"It depends" is one of the most common responses to most poker questions. Rather than sit there like an idiot and give a stereotypical answer like most do because they don't care or know enough to really analyze it themselves thoroughly enough to show why it depends and exactly "what" it depends on, instead I am going to begin the construction of the ultimate "formula" of poker. "solving" poker doesn't mean the game is over by any means. So much of the equation can be analyzed more in depth to come up with more accurate information. Intuition can use how the people are playing and how the table conditions have changed to anticipate hand ranges and thus probabilities. As the information is constantly changing, so must variables of the formula itself.
Poker is a very situational problem with conditional probabilities. IN other words an "IF-THEN" type of programming language of an equation. Or "IF" this happens "THEN" the odds are influenced in this manner and the optimal decision may change. IF you face this opponent who has adapted in such a way to have good odds of playing a particular style THEN they change to something else.
To understand the equation we need to account for multiple variables that potentially can connect with an alternative set of equation. Those wondering if you ever were going to use high school algebra and pre-calc, you got your answer! YES! Now you can see why the F(x) equations come in handy.
What your expectations of your chip stack are in a given hand give us the following equation assuming you plan to open up for a raise. We can go back and modify this later if you have a strategy of occaionally limping with the potential of limp-raising as well based upon the percentage of hands you limp and raise with and what informational advantages/disadvantages you give your opponents by doing so. Nevertheless, here is how we might begin to construct the formula:
Chip stack after result of hand=Current chip stack+(probability hand steals blinds* chips you win if you successfully steal)+(probability it doesnt*equity gained/lost if opponent responds).
Or
Ca=C+(p*B)+((x)*E)
Where
Ca=Chipstack AFTER result of hand
C=Current chip stack
p=probability steal attempt wins
x=probability it doesn't win blinds
B=chips steal attempts gain if successful.
E=equity gained/lost if opponent responds.
In this case, the probability the steal attempt doesn't is equal to (1-p) so we can substitute x for (1-p)
Ca=C+(p*B)+((1-p)*E)
Now what you would do is break down different parts of the equation.
In this case, the probability the steal attempt wins will be a very complicated formula in itself, and really is one of the reasons poker is not really entirely solvable, especially considering your opponent's range will change. Also even knowing information about those who folded ahead of you will provide "variable change" or information that changes the problem to have a different effective probability that the remaining opponents have a particular hand and what your equity is against that hand. Whether or not those left to act have a playable hand to a raise or a playing style/information about you or tournament conditions even that will influence them to raise.
But basically you will need to take 1 minus the probability that each successive opponent will have a playable hand and use that information to determine the probability that one or more of them will either call or raise. We won't be substituting that for now, or considering the possibility of multiway pots just yet. Without getting into all those fancy calculations, you might just intuitively estimate probability of seeing action based upon experience for now.
The chipstack will be known, the chips the steal attempts gain if successful will probably be 1.5 blinds or around 2.5 blinds if there are antes.
That brings us to the "E" part of the equation or "equity gained/lost if opponent responds". This too is entirely opponent dependent. The opponents will dictate what percentage of the time they fold, and if they choose to play, what percentage of the time they play they will raise or just call. Given that they raise, the decision is then back on you to determine based upon your raise hand range, what your equity is and what your fold range and 4bet range or call range is, and then the equity gained from that decision may hinge partially again on what percentage the opponent will call a 4bet shove shove or call a 4bet.
To simplify the equation at some point you have to just figure an all in and use chip sizes and probability of winning if called to determine the results.
BUT I DIGRESS...
E=(Pr*Er)+((1-Pr)*Ec)
as said before E=probability gained/lost if opponent RESPONDS in some way (call/fold)
Pr=Probability opponent raises given he responds
Er=equity gained/lost if he reraises (3bets)
(1-Pr)=probability opponent calls if he doesn't fold
Ec=Equity gained/lost if he calls.
Now I will not yet continue this equation... but just know that the equity gained/lost if at some point there is a "call" will have it's own set of equation with potential actions and probabilities of certain actions on the flop and probabilities of opponents foldings, probability that you will have a hand in your range that you can bet and/or raise on the flop and same thing on the turn and/or river with potential all ins that end the hand along the way with potential drawing equity.
The complicated part is actually not the formula itself with all these huge amounts of contingencies that you keep having to come up with a formula and ultimately plug it into ONE formula, but it's determining the actual values and probabilities based upon the opponents.
With multiple opponents there contains different numbers and thus you actually would have to duplicate the formula for each opponent and weight a probability assigned to each particular opponent and the probability that THEY are the one in the hand when determining value before you raise.
The formula itself is complicated... sure.. but the NUMBERS within the formula depend upon playing style. With completely unknown opponents you can still determine a baseline solid strategy to use that closely enough approximates the information needed to at least have some informational edge. Then as you gain information, the formula itself can be more easily defined.
What will greatly help you find these values is if you have detailed stats on opponents by using some kind of "poker tracker" or hold'em manager" program that does a lot of the leg work for you.
Trying to account for all these subtle nuances is incredibly important, but still very difficult.
YET, what no one has told you yet... is that it CAN be done, and is not all that hard... just very, very time consuming and perhaps actually coming up with the most accurate numbers require very complex calculations that may require advanced knowledge of excel and/or programming skills.
But that's about all i can give you for now!
Edit: This post has since been continued on the post Formula of Poker 2.
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http://nutballpoker.blogspot.com/2013/10/formula-of-poker-2.html
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