Problem #1 - Structured Hand Analysis
I have been working on solving the problem from my last post. I think Harrington used the term "structured hand analysis" in Volume 1 or 2. What it means is when he is confronted with interesting situations during his play, he will often spend a few hours afterwards doing the math to beat the problem to death, and see if he can find a way to play more optimally than before. That is essentually what I am trying to do here. In The Mathmatics of poker they did this for a bunch of very simple situations, and used differential equations and other advanced math to solve them. I don't think they even worked there way up to a problem this difficult where you are solving for the Button, SB, and BB with lots of possible results. I don't quite have the math skillz to extend the mathmatics of poker to a full-up blind steal/defend the blinds type problem. But, I can solve using a trial and error approach with pokestove.
I have not solved the problem yet, but I put the framework together in a spreadsheet. From here it will be a few hours of trail and error to nail the result. Below is what the spreadsheet looks like. Oh yeah, this will be based on 7-handed which specifies the dead antes (not in original problem definition).
In the upper left corner are the hand ranges that I am attempting to solve for. The ranges shown are just an initial guess, and are there as a placeholder for the calculations. Once you have all the players ranges determined, you can get the chances of each situation occurring. This is shown in the lower left corner. Seven possible situations exist for the problem, and you can see that the odds of all the situations sum to 1 (100%), so I did not leave anything out. On the lower right side is the pot equity for each player and each situation. If somebody folds this is obviously zero. If the other two players fold this is 100%. For the situations where there is more than one player all-in, you put the hand ranges into pokerstove, and it will tell you what the equity is for each player. What I am showing here are just guesses at the equity. I have not run anything through pokerstove yet. Lastly, in the lower middle, you can see the EV calculations for everyone involved. The formula for this is (chances of happening) x (Players pot equity) x (pot size - amount invested). The amount invested is 10k for any player that goes all-in. So you calculate the expectation values for all the players/situations and try to find the hand ranges that maximize this for all players. The total expectation value for the Button+SB+BB is $1000. This represents the dead antes from the players that folded to the button. This is not correct now, because I guessed at the pot equity values (guessed high). So you change someones range, run the new ranges through poker stove, and see what it does for everyones expectation value. When everyones expectation value is maximized you have your answer. I will post my answer on Monday.
If you are not in the blogger fantasy football battle at FSL get over there and sign-up. Deposit $100 with bonus code "Blinders", and you will get $20 free for two free trys in the battle. If you don't like it after that, withdraw your full $100 deposit. I thought you guys liked free money? We have already picked up a few more bloggers this week, and it is never to late to play.