ISF's Staircase Theory

NOTE: If you don't think the example used is good look at my second essay's example.

What I'd like to talk about in this post is the theory that all aspects of poker, in terms of profits, is the shape of a staircase. The first thing I will look at is that the graph of “opponent paying you off” on the y axis and VP$IP in the x axis would be shaped like a staircase. Let’s look at a theoretical graph of this very situation.




This is what I would think this graph would look like. Note the actual values of opponent pay off isnt going to be to scale (in english it appears in my graph that your profits quadrouple when you play 35% of hands rather than 20% when obviously I don’t believe that is the case, or the fact that 35% has the highest profits, when its almost impossible to actually tell that that is the case); Don;t focus on the scale, focus on the slope. Don't focus on the example either. If you want a better example then go to my other post down farther.Also Note that we can assume that a 1% VPIP incease consists of worse hands than the previous increase (ie, if the lowest part of our VPIP is 87s, then if we were to play more hands those hands would be worse than 87s)

When a player plays 1%-20% of his hands, his opponent is only paying him off a bit more for each extra hand he plays, and therefore the marginal profit is low. But when the player plays 20% to 35% of his hands, he is gaining a much more significant profit in a very small VPIP range, and after that only gains a tiny bit of profit for the rest. After 35%, the same pattern is repeated.

The numbers itself of the graph aren’t important, but rather the shape of the graph. What the shape is theorizing is the opponents have several “aha!” moments for their perception of you. In a small VPIP to a still small VPIP, 1% to 20% VPIP on this graph, opponents would tag you as a tight player, and would play a player, disregarding all info except VPIP, the exact same between those two small VPIP’s. After a small number, there would be a certain number VPIP where opp has another “aha” moment and tags you as a loose player, and then plays you almost the exact same way between to medium VPIP ranges, in this case 20% to 35% VPIP. Finally right after a large VPIP opp has a final aha moment where you are a “fish” and the pattern is repeated.

At different levels I assume different cases are true. At a very low level the graph will have almost no slope, players suck so bad they don’t even look at your vpip. As you get to a level where opponents have PT and our paying a bit of attention, the graph becomes a staircase. The higher and higher levels you go, the more opponents are paying attention, and the more linear the graph gets (This would be because opponents start adjusting to even one percent changes in vpip).

If the staircase graph is correct, what does it tell you? What it says is that you are losing tons of value for not having a vpip on the vertical part of the staircase. Let me explain this in terms of the graph. At 20%, opp is paying you off approximately the same amount as 35%. At 20%, using only vpip as a factor, we are playing better hands than at 35% but getting paid off the same amount. So therefore, we have the best hand the most at 20% VPIP but getting paid off the same as having the worst hand way more at 35% VPIP. Meaning that at 20-21% VPIP we have the greatest profit!!! Does this make sense? One of the main goals of playing worse hands is to get paid off on our better hands, but that doesn’t happen very much in this range, so we are losing tons of value! So we should strive to find a vertical part of the staircase, or exactly when the opponent has his “aha” moment. In economics, this would be called maximizing our marginal revenue, which is the goal of every buisness and consumer, and for good reason.

If the staircase shape is the shape of the graph for the majority of stats and moves in poker, which I theorize is the case, this could be huge concept.

nt

It's likely an upside-down U. There is a large jump at the beginning, say going from 5-6 hands because you're adding hands with a lot of value. As you start adding more it slows down but keeps growing. You eventually reach a point where it will start to flatten out.

Adding trash hands will not greatly increase your return on better hands, it's pushing smaller and smaller edges. You have to factor in the money you loose from the trash hands to gain with good hands. So T7o may increase the value of AA by 15bb (number pulled out of my arse), but you have to spend 10bb (raise and standard cbet) so the marginal revenue isn't a big jump of 15bb, it's a smaller jump of 5bb. Eventually you will start loosing money from these trash hands, but there is a decently wide range where the effect of adding (or removing) hands on your winrate is small.

Obv this depends a lot on stakes and opponents.

I'm going to write a long and hopefully incredbily insightful post during class so expect one in two hours.

EDIT: Done, look at the OP, and tell me what you think.

I'm a senior in economics and must warn you not take traditional economic theory very seriously. Most of it rests on completely unreasonable assumptions that often directly contradict reality.

Originally Posted by dsaxton

I'm a senior in economics and must warn you not take traditional economic theory very seriously. Most of it rests on completely unreasonable assumptions that often directly contradict reality.

Look at what I just wrote and tell me what you think.

If the spelling and grammer is off, sorry. I didn't proofread at all. Obviously, there are some flaws i assume.

you do nothing to prove why you drew the graph in this way.

Also note if opponents pay off more, then we pay off more too due to our higher vpip.

ISF - the concept is good but the graph is confusing. Maybe not to you because it maps to something in your econ class, but if I ever knew the mathmatical definition of "marginal value" I forgot it.

Your y axis is "chance of opponent paying you off"??? What you want is the graph of profit as a function of V$IP assuming the graph you draw above.

One thing that isn't accounted for (nothing surprising since it's economics we're talking about) is the psychology of showdowns.

Showing down 25o even once vs a thinking opponent in a big pot gets you paid off a lot in later spots vs same villain (no breakthrough here obv). So while the theoretical difference between playing 35% and 20% of your hands may seem negligeable, in practice it allows you to show down significantly more crap, allowing to get paid off more easily by the ppl not using HUDs.

Also, how did you format your graph?

Use the enter key to break up the text. My reply is based off your original post because I don't read huge blocks of text. Also, it makes no sense why you would delete the original after people have already replied to it.

Originally Posted by zenbitz

ISF - the concept is good but the graph is confusing. Maybe not to you because it maps to something in your econ class, but if I ever knew the mathmatical definition of "marginal value" I forgot it.

Marginal is a fancy word for the rate of change (the derivitave, aka slope).

Originally Posted by zenbitz

Your y axis is "chance of opponent paying you off"???

Yes, your graph is way off. You are only considering the increased chance of a big hand being paid off without considering the value lost by playing the trash hands.

What you actually want is a graph that shows how much more you will make vs $VPIP. There is no way total return (let alone marginal return) is increasing from 40 -> 50. If it was then Laggs would be running 50/45 instead of 25/20.

Also your stair step is absurd. You're trying to imply that adding 10 hands has little positive return. Then BAM!!! the 11th hand adds a ton of value?

Originally Posted by Genitruc

Also, how did you format your graph?

Paint

Originally Posted by euphoricism

Also note if opponents pay off more, then we pay off more too due to our higher vpip.

Yeah I agree, so somewhere the graph is going to go down after the optimal VPIP. Also I have no idea why the graph would leap up wards at intervals such as 20%.

If our VPIP is 19% then we are only playing a few hands less than at 20%, so it should just be a stready incline up until the optimum and then a steady decline. Giving us some sort of skewed inverted U-shaped graph, no?

That said I am not an economist, so what do I know!? LOL

Alright here's my second go, hopefully this one makes more sense.
Before I start gabe the reason the graph is a staircase is because of the several "aha" moments opponents have. Better players obviously adjust a lot better and have smaller interval between aha moments, to a point where you can't even call it aha moments they just are changing their range based on every minimal adjustment in VPIP.

The Second Go:
I ended the last thought with if the staircase graph was true for most aspects of poker then it would be very significant.

I realized my first post didn’t really say much at all besides backing the theory of poker as a staircase graph. I realized then another backing point to the staircase theory: people already follow it.

It’s interesting I realized that good players tend to be at certain vpip and pfr levels. 16/12 we have delclared as nitty, and pretty much pay off a 16/12 the same as we pay off a 20/14. Then we have the Tagg, the 22/18, who we pay off with essentially the same range as a 25/19 or 26/20. Then there is the lagg, the 28/24 or such, who we pay off with essentially the same range as a 34/28 or the such. Good players have naturally flocked to the vertical parts of the staircase (or where i'd believe it to be)

Now lets look at a situation where I theorize the staircase effect also holds true. Lets look at one of the more simple plays in texas. You raise preflop, someone threebets you, and you call oop. The flop comes whatever, opponent bets, and we c/r all in. With opponents calling % on the y axis, and % of the time we do this with a non made hand on the x axis, I think the graph would look eerily similar to the VPIP one.




Again, the reason for the staircase is because opponent isn’t realizing you are playing back until their “aha” moment. Every “aha” moment they have increase their calling range, and inbetween each “aha” moment their calling their calling range only goes up a tiny bit.

The significance is the exact same as the VPIP example. We are trying to find the point on the graph with the most marginal value, the vertical part of the staircase. Not being on the vertical parts of the staircase loses us a lot of value.

See the significance now?

If you still don't get it think of it as a gap concept. But that's really all i can do right now.

If we come to agreement on this there's a lot more we can discuss.

Really this has nothing to do with economics I realized. When I first came up with the idea I thought it did but clearly thats not the case.

Also, I'm sorry about the first essay, its so clumpy everything i said could've probably been said in 1/5th the space. I'll redo it sometime soon.

Originally Posted by swiggidy

Originally Posted by zenbitz

ISF - the concept is good but the graph is confusing. Maybe not to you because it maps to something in your econ class, but if I ever knew the mathmatical definition of "marginal value" I forgot it.

Marginal is a fancy word for the rate of change (the derivitave, aka slope).
Meh, marginal value wasn't really correct. I used the term too much, the only place marginal revenue comes into play is that we want the most marginal revenue and therefore the biggest slope (ala vertical slopes of graph)

Originally Posted by zenbitz

Your y axis is "chance of opponent paying you off"???

Yes, your graph is way off. You are only considering the increased chance of a big hand being paid off without considering the value lost by playing the trash hands.

What you actually want is a graph that shows how much more you will make vs $VPIP. There is no way total return (let alone marginal return) is increasing from 40 -> 50. If it was then Laggs would be running 50/45 instead of 25/20.

Also your stair step is absurd. You're trying to imply that adding 10 hands has little positive return. Then BAM!!! the 11th hand adds a ton of value?

My Y axis should actually be "amount of hands opponent takes to SD versus you. " I know it's way too specific but I'm attempting to isolate the variables.

Your right, it is absurd to believe that on the 11th hand everything would change. I didn't mean to have it come off that way, which is why i said ignore the scale. I'm suggesting that there is a small range of VPIP where an opponent tags you as a certain way, in which he pays you off with a certain range of hands. Because of this tag change from tight to loose, the hand range he pays you off with increases in size. But then after that aha moment, your opponent has very little change in the hand range they pay you off with, until the next aha moment. Which is why the graph would be a staircase.

I'm sorry about the sloppiness, I didn't read it over one bit.

I understand what you are saying and sort of agree, but thing you are dramatizing it a ton. You are saying that we often put people into groups where we play against members of a group similarly and sometimes this group covers a wider range of VPIPs than it should. However, I dont know how accurate this is in that explicit sense.

When putting people into "groups" we look at way more than VPIP, maybe I missed this part, but what about PFR? There are HUGE differences between a 25/2 and a 25/22, and Im sure you know this so maybe I just didnt get where you accounted for this. We are also forgetting postflop tendencies with cbets [make them, fold to them, etc.], showdown stats, aggression, etc etc etc. When you analyze all of these things and put people into a "group" we see that we arent really making wide groups of people to play similarly against, instead we are pretty much analyzing the situation depending on each individual opponent.

The only time where we almost solely put people into groups is when somebody replies to a HH where little information is provided. In an actual game, however, Im not sure how much this applies.

Also, how are we supposed to figure out where our opponents "aha" moment [I dont really like that term, why not just like, the height of their stairs or something] are?

I personally think this is a pretty basic concept in general and has been repeated much more simply in this thread; "As you play more hands, you will begin to get paid off more; however, you will also begin playing weaker hands which will offset the times you get paid off and eventually have your profit decrease." Yea, thats great and all, but what does this tell us? I dont think it is realistic to find that special spot on the "staircase" [or that the staircase even exists as described] and even if we theoretically could, that spot is different for every opponent and you cant exactly change your VPIP for each opponent.

I guess what Im saying is that, yea, its great to see people trying to make theory posts here, but this just seems like we are forcing a theoretical discussion about something very intuitive that doesnt have a lot of value outside of that.

I don't think the staircase idea is accurate. I just don't think the average player has these "aha!" moments your describing with there being a discrete VPIP where they will pay you off considerably more than if your VPIP was outside of that narrow range. You're also taking a player's opening action and using it to depict the end result. You're ignoring every subsequent action between them entering the pot and them going to showdown. You're assuming that your opponent ignores every action you make on the flop, turn and river if applicable and only bases their decision to pay your off based upon your VPIP. That is beyond retarded. Your theory might have some merit if you were discussing only hands that are AI pre-flop. The percentage of hands where this happens is very small though and even then there is a lot more than goes into a competent opponents decision than your VPIP.

I think the vpip example is really throwing people because it is so flawed. The second example as far as i know isnt flawed so try focusing on that.
I think in terms of estimating the staircase its not as impossible as you may think.
The place where i disagree is this being a basic concept, its based on the basic concept you point out but its an add on to it. Most people believe poker relationships like these are linear. If they are a staircase shape it means a lot

My interpretation of what your saying is this:

In every situation you have a range. The most simple is preflop. If you are a 2/1 and you raise UTG we can be pretty sure that you have AA or KK. So what people do is widen their ranges so they can get paid off more. However, the amount someone gets paid off goes in a staircase. If you become a 4/2 instead of a 2/1, it isn't going to make that much more profit when you raise UTG. But lets say you play 16/13. Now you have a wider raising range from UTG, and you will get paid off a lot more when you raise UTG.

This applies everywhere, since every action you do has a range. So pretty much, you want your ranges to be on a "beginning of a step." But it seems like determining that is very very hard, and just comes from experience or luck.

An interesting thing that comes to mind when thinking about this is making a weighted range, where you only do something with a certain hand a certain percentage of the time. For example, something i do is raise no gap sc's utg and in MP 40-50% of the time, so i can get more action from my early position raises.

New conclusion:

This makes me think that the beginning of a step isn't a rigid range, but a range that has a few oddball things mixed in. For example (very simple), a rigid range for a c/r on a flop might be a set, two pair, or a good draw. If you wanted to expand that range, you could add weaker draws to that range. But it seems to me now that the best way to expand your range is to throw in bottom pair, mid pair, top pair, an inside straight draw, etc. 10%-15% of the time.

Originally Posted by Massimo

My interpretation of what your saying is this:

In every situation you have a range. The most simple is preflop. If you are a 2/1 and you raise UTG we can be pretty sure that you have AA or KK. So what people do is widen their ranges so they can get paid off more. However, the amount someone gets paid off goes in a staircase. If you become a 4/2 instead of a 2/1, it isn't going to make that much more profit when you raise UTG. But lets say you play 16/13. Now you have a wider raising range from UTG, and you will get paid off a lot more when you raise UTG.

Yes, i think you have it. But just making sure i mean there are certain "steps", such as when ur stats are 16/12 and 22/18, where calling ranges increase much more significantly.

I don't get the second part of what ur saying though.

Originally Posted by IowaSkinsFan

I don't get the second part of what ur saying though.

I'm talking about what the theoretical peak points are, or in your analogy, the beginning of a step or the point where the graph goes from very vertical to very horizontal. We don't know what ranges correspond to those points (and by "those points", i mean the beginning of a step), but i realized that the ranges that corresponds to those points are probably not an evenly weighted range but a range where there is some oddball crap mixed in.

So let me rephrase my example. Lets say that you have a c/r range on a certain type of flop that includes all sets, all two pairs, and all good draws. But you feel that to maximize your EV, to move closer to that peak point (beginning of a step), you need to expand that range. There are many ways to do this, one of which is to add weaker draws into your range. But it seems to me like that best way to expand would be to raise top pair, mid pair, bottom pair, or a very bad draw 10% of the time when you raise in that situation, because then your opponents will be confused about your range and that will lead to mistakes.

Oh I see, interesting points.
However that's kinda tangential to the discussion, I'm not sure everyone agrees with staircase theory yet.
The post you wrote isnt staircase specific either, it's actually a good post on expanding your range in general.
But can you go more indepth on exactly why top pair and midpair is good? Obviously this is only applicable to very good opponents no?