an example of why red zone is good

[pfkaok]

here is a bit of math to show some credibility of the redzone idea. the contention here, is that having a redzone stack offers you more very profitable options than being a midsized stack. my example here shows how having an M of 4 can yeild big +EV in a spot where you couldn't possibly get the same profit from a middling stack.

I'm looking at a particular situation, but from it you should be able to see how the idea will apply to other, similar spots. Here, we're 2 off the button at a 9 handed table, and the first 4 people fold.

In this spot, we have an M of 4, and there's antees, so this means that our stack is 8 BB. first i'll look at what happens if we push with our top 33% of hands (in pokerstoves range)

the calling ranges of your opponents will depend on lots of different factors, but for this example i'll just say that the first 3 players are fairly tight, and the BB is a bit looser. With an 8x BB stack, most people will still be fairly tight against you, since they don't want to call off a lot of their stack, and in most situations, obviously the BB will be able to call with a wider range. of course, the SB will probably call slightly looser, than button, who will be slightly looser than CO, but this is just an approximation. and also, we're ignoring the possibility of being called in 2 spots which is also pretty rare.

Assume CO - SB call with 77+, AQo+. BB will call with 22+, AJo+, KQo. if you guys really think these ranges are way off then let me know, and i can redo the numbers, but i think the for the most part its pretty decent, given that online this call will likely be for at least close to half their stack.

So, when this happens, you'll get called by CO - SB 17% of the time, BB 9% of the time, and the other 74% you'll take down the pot. When called by one of the tighter players, you'll win 35.5% of the time, and against the BB, you'll win 40.5% of the time.

When you get called by the tighter players, you lose 8 BB 64.5% of the time, or 8 (.645) = 5.16, and you win 9.83 BB(average of 10, 10, and 9.5 when SB calls) 35.5% of the time, or 9.83 (.355) = 3.49, so overall, 3.49 - 5.16 = -1.67BB when you're called

When you get called by the BB, you lose 8 BB 59.5%, 8(.595) = 4.76, and you win 9 BB 40.5%, 9(.405) = 3.64. overall 3.64 - 4.76 = -1.12

when you take down the pot, obviously you win 2 BB.

so, all together you have .17(-1.67) + .09(-1.12) + .74(2) = 1.1 BB. and you're pushing 33% of hands, so you're making .33(1.1) = .36 BB per hand when its folded to you. Over 1/3 BB is a LOT in this spot, and i doubt you could get anywhere near that, no matter how brilliantly you play with a stack twice that size.

another thing is that this style is pretty safe too, b/c you are only going to bust out about 16% of the times you push, or just over 5% of the times overall. thats not a whole lot of risk with such a short stack. you COULD eek out more profit if you wanted to push, say 50% of your hands, but it would be a higher risk, and also would have the long term effect of causing others to make looser calls against you which would kill your profits. but lets just say for this example that you haven't had any good spots come around, so you've been folding lots, and your push will still have credibility. see what happens if you extend your range:

since you haven't been pushing much, and still have respect, you still have the same calling ranges for the other players. but now, when the tighter ones call you're only going to win 32.3%. the math shows, .323(9.83) - .677(8) = -2.24

When the BB calls, you're now only going to win 38% of the time. so the numbers are, .38(9) - .62(8) = -1.54

winning the blinds is still of course 2BB

Overall, when pushing with half the hands, we get

.17(-2.24) + .09(-1.54) + 2(.74) = .96

Less per hand, but since we're pushing more, we now get .96(.5) .48BB per hand, which is A TON. of course the downside of this is that now we bust out almost 9% of the time, which is close to twice as much as in the other example. I won't get into the sacrificing EV for variance issue now b/c thats an entirely different, but very debatable issue. but it is very important to consider table image, and play cautiously in these spots if you've been pushing a lot.


All in all though, at least for the issue of EV, I challenge anyone to contend that they think they can make .48 BB/hand in that scenario described above with an M of, say 8. I highly doubt that you can even make more than the .36 from the first example, but i'm almost certain you coudn't have more profit than the .48 with a midsized stack, unless the players behind you were actually trying to lose their money to you.

Of course this is merely a singular example, but I believe that it does at least prove something. it shows that in this particular spot, you CAN make more profit with a smaller M. the next step would be going around an entire orbit, showing what you can make from each position, and under different sets of conditions (folded to you, raised already, limpers) and do weighted averages. this would take a lot more time, and possibly computer simulations, but i'm pretty sure it would show that you earn more in an orbit, on average, with an M of 4 than with an M of 8.

One very important point, which i think most are missing, is that "tough" decisions for good players, usually are "tough" b/c they're borderline, and if any EV is to be found, its VERY, VERY margainal. When you have an M of 8, a lot of times its best to fold b/c every other option would give a negative expectation. and sometimes, in those same spots a push would be a +EV play with a smaller M. also, a lot of times you could play the hand for a profit with a larger stack.

Its true that an expert will make the most with a huge stack when he has tons of options, and can use his skill to outplay the opponents on multiple streets. however, with a stack of say, around 8 M, you don't really have that many options. if you push preflop its usually quite an overbet, and not great risk/reward ratio. if you make a standard raise, unless your hand is very strong, you usually won't be presented with too many possible +EV moves the rest of the hand. this becomes more true when playing against aggressive players who at least have a clue what they're doing.