Path Theory - Pooh Bah Dissertation

[krishanleong]

Mason didn't like it. So here it is.


Path Theory

Pooh-Bah Dissertation


I’ve come up with a new way of looking at a hand called path theory. The basic idea is every decision you make in hold-em causes can cause different results. I refer to each possibility as Paths. The paths can be bright green (very profitable) to bright red (very unprofitable). As the community cards are dealt, the numbers of each kind of path change. For instance when you raise AK UTG in a full ring game, there are lots of green +EV paths. Some paths, like stealing the blinds, are only slightly green. The path representing a QJT flop against a player with QQ is blindingly green. The flop top pair top kicker against a set is very very red path(-EV). When we play, we naturally adjust the value of our hand based on the community cards and our opponent’s actions. Path theory doesn’t help us much in those situations. What path theory helps the most with are decisions that are contingent on a particular outcome.

We all raise AK UTG at full ring because we know it gives us more and greener paths than limping does. But what happens when we miss the flop? If you get 3 cold callers and 2 blinds calling your raise preflop and the flop comes Js8s9h, most of the green paths have dried up. We could also say your overcard outs are tainted, you have no backdoor draws. We could be drawing dead to a straight or reverse dominated by AJ, A9, or A8. You are left with a very small number of green paths. This is why against 5 players with AK on a flop such as this, most good players will check-call or check fold. The paths available are no longer +EV.

Another example would be calling a turn bet with a gutshot. If you have 4 outs to the nut straight and are HU in an 11 BB pot, it’s an easy call. There are 4 bright green paths where you win and all other paths are red. Playing a hand such as this is simple but also illustrates the main path theory concept.

The advantage of thinking about poker hands in a path theory context is because it helps you release hands when you are likely behind. This applies especially to hands that are contingent on a specific action from your opponent. Suppose you are dealt A2o in a 6-max game on the button. Every folds to you. Most shorthand players know its +EV to raise. There are a reasonable number of green paths. So you raise and the small blind folds and the big blind call. Before the flop is dealt, a moderate % of those green paths have disappeared. Some of the value of open-raising A2o from the button is the chance to steal the blinds. When the BB blinds call you have to remove the value you gave A2o preflop because of steal equity. You are now left with a high card, no kicker and position which is quite a strong position. If both blinds called you would have even less green paths to choose from.

Back to the hand, the flop comes QsJs6h and the BB checks. This isn’t a great flop since it has flush and straight possibilities, but a bet here will often take the pot of the BB has nothing. This is one of the green paths left after the BB called your preflop raise before the flop had been laid out. There are lots of flops that will miss both players where a single bet will take down the pot. There are also some flops where you hit an ace or trip 2s that result in nicely green paths.

Now consider the same flop is the BB bets. A lot of people have a hard time releasing this. This is an easy fold. It’s possible that you have the best hand and he is on a spade flush draw or a gutshot. But if you think about the hand from a path theory perspective it becomes easier to fold. First you raise preflop to try and steal the blinds. We know this is a +EV raise. When you got called some green paths withered. If you made that same raise 1,000 times, a significant % of time you would win the blinds outright. Then the flop came QsJs6h. This isn’t a great from since you didn’t pair and there are a lot of flush and straight draws on the board. On the flop you were bet into. This futher reduces the green paths you had on flop before the BB bet. At that point there were still many paths where a single bet would take down the pot. All those paths are gone once the BB bets. At this point, the number of red –EV paths greatly outweighs the number of green paths. You should fold.

Here is another example of a hand path theory would have helped me with. I’m in the BB with 88 at a 6-max table. UTG raises, MP calls, CO raises, Button folds, sb caps. I’m left with the choice of paying 3sb to see a flop. I think most players will agree that this is a profitable situation because if you hit your set, you should get paid off nicely. It’s likely that there is at least one premium pair. Given this logic, I call and intend to play fit or fold with the hopes of hitting a set. The flop comes 477 and the sb leads. Given my earlier read, I should dump this hand now. The green paths were all depended on me hitting a set on the flop. Since I didn’t, I’m not left with nearly all –EV paths. Despite this, I raise my weak overpair, get 3-bet and then call down based on pot odds until the river. On the river I fold and am shown KK and AA. If only I’d worked it out from a path theory perspective and been able to fold to the single flop bet.

A final example is from this hand. I had AA UTG at 5/10 6 max. I raise and am called by a loose passive BB. The flop comes JT4 suits unimportant. I bet and he calls. The turn is a 3. I bet and am check raised. Given that my opponent was loose passive, I knew he wouldn’t check raise the turn without having 1 pair beat. His most likely hand is two pair against which I have outs to continue. Calling the turn raise in this situation is pretty easy. The river is a 2. BB bets out. Lots of people make the crying call here saying the pot is large and you have an overpair. This is a mistake. If the turn call is correct because you had sufficient odds, the river fold is also absolutely correct. Think about all the green paths you had when you called the turn and all the red paths you have available to you now and fold.

To recap, use path theory when a certain response from your opponent generates the EV. When your opponent does not react the way you expect, you have to adjust value of the hand you are playing to account for your opponent’s action. Path theory works best in blind steal situation and situations involving fold equity.

Thanks for reading.

Krishan

[krishanleong]

Mason's response, [ QUOTE ]
Hi Krishan:

I've read your article and while interesting it has a major flaw which is that not all paths are created equal. The way the article is constructed you don't bring this out. So it's possible that while a hand may have more negative paths than positive ones, that positive one may be so strong that it still determines your choice. If you want to address this in the article feel free to resubmit.

Best wishes,

Mason Malmuth
Publisher
Two Plus Two Publishing LLC

[/ QUOTE ]

[RunDownHouse]

I was going to make a comment similar to Mason's. Its an interesting visualization, and no doubt helpful, but its usefulness is limited by the lack of quantification of the paths.

[helpmeout]

Nice post, you are pretty much saying dont get married to your hands.

[wowacedude]

Nice post.
AA, AK, KK is my lovers and i cant leave them for 1 BB after calling a turn CR knowing im beat.

Thanks.

[krishanleong]

[ QUOTE ]
Nice post, you are pretty much saying dont get married to your hands.

[/ QUOTE ]

Mostly. But the reason not get married to a hand is part of it's value is lost. I mostly think of this in blind stealing and blind defense situations.

Krishan

[stigmata]

I like it - what's nice is that it is a visual way of exploring something that is normally approached mathematically. It definately helps to think through things in a different way, and it helped me think about a couple of recent problematic hands.

I think "Mason didn't like it" is a little strong, and you did touch on the criticism through your analogy of path "brightness".

I don't know if you could expand on this somehow. E.g What are paths made up from? I'm thinking they have two assets - one is the probabilty that this is the "true path" given villains previous actions and the choice of your next action. The other property is your pot equity given this path.

So coming back to your example of A2o on QsJs6h, and the BB check to you. If you bet, there are several paths, one of which is that your opponent folds. Whilst the probability of this path is low (small size path), the brighness (e.g. equity) is high, for example.

So in a simple blind stealing type situation, could you use e.g. pokerstove (given the range of villain defending hands) to work out the probabilities and equities of each given branch on the tree.

I think a clear, simple "tree" diagram with these plotted might address Mason's criticisms, although I'm not sure.

Rambling now, just my immediate thoughts....

[Lost Wages]

I think Mason hit it on the head. Take the gutshot on the turn example from your original post. There are 4 green paths and 42 red yet calling is correct. Path theory doesn't really give you any more than a general feel of what to do, i.e. there are a lot of green paths or there are few green paths.

For that reason, I think that path theory is inferior to an EV perspective. As long as EV is reevaluated as new information is received.

On the other hand, I like the idea of Pooh Bah dissertations.

Lost Wages

[krishanleong]

[ QUOTE ]
I think Mason hit it on the head. Take the gutshot on the turn example from your original post. There are 4 green paths and 42 red yet calling is correct.

[/ QUOTE ]

Yeah, I hate that I included that hand. That hand was just suppose to be an example of how path theory would work in ordinary situations. It's not useful for ordinary situations. It's only useful in situations where you have fold-equity, bluff-equity, and other opponent response contingent plays.

Consider a turn semi-bluff lead. When people get called they frequently feel obligated to bluff the river when they miss. I'm saying this is wrong because part of the value of a semibluff on the turn was taking it down right there. The other part comes from making your draw. On the river, you are left with nothing and should check-fold in most cases.

Krishan

[krishanleong]

[ QUOTE ]
For that reason, I think that path theory is inferior to an EV perspective. As long as EV is reevaluated as new information is received.

[/ QUOTE ]

It's meant more as a complimentry system of analysis. Kinda like implied odds and pot odds. You use it to determine the EV of a situation.

Krishan