I'm not much of a fan of discussing individual hands as is common here. Bryce Paradis, in his article "What is good poker?"
expresses many sentiments I agree with:
The problem with these sorts of questions is that, in general, the ability to accurately quantify the likelihood of your opponent’s actions and future actions is not something that can be effectively taught. While it is perhaps better to have asked this sort of question and received an answer than not to have asked at all, even players at very advanced levels spend too much time debating these largely circumstantial issues, although it makes little sense to ask those who did not have access to all of the available information (e.g., previous hands) what they thought your opponent’s actions were likely to represent. Ultimately, a player has to accept that this is an analysis skill acquired through experience, and he will have to play some measure of poker in order to accurately determine the likelihood of his opponent’s actions.
Some of the most productive poker discussion stems from people sharing methods to quickly and easily “solve” common mathematical questions
When we play poker, we make use of heuristics (rules of thumb) to alleviate for the difficulty of making thorough evaluations of our expected value in the limited time frame we are allocated for each hand. Unfortunately, many common poker heuristics do not have any underlying mathematical basis. (Consider my debunking below of the notion that calling cannot be the most profitable play when a raise is for a large portion of your stack: "shove or fold"). Hence, I would like to work on developing some better heuristics.
The first type of heuristic is for simplifying problems that are theoretically easy, but are often too time-consuming to do on the fly. An example is the decision to open-shove preflop (in this case, relative to folding). Given your assumptions about your opponent's calling range, the EV equation is simple enough, but it can be difficult to come up with a good estimation of your hand's equity against your opponent's calling range and plug it into the equation, all in your head and in less than 15 seconds. In this case I think graphs are helpful (see deviations from Nash thread below), as memorising the basic look of a graph can help you decide approximately what to do in a multitude of situations. But other forms of heuristics are possible.
Secondly, I would like to work on approximations for problems that are mostly intractable even in theory. I speak of evaluating expected value when many future actions are possible. On the flop, for instance, there are 2162 possible turn and river combinations. It is much too difficult to go through all of these and estimate your opponent's actions with every hand. Instead, it is better to come up with a good approximation of the expected value of the subgame that results from a certain action and compare it with an approximation of the value of the subgame that results from another action. One way to do it is estimate your all-in equity, and then adjust it upwards or downwards depending on how much you figure you are likely to gain or lose from the existence of future betting rounds. This depends on your hand, your opponent's range, and some basic assumptions of how your opponent is likely to play on future streets (which may depend on your actions).
I would like to discuss this on AIM or MSN (PM me), however this thread is fine too. I am not interested in "coaching", so please try to make a contribution rather than just ask me about my ideas. It would be particularly helpful (although by no means necessary) if you are competent at higher math and especially programming. I'm pretty bad at math above elementary calculus and know nothing about programming, which is mostly the reason that some of these threads below are "incomplete".
A digest of my strategy posting + ideas I would like to see to fruition.
Why the notion of "shove or fold" is a myth
Graphing optimal deviations from the Nash equilibrium shove or fold chart (only one graph completed)
Graph of minimum equity needed to minraise, the equation
The latter part of this thread contains a number of musings on the correlation between all-in equity and expected value
Analysis of card removal effects
Rake, tournament equity and hourly rate
Optimisation against open-shover (incomplete)
Optimising the balance between interest-bearing accounts and transaction costs (incomplete)