Monday, September 17, 2007

The Solution

The solution to the problem is above. It is accurate to 5% hand ranges, but with these type ranges you would be nit-picking to care if it is a 50 or 52% range. In case you do not remember the set-up is an MTT late, 7 handed. Blinds 1000/2000, ante 250. It folds to the button. Remaining players have 10k to start the hand and may only push or fold. What is everyones optimal range to maximize chip expectation value.


1) The button pushes with his top 50% range, or 1/2 of the time. The problem revolves around the competition for the dead $1000 in antes. This is the EV to be divided up between the three players. The button obviously would push more often than 33%, and because he is not posting any blinds, should be +EV overall for this problem.

2) The SB calls the buttons push with his top 45% range. This may seem high on the surface, but once the SB figures out the BB pushes 50% here, the SB goes ahead and calls with top 45%, also knowing that the BB is will come along about 40% of the time as well. So the SB is competing for the dead antes, defending his SB, and staying in the running for the BB. Again, three handed you would need to be calling at least your top 33% here.

3) The SB pushes his top 75% of hands if it folds to him. Smokkee guessed 100% which is close. The BB will adjust and call a huge portion (top 65%) of his hands because of this. The SB only needs to get through the BB to get the $1250 in dead antes and the BB. If he pushes 100%, the BB can pick and choose a little bit to hurt the SB. At top 75% there is nothing the BB can really do.

4) BB calls both 40%, SB only 65%, and BB only 70% of the time. This also seems high, but these guys are attacking the BB with the button raising top 50% and the SB raising 75% when the button folds. The BB must compete for the antes and defend his blind. Folding 100% is -2250 EV. The aggressive calling strategy is only -113 EV which is a huge improvement.

EV for the situation in the problem

Button +804
SB +308
BB -113
Rest of Table -1000

Total EV = 0

So does the EV make sense? The button is actually out of position for the problem, because this is push or fold preflop. That's a pretty big disadvantage, but he has the advantage of not posting a blind. The SB has positional advantage over the button, but must post the $1000. The BB has positional advantage over both players, but is putting the most into the pot blind. It makes sense that the Button would have the highest EV, and that the BB would have a negative EV based on the situation with the SB somewhere in between.

The interesting thing about hand ranges is how they are relative. A 35%/40% match up is almost the same as a 65%/70% match up. Once you figure out that the SB calls slightly less than the Button raises, and the BB calls them both slightly less of the than the SB calls, as well as the other two match-ups, it becomes a question of how often does the button steal.
The button acts first, and is choosing to maximize his expectation value (cut of the dead antes), so assuming the others will adjust to him, he finds that "stealing" with the top 50% of his hands maximizes EV at about 80% of the dead 1k. Once he makes this choice, everyone else reacts to what he did, and tries to maximize their own EV as best they can. The SB does this by aggressively attacking the BB, and the BB by aggressively calling them down.

These ranges are for optimal play. Obviously many do not play optimally during the end games of MTTs. This also may not be optimal in some cases from a "tournament equity" perspective where you are trying to maximize your expected cash in the MTT. If you feel that others will not play optimally and call down with these wide ranges, you pretty much have to raise these types of ranges, and your EV will be even higher than whats shown here. Below are some other, non-ideal solutions for the problem.
Initial Guess at Solution

Raising the BB Call Both Percentage Helps BB

BB calling the SB and button more often helps
Button Raising a higher percentage helps, but this was a bit too much

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At 12:30 PM, Blogger Hammer Player a.k.a Hoyazo said...

My only issue with all this game theory and optimal play stuff I'm reading about lately is that I've never met one single person who actually plays "optimally". Not one. So all the game theoreticians it seems to me make theoretical answers to theoretical problems, that have absolutely zero impact on real world poker.

I feel the same way about micro and macro economics generally, btw. Just a bunch of theoretical hooey that has no relevance to actual life.

At 1:39 PM, Blogger Blinders said...


It actually does help quite a bit. Lets say that you are the SB here after the button folds. It is not obvious you should push 75% of the time, but lets say you start doing so now. Lets say the BB does not play optimally and call the 65%, but calls top 30%. Then you make even more EV, because they have to call 65% to minimize your EV and maximize thiers. So here you are taking advantage of people not playing optimally. Now if you are the BB, and the SB is not pushing 75% to maximize his equity, you are making tons of EV becuase he is folding the pot to you too much. You obviously need to ratchet down your call%, but the sensitivities of the problem show you that you call the SB with a slightly smaller range than the SB is pushing with reguardless of the range. So you picked that detail up from the analysis.

The bigest problem with a game theory approach like this is that it is stack size dependent. You will get a much different answer for deeper stacks. Late MTTs with Ms around 3 are semi-common, and this answer is only for that exact situation.


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