Pooh-Bah Post: 3bet shoving EV math
09-04-2009
, 06:29 PM
First, a bit of history on how this turned into what it is:
I've been thinking for quite a while about what to write for my pooh-bah post (for those of you who don't know, it's somewhat of a tradition here to write something of substance when you reach the poster title of "Pooh-Bah", which is 3500 posts). I had a couple ideas kicking around, but then one of spamz0r's old posts about shallow-stack play was bumped and talked about quite a bit, and one of the things he talked about was Sklansky-Chubukov tables for shoving from the button. If you're not familiar with Sklansky-Chubukov, the basic idea is to figure out how shallow stacks need to be before open-shoving any hand is +EV, even if you turn your hand faceup and your opponent calls perfectly. (meaning they call when they have 50%+ equity against your specific hand, and fold when they don't)
I think I had heard about these numbers before, but never really given them any thought as a preflop shoving strategy, for the same reason that I haven't put any effort at all into memorizing nash equilibrium tables, or even SAGE for that matter (and yes, as a matter of fact, I *can* feel the waves of shock and disdain that just flooded my direction for saying that). But realistically speaking, your opponents are almost always playing way too poorly for those tables to have any meaning. For example, I generally continue minraising until my opponent starts shoving too often, even down into the 4-6bb area. If somebody is willing to flat a minraise with 6bbs, and play fit-or-fold post flop (and frankly, you might be very surprised to see how many opponents do exactly this), then shoving loses out on a ton of value, even if they're folding way too much, which is also generally true.
So, in general, I don't give preflop shoving strategies much weight. But in looking over Sklansky-Chubukov numbers, it occured to me that it really is useful to have some idea of where those numbers lie, because no matter what your preflop strategy is, or what your opponent's preflop strategy is, it is *always* better to shove when the stacks are below the Sklansky-Chubukov number than it is to fold. So even in those areas where I'm minraising much longer than most people would, it's still useful to know a range of hands that I shouldn't ever consider folding even if I don't want to minraise them, because they are +EV to shove even if I turn my cards face-up and my opponent calls perfectly. (and obviously, if they don't call perfectly, it will be even *more* +EV, because they're either folding hands that are in the lead, or calling with hands that are an underdog) For example, with 6bb stacks, I would typically fold a hand like J6 offsuit, because it just really can't make much of anything post-flop, so you're almost always bluffing post-flop. Now, in the situations where I'm still minraising with shallow stacks, it probably is profitable to minraise 100% of hands and cbet a large percentage of the time, but I don't really want to make it extremely obvious that's what I'm doing, so I dump the hands in the middle of my range that don't hit very many flops, like J6o. But since it's profitable to shove J6o face-up at up to 6.1bbs, obviously I should not ever be folding J6, and should be shoving instead.
Then as I was thinking about Sklansky-Chubukov numbers, it occured to me that it might be really interesting to see a similar table for 3-bet shoving. So I started working on software to calculate tables for that.
Obviously, it's a lot more complicated than the Sklansky-Chubukov table, because you have to take your opponent's opening range into account, as well as the size of their open, whether it be a limp, minraise or standard 3x raise. Since there are just simply way too many possibilites even with just those two variables to list out tables, I ended up writing an application that lets you enter ranges, and generates an EV table based on either a specified calling range, or a "perfect" calling range.
To be perfectly honest, this application turned out being a *lot* more work than I originally though it was going to be, and it also turned out to be pretty slick, (think PokerStove for preflop play, but on steroids) so I'm exploring a few other options for maybe getting compensated for the work I did on it before I think about just giving it away for free.
What I am going to do here though, is go through the 3betting math that it uses, and give some examples.
Lets start with a general EV equation for 3-bet shoving.
Given the following variables:
C = the percentage of the time your 3bet gets called
E = your equity when you get called
B = the button bet size (in big blinds, where a limp = 1, a minraise = 2, etc)
M = effective remaining stacks (measured in big blinds, and before blinds are posted)
You can calculate the EV of a 3bet shove using the following equation:
EV(fold) = (1-C)*(B+1) (when villain folds, you win whatever he put in the pot plus your posted big blind)
EV(call) = C * (E * 2 * M - ( M - 1 )) (when villain calls you risk M-1 chips and get back the effective stack * 2 * your equity)
Giving a total of:
EV(total) = 1 - CB - C + B + 2CEM - CM + C
= 2CEM - CM - CB + B + 1
For example, with even stacks at 30BBs, if your opponent is raising the top 60% of hands to 3x on the button, and calls the top 15% when you 3bet shove (meaning he folds 75% of the time when you shove), you can calculate the EV of shoving individual hands as long as you know the equity against the range that he calls with.
So for this simple example, if we use PokerStove's "top x%" ranges, we can calculate the EV of shoving AKo, K4s, 77, J5o, and 53o as follows:
PokerStove lists the following range as "top 60%": 22+,A2s+,K2s+,Q2s+,J2s+,T3s+,95s+,85s+,75s+,64s+,5 4s,A2o+,K3o+,Q5o+,J7o+,T7o+,97o+,87o
and this range as "top 15%": 77+,A7s+,K9s+,QTs+,JTs,ATo+,KTo+,QJo
AKo has 61.356% equity against "top 15%" range, giving us:
EV(total) = 2*0.25*0.61356*30 - 0.25*30 - 0.25*3 + 3 + 1
So the EV for shoving AKo against this calling range is 4.9534 big blinds.
Given that K4s has 35.903% equity, 77 has 45.849%, J4o has 28.021%, and 53o has 30.697%:
EV(K4s) = 2*0.25*0.35903*30 - 0.25*30 - 0.25*3 + 3 + 1 = 1.135 bbs
EV(77) = 2*0.25*0.45849*30 - 0.25*30 - 0.25*3 + 3 + 1 = 2.627 bbs
EV(J4o) = 2*0.25*0.28021*30 - 0.25*30 - 0.25*3 + 3 + 1 = -0.04685 bbs
EV(53o) = 2*0.25*0.30697*30 - 0.25*30 - 0.25*3 + 3 + 1 = 0.35455 bbs
Obviously, if shoving 53o is +EV against this opponent, we can 3bet shove very liberally, but knowing that he is folding 75% of the time that he raises makes this fairly obvious.
Simple enough, right? Here's a screenshot of the full table given a raising range of Pokerstove's "Top 60%" and a calling range of PokerStove's "Top 15%":
The first thing to notice is that the numbers don't match! We figured K4s was worth 1.135 big blinds, and the table shows 1.2 bbs. The reason for this is twofold. First of all, When we figured out how often we were going to get called, we just said 15% vs 60%, meaning we only get called 25% of the time. But pokerstove's "top 60%" is not really exactly 60%. It's actually 59.57%, or 790 combinations out of a possible 1326. And "top 15%" isn't really 15%, but 15.08%, or 200 out of 1326.
In addition to this, if villain is raising the "top 60%" (or "top 59.57%") of hands, some of those hands include kings and 4s. If we're holding K4s, then there are really only 727 combinations that villain can be holding in the specified range, rather than 790, and he's only calling with 180 combinations, as opposed to 200. This gives us a true C of 24.759%, meaning:
EV(K4s) = 2*0.24759*0.35903*30 - 0.24759*30 - 0.24759*3 + 3 + 1 = 1.163, or 1.2 if you round up.
In the case of 53o, the button is raising 754 combinations rather than 790, but still calling 200 combinations, because 53o doesn't block any hands that are in the "top 15%" range, since none of them contain a 3 or a 5. This gives us a true C of 26.525%, meaning:
EV(53o) = 2*0.26525*0.30697*30 - 0.26525*30 - 0.26525*3 + 3 + 1 = 0.132bbs, or 0.1 rounding to the nearest tenth of a big blind.
Please note that I'm not actually recommending that you start 3bet shoving at 30 big blinds. This is purely an example that illustrates the math. First of all, it's very difficult to approximate both opening and calling ranges for pretty much any opponent. In addition, it's almost certain that there are a ton of hands that would be way more +ev to 3bet smaller, or flat call, than it would be to shove. Then there's also the fact that some hands that are slightly +EV have you risking a relatively big stack for a very small amount of EV.
This is a also a very simplistic view of opening and shoving ranges, because for the above math to be valid, we're saying that our opponent always raises 3x, always does it with exactly the top 60% of hands, and always folds everything but the top 15% of hands. These assumptions are all pretty unrealistic. Opponents will sometimes 3x, sometimes minraise, sometimes limp, and sometimes just open-shove. Their calling ranges are equally fluid. But hey, that's the game of heads-up. It's a guessing game most of the time. But still, knowing the math behind it all can help you to make better decisions.
So that's great. We now know that, given a pretty unrealistic set of assumptions, shoving is +EV with almost any two cards if our opponent is folding a lot. Gee, aren't I a genius? Cause I'm sure you didn't know that already.
Also, the original point of this was to explore an equivalent to Sklansky-Chubukov numbers for 3bet shoving. I've done that, but unfortunately, as I mentioned, there's not really a way to distribute that information, since the results are different for every combination of your opponent's opening range, and his opening bet size.
But I will go through the math used to figure out what stack sizes you can profitable shove, given a specific range and open bet size.
Our original equation was:
EV = 2CEM - CM - CB + B + 1
But what we really want to know is at what stacksize (M) is it profitable to shove against a given opening range, even if we turn our hand faceup and our opponent calls perfectly?
To do this, we set the EV equal to 0, in order to find the break-even point, and then solve for M.
0 = M(2CE - C) - CB + B + 1
M(2CE - C) = CB - B - 1
M = (CB - B - 1)/(2CE - C)
Unfortunately, we still need two things that are rather difficult to get at without running a simulation for the ranges involved. We need to know E, which represents the equity our hand has when we get called, and we need to know C, which is the percentage of the time that our hand gets called. (which, since our opponent is calling perfectly, obviously relies directly on what our exact hand is!)
Here are a couple examples, based on situations similar to the ones in the previous example.
If our opponent is raising to 2x with the top 60% of his hands, and calling perfectly when we 3bet shove, we get the following table:
Unlike the last table, which showed EV in big blinds, this one shows the effective stack size (measured in big blinds, and at the start of the hand, before blinds are posted) at which it's neutral EV to 3bet shove, given your opponent's open size and opening range.
From this, we can see that if our opponent is raising top 60%, we should almost never consider folding Ax when stacks get shallow, because shoving will be +EV even if our opponent calls perfectly.
But there are a few important things to note here. First, and most obvious, is that it assumes villain is calling perfectly. That is definitely not every really going to be true. Second, this table is obviously going to undervalue hands like 75s that have a lot of equity when called by a realistic range, but not so much when they get called by a "perfect" range. A lot of the equity in 3bet-shoving something like 75s comes from the fact that villain is folding a lot of better hands (like 95s for example), and the stack size shown here for 75s assumes that every better hand in villain's opening range that's ahead of is will call. On the flip side, bad hands with one big card (like K6o for example) will be a lot closer to their actual value, because villain will call with fewer worse hands, but won't fold very many hands that are better.
Of course, the most important thing to note is that we're again dealing with a set of assumptions that are fairly unrealistic: that villain always minraises top 60%, and never plays any other hand in that range any other way. Unfortunately this isn't really a problem with simple answers.
But hopefully you now have the tools to start figuring things out on your own, if you have the desire. If not, well, maybe you can just buy the tool I developed to generate these tables, or download it if I decide not to try to sell it.
3500th!
I've been thinking for quite a while about what to write for my pooh-bah post (for those of you who don't know, it's somewhat of a tradition here to write something of substance when you reach the poster title of "Pooh-Bah", which is 3500 posts). I had a couple ideas kicking around, but then one of spamz0r's old posts about shallow-stack play was bumped and talked about quite a bit, and one of the things he talked about was Sklansky-Chubukov tables for shoving from the button. If you're not familiar with Sklansky-Chubukov, the basic idea is to figure out how shallow stacks need to be before open-shoving any hand is +EV, even if you turn your hand faceup and your opponent calls perfectly. (meaning they call when they have 50%+ equity against your specific hand, and fold when they don't)
I think I had heard about these numbers before, but never really given them any thought as a preflop shoving strategy, for the same reason that I haven't put any effort at all into memorizing nash equilibrium tables, or even SAGE for that matter (and yes, as a matter of fact, I *can* feel the waves of shock and disdain that just flooded my direction for saying that). But realistically speaking, your opponents are almost always playing way too poorly for those tables to have any meaning. For example, I generally continue minraising until my opponent starts shoving too often, even down into the 4-6bb area. If somebody is willing to flat a minraise with 6bbs, and play fit-or-fold post flop (and frankly, you might be very surprised to see how many opponents do exactly this), then shoving loses out on a ton of value, even if they're folding way too much, which is also generally true.
So, in general, I don't give preflop shoving strategies much weight. But in looking over Sklansky-Chubukov numbers, it occured to me that it really is useful to have some idea of where those numbers lie, because no matter what your preflop strategy is, or what your opponent's preflop strategy is, it is *always* better to shove when the stacks are below the Sklansky-Chubukov number than it is to fold. So even in those areas where I'm minraising much longer than most people would, it's still useful to know a range of hands that I shouldn't ever consider folding even if I don't want to minraise them, because they are +EV to shove even if I turn my cards face-up and my opponent calls perfectly. (and obviously, if they don't call perfectly, it will be even *more* +EV, because they're either folding hands that are in the lead, or calling with hands that are an underdog) For example, with 6bb stacks, I would typically fold a hand like J6 offsuit, because it just really can't make much of anything post-flop, so you're almost always bluffing post-flop. Now, in the situations where I'm still minraising with shallow stacks, it probably is profitable to minraise 100% of hands and cbet a large percentage of the time, but I don't really want to make it extremely obvious that's what I'm doing, so I dump the hands in the middle of my range that don't hit very many flops, like J6o. But since it's profitable to shove J6o face-up at up to 6.1bbs, obviously I should not ever be folding J6, and should be shoving instead.
Then as I was thinking about Sklansky-Chubukov numbers, it occured to me that it might be really interesting to see a similar table for 3-bet shoving. So I started working on software to calculate tables for that.
Obviously, it's a lot more complicated than the Sklansky-Chubukov table, because you have to take your opponent's opening range into account, as well as the size of their open, whether it be a limp, minraise or standard 3x raise. Since there are just simply way too many possibilites even with just those two variables to list out tables, I ended up writing an application that lets you enter ranges, and generates an EV table based on either a specified calling range, or a "perfect" calling range.
To be perfectly honest, this application turned out being a *lot* more work than I originally though it was going to be, and it also turned out to be pretty slick, (think PokerStove for preflop play, but on steroids) so I'm exploring a few other options for maybe getting compensated for the work I did on it before I think about just giving it away for free.
What I am going to do here though, is go through the 3betting math that it uses, and give some examples.
Lets start with a general EV equation for 3-bet shoving.
Given the following variables:
C = the percentage of the time your 3bet gets called
E = your equity when you get called
B = the button bet size (in big blinds, where a limp = 1, a minraise = 2, etc)
M = effective remaining stacks (measured in big blinds, and before blinds are posted)
You can calculate the EV of a 3bet shove using the following equation:
EV(fold) = (1-C)*(B+1) (when villain folds, you win whatever he put in the pot plus your posted big blind)
EV(call) = C * (E * 2 * M - ( M - 1 )) (when villain calls you risk M-1 chips and get back the effective stack * 2 * your equity)
Giving a total of:
EV(total) = 1 - CB - C + B + 2CEM - CM + C
= 2CEM - CM - CB + B + 1
For example, with even stacks at 30BBs, if your opponent is raising the top 60% of hands to 3x on the button, and calls the top 15% when you 3bet shove (meaning he folds 75% of the time when you shove), you can calculate the EV of shoving individual hands as long as you know the equity against the range that he calls with.
So for this simple example, if we use PokerStove's "top x%" ranges, we can calculate the EV of shoving AKo, K4s, 77, J5o, and 53o as follows:
PokerStove lists the following range as "top 60%": 22+,A2s+,K2s+,Q2s+,J2s+,T3s+,95s+,85s+,75s+,64s+,5 4s,A2o+,K3o+,Q5o+,J7o+,T7o+,97o+,87o
and this range as "top 15%": 77+,A7s+,K9s+,QTs+,JTs,ATo+,KTo+,QJo
AKo has 61.356% equity against "top 15%" range, giving us:
EV(total) = 2*0.25*0.61356*30 - 0.25*30 - 0.25*3 + 3 + 1
So the EV for shoving AKo against this calling range is 4.9534 big blinds.
Given that K4s has 35.903% equity, 77 has 45.849%, J4o has 28.021%, and 53o has 30.697%:
EV(K4s) = 2*0.25*0.35903*30 - 0.25*30 - 0.25*3 + 3 + 1 = 1.135 bbs
EV(77) = 2*0.25*0.45849*30 - 0.25*30 - 0.25*3 + 3 + 1 = 2.627 bbs
EV(J4o) = 2*0.25*0.28021*30 - 0.25*30 - 0.25*3 + 3 + 1 = -0.04685 bbs
EV(53o) = 2*0.25*0.30697*30 - 0.25*30 - 0.25*3 + 3 + 1 = 0.35455 bbs
Obviously, if shoving 53o is +EV against this opponent, we can 3bet shove very liberally, but knowing that he is folding 75% of the time that he raises makes this fairly obvious.
Simple enough, right? Here's a screenshot of the full table given a raising range of Pokerstove's "Top 60%" and a calling range of PokerStove's "Top 15%":
The first thing to notice is that the numbers don't match! We figured K4s was worth 1.135 big blinds, and the table shows 1.2 bbs. The reason for this is twofold. First of all, When we figured out how often we were going to get called, we just said 15% vs 60%, meaning we only get called 25% of the time. But pokerstove's "top 60%" is not really exactly 60%. It's actually 59.57%, or 790 combinations out of a possible 1326. And "top 15%" isn't really 15%, but 15.08%, or 200 out of 1326.
In addition to this, if villain is raising the "top 60%" (or "top 59.57%") of hands, some of those hands include kings and 4s. If we're holding K4s, then there are really only 727 combinations that villain can be holding in the specified range, rather than 790, and he's only calling with 180 combinations, as opposed to 200. This gives us a true C of 24.759%, meaning:
EV(K4s) = 2*0.24759*0.35903*30 - 0.24759*30 - 0.24759*3 + 3 + 1 = 1.163, or 1.2 if you round up.
In the case of 53o, the button is raising 754 combinations rather than 790, but still calling 200 combinations, because 53o doesn't block any hands that are in the "top 15%" range, since none of them contain a 3 or a 5. This gives us a true C of 26.525%, meaning:
EV(53o) = 2*0.26525*0.30697*30 - 0.26525*30 - 0.26525*3 + 3 + 1 = 0.132bbs, or 0.1 rounding to the nearest tenth of a big blind.
Please note that I'm not actually recommending that you start 3bet shoving at 30 big blinds. This is purely an example that illustrates the math. First of all, it's very difficult to approximate both opening and calling ranges for pretty much any opponent. In addition, it's almost certain that there are a ton of hands that would be way more +ev to 3bet smaller, or flat call, than it would be to shove. Then there's also the fact that some hands that are slightly +EV have you risking a relatively big stack for a very small amount of EV.
This is a also a very simplistic view of opening and shoving ranges, because for the above math to be valid, we're saying that our opponent always raises 3x, always does it with exactly the top 60% of hands, and always folds everything but the top 15% of hands. These assumptions are all pretty unrealistic. Opponents will sometimes 3x, sometimes minraise, sometimes limp, and sometimes just open-shove. Their calling ranges are equally fluid. But hey, that's the game of heads-up. It's a guessing game most of the time. But still, knowing the math behind it all can help you to make better decisions.
So that's great. We now know that, given a pretty unrealistic set of assumptions, shoving is +EV with almost any two cards if our opponent is folding a lot. Gee, aren't I a genius? Cause I'm sure you didn't know that already.
Also, the original point of this was to explore an equivalent to Sklansky-Chubukov numbers for 3bet shoving. I've done that, but unfortunately, as I mentioned, there's not really a way to distribute that information, since the results are different for every combination of your opponent's opening range, and his opening bet size.But I will go through the math used to figure out what stack sizes you can profitable shove, given a specific range and open bet size.
Our original equation was:
EV = 2CEM - CM - CB + B + 1
But what we really want to know is at what stacksize (M) is it profitable to shove against a given opening range, even if we turn our hand faceup and our opponent calls perfectly?
To do this, we set the EV equal to 0, in order to find the break-even point, and then solve for M.
0 = M(2CE - C) - CB + B + 1
M(2CE - C) = CB - B - 1
M = (CB - B - 1)/(2CE - C)
Unfortunately, we still need two things that are rather difficult to get at without running a simulation for the ranges involved. We need to know E, which represents the equity our hand has when we get called, and we need to know C, which is the percentage of the time that our hand gets called. (which, since our opponent is calling perfectly, obviously relies directly on what our exact hand is!)
Here are a couple examples, based on situations similar to the ones in the previous example.
If our opponent is raising to 2x with the top 60% of his hands, and calling perfectly when we 3bet shove, we get the following table:
Unlike the last table, which showed EV in big blinds, this one shows the effective stack size (measured in big blinds, and at the start of the hand, before blinds are posted) at which it's neutral EV to 3bet shove, given your opponent's open size and opening range.
From this, we can see that if our opponent is raising top 60%, we should almost never consider folding Ax when stacks get shallow, because shoving will be +EV even if our opponent calls perfectly.
But there are a few important things to note here. First, and most obvious, is that it assumes villain is calling perfectly. That is definitely not every really going to be true. Second, this table is obviously going to undervalue hands like 75s that have a lot of equity when called by a realistic range, but not so much when they get called by a "perfect" range. A lot of the equity in 3bet-shoving something like 75s comes from the fact that villain is folding a lot of better hands (like 95s for example), and the stack size shown here for 75s assumes that every better hand in villain's opening range that's ahead of is will call. On the flip side, bad hands with one big card (like K6o for example) will be a lot closer to their actual value, because villain will call with fewer worse hands, but won't fold very many hands that are better.
Of course, the most important thing to note is that we're again dealing with a set of assumptions that are fairly unrealistic: that villain always minraises top 60%, and never plays any other hand in that range any other way. Unfortunately this isn't really a problem with simple answers.
But hopefully you now have the tools to start figuring things out on your own, if you have the desire. If not, well, maybe you can just buy the tool I developed to generate these tables, or download it if I decide not to try to sell it.
3500th!
09-04-2009
, 06:36 PM
09-04-2009
, 06:59 PM
09-04-2009
, 10:19 PM
09-05-2009
, 03:00 AM
Quote:
Can you define "calling perfectly" for me? Does this mean; Given that hero turns his hand face up, villain calls with whatever has the necessary equity. Or, does it mean; Given that hero only jams and jams each of x hands, villain calls with whatever has the necessary equity.
some weird spots can come up like this obv, openshoving K2o on the button for example against JTs, villain will have pot odds to call (assuming we're talking about shallow stacks obv) since he has a little over 49% equity but cant call according to the defenition of the chubukov number
same with like shoving 96s vs 33 etc etc
if we shove over a minraise though villain will defenitly get way better pot odds to call when shallow; thing is that if villain calls perfectly according to pot odds i think some hands can actually shift a bit in "amount of bb's where pushing is ev+"
let's just look at the chubukov number to openshove button with K2o; in the chart it says 19.999415, which is a little under 10bb's; if villain calls with hands which have enough equity however, the equity when called goes up but the amount of folds we get goes down... obv these two cancel out each other a bit but not perfectly obv and it will defenitly change the number at least a bit
or maybe i'm too tired and this doesnt make sense atm hahastill havent read/checked everything in OP, will defenitly do when i wake up and check all the numbers/calc then
interesting post though, nice poohbah, tool looks awesome as well btw
only thing you really have to take into consideration here (not looking at math or anything now) is the fact that metagame/gameflow will change hugely if you start shoving over minraises frequently and/or pretty early on
PS: i just thought of something weird in chubukov's numbers which can lead to something which goes into an infinite loop into your program if your defenition is like this imo; let's just say we have a hand X and we can openshove button for 50bb's; let's just say there's a certain hand Y with 49% equity and villain has it, since villain needs to call 49 into total of 100 pot if we openshove exactly 50bb's, he has odds to call here, k? but now, because villain calls with this hand instead of folds, we can openshove 50.1bb's because our equity when called goes up a bit; but now we can openshove 50.1bb's but hand Y isnt included in the calling range anymore since it's not getting the correct pot odds... wtf? infinite loophole? maybe i'm just too tired lol, but interesting point skates
09-05-2009
, 11:12 AM
Pretty sure 'calling correctly' is making the call with the necessary equity for it to be +EV right? as opposed to equity>50%? I might be wrong tho... ill dig out my book when i get the chance...
and yeah spamz +1 on the gameflow changing significantly etc (don't know how to quote only part of a post yet)
Any how pretty sweet post Nixon... hope it sells glglgl
and yeah spamz +1 on the gameflow changing significantly etc (don't know how to quote only part of a post yet
) Any how pretty sweet post Nixon... hope it sells glglgl
09-05-2009
, 03:49 PM
