实在是因为我打拼音太慢了...我打POKER也不是什么SHARK.  总共打了7个月在FTP(以前只知牌的大小).. 只是混迹在MICROSTAKE里总共小小赢利 1千不到(刚开始输了一百多)..  更多时候在冲着屏幕上的DONKEY骂娘.  也经常问候FTP他姥姥.. [s:167]  Poker is a hobby for me. I do notice that in microstakes (sng, mtt, cash), most of the players are completely clueless.  All you need is some common sense to have a profit.

先赞一下windstormm的思路，非常清晰，论点也很鲜明。虽然是在反驳我的观点，但我必须要说我是同意并赞赏其中多数内容的。

mathematical models are all based on assumptions.  Most of time they are not exactly the same in real cases.

I am guessing Howard averaged all odds against other hands to calculate the overall odds for each hand. This is assuming all hands in the list are uniformly distributed, meaning every hand has equal chances to call (he actually use a special case of that, 100% chance for each hand to call).  This is clearly not an accurate assumption.  A 7 A 8 will not call your all in all the time and 10 10, 9 9, 8 8 will not either but will call more often than not, because people over play small pocket pairs much  more often.  This is especially true when the pusher's stack is larger. Are you going to call off half of your stack with A 7 or 99 88?  This is also why in my opinion KQ is much more appealing than 33.  To compute an exact methematical model, you have to give out the probability of each hand that might call your all in and computer the weighted odds instead of the average of all odds.  Is that easy? absolutely not. Because different people on your table will give you totally different probabilities.  

In all, using a simple almost naive math model to rank the hand is not only inaccurate but also misleading.   I would use this list with extreme caution.

我那篇《排名》所用的方法，的确如你所说，是先假定一个range，假定有且仅有一个对手，然后再假定他在这个range中100%会call，不在range则100% fold。注意这并不是说对手在这个range中uniformly distributed，因为我假设你是KQ的时候对方有16种可能AJ，却只有12种可能AQ。可以算作naturally distributed吧。当然最终赢率更不是针对单手牌算出来后再平均，而是考虑了这个naturally distributed的权重平均。

这些假设有一个问题，windstormm已经提出，就是不符合现实牌手的行为方式。他即使在range内，也未必要100%跟；即使不在range，也未必100%扔。他跟牌可能性的大小跟牌力有关，range里面AA/KK等牌，他一定要跟，AJ，99之类中等牌力，可能只有70%可能性，再弱一点到33，KJ，可能只有50%了；跟筹码绝对大小也有关，他跟一下只损耗他筹码的的10%，他跟的可能性就大，如果要伤筋动骨，就小；跟筹码相对大小也有关，他已经是绝望的最小筹码，他很可能跟，如果是中等大小，很可能就扔；跟他的风格有关，松就跟的多紧跟少；跟他上几次比赛的经历也有关，如果他连续3次比赛都是99最后一手牌，这次他再见了99可能有心理障碍会扔，88反而会跟；跟他对这场比赛的期望值有关，他只想混进钱圈，还是想争取夺冠；跟其他大量的随机因素也有关，比如他最近的心情，他老婆跟他的关系，他最近的经济状况，他的狗是不是生病。。。。

这些无数的因素，要都考虑进去才能做出“完美”的模型。有些因素是可以考虑进去的比如说筹码，但是有些因素根本无法量化。不过这并不妨碍我们做一大堆的假设，然后从最容易的角度入手考虑问题，就可以得到一个近似的结果，至少是一个方向。这有点像中学物理力学，总是假设光滑平面，空气阻力，摩擦力均为零，完美球体完美平面，任何弹簧/滑轮/铰链总是完美设计，现实中有这样的东东吗？没有。但是，这些假设是必要的，不做这些假设，就根本连分析问题的“入手点”都没有，只能“凭感觉”了。这些假设是不是“simple and naive”？我觉得也可以说是，但是我还真就喜欢这种naive的方法，呵呵，它毕竟是研究任何复杂问题都绕不开的第一步。

"I would use this list with extreme caution"，这个我是绝对同意的。我也觉得没有人会背诵这个list然后按照它行事，更何况list中也没有说你什么情况下应该推前百分之多少的牌。它只不过是针对特定range的排名而已。这个list有没有改进的余地？太有了。我做这个list的意图只不过想表明，在特定对手range下，你先push时的牌的好坏有时未必如你想象。比如，中等suited connector好于A-rag。

I also disagree with only trying to maximize your chip gain in touney like you do in cash.  In cash, you go all in every time when you are with 51% chance vs 49% chance you will always be winning. Totally true. But in touney, totally not true.  Let us assume you always go all in AA vs 22. you have 80% chance to win the pot. Great. Now if you do that 6 times in a touney (totally possible for large MTTS 1000 people because there is always bigger stack waiting for you out there), you have 0.8^6 time chance to stay after all in 6 times. What chance does that give you to still be in alive?  0.2621!  did you imagine this number?  If you always wait for good hand then go all in with it, you have small chance to make it to the end even with 6 AAs.  Of course this model is naive as well because of the simple assumption but it gives you an idea. Now do you think it is still wise to go all in with 55% vs 45% all the time?   I would be pushing with 45% of chance winning, but prefer not to calling with 55% chance winning if it is for my entire stack. Because this fold equity is so important in the game, it alters the touney play completely vs cash game.

你also disagree的这个论点我also disagree。如果你是在disagree我，我想可能是误会了，我从来没有说过maximize your chip gain是比赛中的要务，相反我在以前的文章中多次提到比赛合适机会甚至可以fold AA。我的模型是按照chip EV计算的，只不过是因为，money EV没法算，要考虑别人的筹码分布等等，可以说又是一个naive的假设把。

fold equity也是考虑到了的。如果你先推，首先有一点很明确，那就是无论你推72，还是推AA，对手给你的fold equity都是完全一致的。所以你不必考虑他fold的情况，只考虑他的calling range就够了。这也是为什么中等suited connector好于A-rag。因为他只要跟你，多半就大米呢特了你的A-rag，即使在单挑中，A-rag是比中等suited connector好得多的“hot-and-cold”的牌。

windstormm 写一手漂亮的英文，观点也非常鲜明，是本论坛新发现的人才。
我本来一看英语就躲远远的。如果在英文论坛里，没办法，只好捏着鼻子看了。但在中文里，一般英语就躲开。刚才看了你最后的回帖，觉得你的英文真是很棒，文章内容也相当好。欢迎来坛里，可否自我介绍一下。

mathematical models are all based on assumptions.  Most of time they are not exactly the same in real cases.

I am guessing Howard averaged all odds against other hands to calculate the overall odds for each hand. This is assuming all hands in the list are uniformly distributed, meaning every hand has equal chances to call (he actually use a special case of that, 100% chance for each hand to call).  This is clearly not an accurate assumption.  A 7 A 8 will not call your all in all the time and 10 10, 9 9, 8 8 will not either but will call more often than not, because people over play small pocket pairs much  more often.  This is especially true when the pusher's stack is larger. Are you going to call off half of your stack with A 7 or 99 88?  This is also why in my opinion KQ is much more appealing than 33.  To compute an exact methematical model, you have to give out the probability of each hand that might call your all in and computer the weighted odds instead of the average of all odds.  Is that easy? absolutely not. Because different people on your table will give you totally different probabilities.  

In all, using a simple almost naive math model to rank the hand is not only inaccurate but also misleading.   I would use this list with extreme caution.

I also disagree with only trying to maximize your chip gain in touney like you do in cash.  In cash, you go all in every time when you are with 51% chance vs 49% chance you will always be winning. Totally true. But in touney, totally not true.  Let us assume you always go all in AA vs 22. you have 80% chance to win the pot. Great. Now if you do that 6 times in a touney (totally possible for large MTTS 1000 people because there is always bigger stack waiting for you out there), you have 0.8^6 time chance to stay after all in 6 times. What chance does that give you to still be in alive?  0.2621!  did you imagine this number?  If you always wait for good hand then go all in with it, you have small chance to make it to the end even with 6 AAs.  Of course this model is naive as well because of the simple assumption but it gives you an idea. Now do you think it is still wise to go all in with 55% vs 45% all the time?   I would be pushing with 45% of chance winning, but prefer not to calling with 55% chance winning if it is for my entire stack. Because this fold equity is so important in the game, it alters the touney play completely vs cash game.

这种帖子只能干看，也插不上嘴，知道是好东西，但就是消化不了。

顶贴纯支持吧~

你文中有一句话我不能同意“tournament活命是第一要务”。我认为，tournament跟cash game一样，Max Money EV才是第一要务，而且是唯一要务。

比赛里面Max Money EV也能细分两种人，一种是Max hourly EV，另一种是Max per tournament EV。两种打法在某些地方会有细微的差别。

我承认在比赛里面有无数跟你所说一样的人，以活命为第一要务。这种人带给你额外的fold equity，如不利用，会遭天谴。

此外，达米内特与非达米内特并无本质区别，不是黑与白那样。比如ATo对JTs，要算做达米内特，但是JTs有33%以上的EV。A8对77，说来也算coinflip，但他们是43%对57%的巨大差异。一个所谓coinflip中，如果能差别出14%都可以忽略不计，那么，我被达米内特一下下也没什么大不了的。

tournament玩的盘数远远小于玩cash的手数，并不是忽略“小优势”的理由。小优势也是优势，豆芽也是菜。巴菲特告诉我们，如果一个股票你认为它5年后不会比现在更值钱，那么就一天也不要持有。扑克也是投资，不是投机，一样的。

你说的正是我犯迷糊的地方。
我感觉，这里涉及到对手的fold equity、bubble time、stacksize等等等等。
这些因素都要考虑进去，再加上手上持有的牌之类，太他妈复杂了。
我无法考虑清楚，包括你们有不同看法的KQs和small pair。
最后，就只有信感觉了。（这个习惯一点也不好，但好像没有正确答案就只有靠感觉了。）

其实我蛮想知道严密的数理统计的结果。