Gaining Information from Folded Opponents

I was trying to explain how I read hands to an aspiring poker newbie friend of mine the other day when an interesting point came up.


Basically, I was explaining the Sherlock Holmes approach ... "when you have eliminated the impossible, whatever remains, however improbable, must be the truth." In practice, this corresponds to having a rough idea of the range of hands your opponent could be playing, and then eliminating those which are inconsistent with his actions in the hand to date. Nothing new here, right?


The thought then occurred to me ... we spend a great deal of thought trying to put our active opponents on hands. What about those opponents who have already folded?


Now, this is, in part, motivated by the fact that I tend to play in relatively loose games. Five or six players seeing a flop is not an altogether rare event. What is rare, however, is when the whole field calls to see the turn. Usually, you lose two or three players on the flop.


Isn't there a fair amount of information to be mined in this situation as well? The flop did not hit these opponents' hands hard enough for them to continue playing. If you know your opponents and their playing styles well enough, you could conceivably remove up to 2*n cards (where n is the number of opponents who folded on the flop) from the pool of "unknown" cards. In some cases, this could result in a significant shift in the odds for calling/folding later on in the hand.


Here's an example. In a loose game, six players see the flop for a single bet. The flop comes Qh 7s 2h. There are a bet, a call and three folds to you on the button. What do we know?


Well, if the three players who folded are average players, we can certainly remove at least 3, and possibly as many as 6 non-heart cards from the pool of unknown cards. Why? Because it is very likely that they would at least call with any two hearts in their hand, hoping to hit the flush. So at least one of the cards in their hand had to be something other than a heart. (If the players were truly awful, and would call for the runner runner draws, then the removal of 6 non-hearts cards becomes possible.) We can also probably make some assumptions that they likely do not hold queens or sevens, overcards, etc.


Taking our hearts draw example a bit further, let's say that we make the minimal assumption that we can eliminate 3 non-hearts cards from the unknown pool. How are our odds affected?


Well, not taking into acount the 3 cards we "know" about, we have 47 unknown cards, of which 38 are not hearts, and our probability of not hitting our flush by the river are simply: 38/47(37/46) = 0.6503, which corresponds to the 1.86 to 1 result we all know and love.


Factoring in our belief that there are, in fact, only 44 unknown cards out there, of which 35 are not hearts, we now have the probability of missing being: (35/44)(34/43) = 0.6290, which gives new odds of 1.70 to 1.


Now, in this particular scenario, the shift in odds is not very dramatic. I think if you found yourself on the button with two halfways decent hearts in this situation you'd certainly call at the very minimum in any case. The improved odds might affect your willingness to "jam" lesser flush draws than you otherwise might in this position, however.


A few observations:


First, this is certainly a tough exercise in conditional probabilities, and is very prone to misreads. You have to know your opponents very well before there would be much value in any information that you manage to glean from the fact that an opponent has folded. It may also be difficult to immediately see what effect knowing what an opponent did not have might have on the decision facing you at the table.


Second, the shift in odds is, in general, going to be small, since you will only be able to eliminate a relatively small number of cards "for certain." However, in situations where you are considering a marginal implied odds call, being able to deduce that your opponents have folded one or two cards which do not help you might be enough to shift things from a fold to a call.


At any rate, I thought this idea was kind of interesting, and one I had not seen on the forum. What are your thoughts on all of this? Is this something you consider at the table, or do you even think this idea has value at all?


All comments/criticisms welcome,


Dave

Several moons gao., Dan Hanson (who is also an Edmontonian like you) posted an Omaha problem that illustrates this concept drmatically. I may have the details wrong but it went something like this:


All 10 players pay to see the flop.


You have K88x on the button (don't ask why you played it...it's Omaha high only...maybe it's playable...who knows).


Flop: KK8


Small blind bets out. Everyone folds. You raise. He calls.


Turn: Ace


Small blind checkraises. The pot now has $200 in this 10-20 game. You ponder your move. He then shows AA56 'cause he's your buddy.


Because of the other players folding, your odds of hitting Quad Kings are now only 1 in 8 and not 1 in 40. You therefore have to call based on that possibility alone. Of course, you also have a 1 in 40 shot of hitting Quad eights.


...anywawy, yours is an interesting post which brings up an interesting and rarely mentioned concept.

if you can read your opponents that well, then you should be able to read the remaining opponents in the hand to know where they are at as well. this, i think, is more valuable than knowing the significant change in remaining odds with respect to the cards left to come. however, this is going to be tough to do, because, unless you can read your opponents very very well, there are just too many hands that you can put your opponents on to narrow it down enough to be significant.

I think an obvious application that we all use subconciously is the case where an opponent is visibly or audibly pissed because his longshot came in on the turn or river. As an example, you have AA (or any other big overpair) and the flop comes something like Q-8-5. four called BTF, and two call on the flop when you bet (say the blinds fold). turn is a J, you get checkraised by your thinking opponent UTG and only you call. Now the river pairs the 8 or 5. Sometimes one or more opponents will slam their fists down on the table in anger when they realize that they would have had trips. You can't put the checkraiser on a huge hand because he probably wouldn't shut the field out like that. I think here you can bet or raise with impunity because his likely hand is top two and you just sucked out. The fact that one or more opponents who're now out of the hand probably had a 5 makes this read more likely to be correct, I think.

Hi baggins,


You raise a couple of good points.


I agree that you have to be very good at reading your opponents' playing styles to gain enough information from the way they play their hands to significantly alter your preferred line of action.


You also tend to get more information from your active opponents simply because you get to see them make more decisions. At the river, you can often call their exact hand or class of hands ... which is often useful for either saving bets or making good value bets on the end.


On the other hand, keep in mind that the number of folded opponents is most often larger than the number of active opponents on the turn and river. While less information might be available per player, the total amount of information from the folded players could be comparable to what we know from those players still active in the hand. (skp quotes an Omaha scenario created by Dan Hanson which illustrates this far better than my wimpy example above.)


Ultimately the decision you make at the table should be based on all the information available to you. Thinking about what your opponents might have folded will probably affect your correct choice of play in certain situations.


A few scattered thoughts on my part, at any rate. Thanks for contributing,


Dave Shaw

I think you are entering a realm where you have to consider the computational complexity of what you are doing, and whether you have enough time or brainpower to do these extra eliminations to make it worth the effort.


What I am saying is that if I were writing a book on computer poker or writing a computer poker program, topics along the lines you are suggesting would be mentioned. Otherwise, I would consider the problem too hard.


In general, the problem is that a computer may hold a record of each possible opponent holding along with the probability that an opponent will play a holding a given way to a given point in a hand. So a computer will find it easier (maybe even easy) to update the conditional probabilities than a human would.


Compare this to how a human considers his opponents' holdings. Instead of holding a record of each possible opponent holding and the probability of his playing a holding a given way; we're likely to invent certain classes of holdings along with some general probabilities that an opponent will play a whole class of holdings a certain way. I don't think there are a lot of humans who can consider enough 'classes' in a short amount of time for each of his folded opponents and then update conditional probabilities for his active opponents to be holding what is usually an entirely different set of classes of holdings to make this sort of idea useful in live play.


Maybe you can reduce an opponent to one or two holdings, watch him fold to a bet on the river, and then decide a little better if the bettor is bluffing or not. That might be useful.

maybe this can be applied better to stud??