Final Table Bubble Nittery

Let's say you're on the final table bubble with KJ on the cutoff. It folds to you and you have 10 big blinds. In most situations, it should obviously be a shove. But you have a feeling you'll get called with nearly 20% of hands. A range of 77+,A2s+,KTs+,QTs+,JTs,A7o+,K9o+,QJo vs KJo is going to be behind with 42.665% to win. Foretunately, you have fold equity still. If 3 opponents fold 80% of the time each, that's about a .80^3=~51% chance they all fold.

However, on the bubble there are bubble factors you're supposed to adjust for. The chips you lose are supposed to be worth about 1.6 times more than the chips you win because losing chips threatens you being unable to fold up the money and make considerably more.

So when you lose, you aren't losing 10, you're losing 16.
If you get folds 51% of the time and pick up 2.25 (EV of 2.25*.51=1.1475), and you get called and win 12.25 .49*.42665 or expected value of 12.25*.49*.42665=~2.56 and called and lose 16 for EV of -16*.49*.57335=~-4.5, the total sum is 1.15+2.56-4.5=-.79. So the individual result of shoving with KJo is actually losing money. While technically acceptable to shove with some losing hands to force opponent to widen call off range and collectively make more shoving WITH KJo than without it, I think it's a fair assumption that opponents are highly unlikely to adjust very much to you.
Also, opponents probably SHOULD be calling you much tighter than this due to bubble factors themselves, but if you think they probably won't, then shoving becomes incorrect, because their decision to call with hands like A7 and QJ ends up hurting BOTH of you.

A standard "push/fold" chart that doesn't account bubble factors will say to push with
22+ A2s+ A2o+ K6s+ KTo+ Q8s+ QTo+ J8s+ JTo T7s+ 97s+ 87s 76s

But I believe pushing this wide on the bubble is almost certainly a mistake.

Nevertheless, there is a very strange concept with regard to tournaments. If everyone is pushing with wider and calling with wieder, the probability of you being able to continue without having to withstand multiple all ins is low. If you preserve a raise/fold range you can either play very tightly and call off appropriately according to the bubble factors and rarely have an opportunity to steal. This is because the assumption is that "all skill is equal" and thus, all players will play the same way. This isn't the case, so what are the implications? The implications are now that either you have to wait for the best spot you can afford to wait for, or you have to try to play much looser before you have 10BBs.

If tight play forces you to go into this ultra tight blind yourself to death strategy or close to it, it would seem that it's of a huge advantage to eliminate the skill factor of short stacks and put them at the mercy of their cards by raising more widely, particularly in early position than any "chart" might suggest.

There's a reason Phil Helmuth has so many bracelets, so you have to at least entertain some of the ways he plays. When he has 8 big blinds, he will still minraise/fold. That is on much slower structures, mind you, but nevertheless, I think the idea of raise/folding down to 9 big blinds isn't all that bad if
1)You are much more likely to have a monster
2)Opponents are unlikely to really adjust their call range all that much between a minraise and a shove
3)You are close to the final table and thus survival is important, but the difference in chips is not nearly as important as the difference between living to see another hand... A double up will make you so much more likely to final table, and a steal will buy you a lot more hands that will make a difference in how other short stacks are now forced to make a move (and they probably are more likely to move in and bust than you). Consider buying 9 more hands when you win, and pretty much having roughly the sam amount of fold equity if you fold and have 6 big blinds or under 4 if you pay big and small blind and
4)You can raise without having the pot odds adjusted for the bubble factor to call. This is tricky. I tend to have a very polarized range. Either I am getting desperate and will raise with any two from UTG or UTG+1 but will do so infrequently enough that there's a real good chance I have a very strong hand, or I have the very strong hand that I've been waiting for. For example, in 3 rotationsI might try this once with air. If there's 7 opponents that's 21 hands. In 20 hands you have about a 50% chance to get a hand in the TT+, AQs+ range (not including AKo). In 30 hands, JJ+, Aks. Since you're also going to be raising with some hands like 88,99 and widening that range in later position when it folds to you, there's a more likely chance you have a premium hand or strong enough hand to call than a poor hand.

Let's look at the math.There are many different ways to dissect this problem depending on how you view bubble factors and whether or not you want to include your equity from raising preflop. I want to just isolate making the raise from the decision once you are raised.

So your risk is worth 1.6 times more than you gain. You are not risking 6 to win 8+2.25+ your 2 back or risking 6 to win 12.25. Instead you are risking 9.6 to win 12.25. With 1.276 to 1 odds adjusted for bubble factors, you are going to have to be 44% to win. With a hand as bad as 23o, you most certainly are not, so there is an argument to folding. You have to then argue why raise at all? Well, if you only raise about twice every 21 hands with a polarized range of either "any two" or TT+,AKo,AQs+  your opponent is pretty much going to have to all in or fold. So he either risks 8 and is against a very tight range where he is behind, or he picks up 4.25. He is risking 8 (and then some because others could wake up with a hand) to win 4.25. If you adjust his decision for bubble factors, he is risking 12.8 to pick up 4.25. But of course when he shoves and gets called he can still win. So if your opponent gets called 60% of the time that he shoves, being 35% to win when he gets called isn't enough. He needs to be 35.7% vs your range of TT+,AQs+,AKo. If we want to get really accurate, we can say there's a 58.2347% chance that you get dealt TT+,AQs+,AKo+ in 21 hands. If you don't raise in 21 hands you will have to raise with whatever you can come up with. So if you factor that in, he only needs to be 34.25%. I actually think his chances are much worse since you probably will widen range when folded to in later position, so the probability of you not raising within 21 hands is probably much lower.

In theory, he should require the exact same range as you to be able to move in on your raising range, and probably tighter because he has to concern himself with other opponents.

So if there are 6 opponents left to act and they need TT+,AQs+,AKo, what's the chances that you get reraised? In reality it SHOULD be only 22% . Slightly more because opponents that have you covered can call off lighter than you as bubble factors are smaller to them than they are to you if they have you covered in chips.

Raising "any two" vs these opponents should be profitable since 77.9% of the time you pick up 2.25 and 22% you lose 2*1.6 or 3.2. That means you can pick up 1.046 big blinds every 21 hands. Hopefully you get a walk, or find another spot, or also at some point find a spot to get your money in good by opponent that makes a mistake, but losing chips at less than 2 big blinds per rotation on average when you are on the final table bubble is not all that bad. You are losing about .26444 per rotation which means if blinds stay the same, you can survive in this manor an average of 45.37 hands if you have 12 big blinds. Obviously blinds go up and as you become shorter, decisions become more desperate. But even so, everyone else will grow more desperate, and you are still giving yourself a good chance of making the final table and make a lot of money from moving up the money.

You can look at this "hands left" and adjust for the blinds going up by calculating your "burn rate" based upon your strategy at the given level and adjust as you become more desperate. I would say when you move to all in or fold mode to just calculate being on autofold from there.
With 45 hands left or even 35 hands left you can afford to wait for JJ,QQ, KK or AA...

If you can find an equivalent spot where your overall expected value is as good as getting your money in with jacks you should go for it. If it's less, you may have to reconsider even if you have JJ+ if the supernit shoves.

But since there's added value in raising, or having fold equity PLUS being ahead when called, you can widen the range, and adjust for spots. However, what you cannot do is try to justify waiting for any better than this.

3Betting Vs a Static Opponent Given Certain Bubble Factors

In poker, aggression is important, but it's also important not to take things too far since you can never get 100% of the prize pool and you are simply helping your opponents make money if you are always too aggressive. Tournaments have bubble factors that should adjust your decisions as we've shown. We've even shown a strategy for unexploitable opening ranges with 20 and 30 big blinds at different bubble factors and vs different 3bet ranges.

But what about your aggression? What about when someone raises 2.2 BBs and you are trying to decide whether or not to jam? How should bubble factors affect your jam range given certain assumptions about your opponents?

Let's say opponent has a "static" or unchanging strategy. He does not adjust at all to bubble factors, and he is predictable in always open raising when it folds to him with 10% of hands regardless of position. He will call 3bet shoves with 99+,AK,AQs (calls with 42.2% of his range). If these assumptions are correct, how often can we Jam vs this player from the button if both the blinds are on autofold?

First we determine how often opponent calls a shove. This is 4.2232/10 since he calls with  4.22% of hands out of 10% or 42.23%. We then run the math assuming we gain 4.5BBs when he folds, and when called we win 22.25BBs when we're called AND win, and lose 20 when we lose. We then multiply that total expected loss by the bubble factor since chips we stand to lose are not equal to the chips we gain. We can adjust until we find the break even point. Based upon these percentages we can find which hands a shove is break even with or better. The real optimal solution may have some flat calls in there, and thus might shove with more premium hands and a small percentage of bluffs and just call with the remainder, but it's not a huge mistake to ignore these and jam or fold.

Jam with:
Bubble Factor of 0.8 jam such that you are 25.74% to win when called:
22+,A2o+,KTo+,QTo+,JTo,A2s+,K2s+,Q2s+,J3s+,T6s+,97s+,86s+,75s+,64s+,53s+
Bubble Factor of 1.0 jam such that you are 32.77% to win when called
TT+,AQo+,AJs+,KJs+,QJs
Bubble Factor 1.2 Jam such that you are 38.58% to win when called
TT+,AKo,AKs
Bubble Factor 1.4 Jam such that you are 43.47% to win when called
JJ+,AKs
Bubble Factor 1.6 Jam such that you are 47.64%  to win when called
JJ+

Charted neatly below to eliminate the wording
bubble factor|% to win| hand range:
0.8 25.74% 22+,A2o+,KTo+,QTo+,JTo,A2s+,K2s+,Q2s+,J3s+,T6s+,97s+,86s+,75s+,64s+,53s+
1.0 32.77% TT+,AQo+,AJs+,KJs+,QJs
1.2 38.58% TT+,AKo,AKs
1.4 43.47% JJ+,AKs
1.6 47.64% JJ+

As our opponent folds more than 57.8% of his hands we can push with a lower percentage to win vs whatever calling range he calls off with.

With a larger number of big blinds like 30, you need to Jam with a tighter range of hands since you are giving your opponent a better payout when he calls, and gaining less proportionally to what you lose when you are called and behind.

Obviously a tight player isn't the ideal target to 3bet, but if you have a decent hand you can certainly go for it.

What about other player types? Although I haven't gone through to do the work to define the exact hands, I have came up with the percentages of a few scenarios.

20BB 3 betting over
% to win needed when called if opponent opens with

35% of hands and only calls with 99+,AQ+,AJs,KQs (calls 13% of the time he opens)
bubble factors
0.8  0% any two
1.0  0% any two
1.2  6.55% any two
1.4  13.99% any two
1.6  20.34% any two

30% of hands and only calls with 88+,AJ+,KQ (calls 20.61% of the time he opens)
bubble factors
0.8  0% any two
1.0  6.32% any two
1.2  14.42% any two
1.4  21.23% any two
1.6  27.04%

opens 25% hands and calls with 88+,AJ+,KQ (calls 24.7% of the time he opens)
bubble factors
0.8  any two
1.0  any two
1.2  22.29% any two
1.4  28.47%
1.6  33.75%

Opens 15% of hands and calls with 55+,AT+,KQ (calls 70% of the time he opens)
0.8 36.88%
1.0 42.86%
1.2 47.80%
1.4 51.95%
1.6 55.50%

Opens 20% and calls with 55+,AJ,KQs (calls with 42.2% of the time he opens)
0.8 25.74%
1.0 32.77%
1.2 38.58%
1.4 43.47%
1.6 47.64%

Opens 10% and calls with 99+,AK,AQs (calls with 42.2% of the time he opens)
Jam with
0.8 25.74% 22+,A2o+,KTo+,QTo+,JTo,A2s+,K2s+,Q2s+,J3s+,T6s+,97s+,86s+,75s+,64s+,53s+
1.0 32.77% TT+,AQo+,AJs+,KJs+,QJs
1.2 38.58% TT+,AKo,AKs
1.4 43.47% JJ+,AKs
1.6 47.64% JJ+

It's important to notice that opponents SHOULD call off much tighter as bubble factors increase, but if they don't, you'll have to shove much tighter yourself.

Against an opponent who calls off with the SAME range of hands and raises with the SAME range of hands, you are going to have to tighten up dramatically as bubble factor increases both in your opening range, your call off range and your 3bet range.

However, what tends to happen is opponents might try to exploit by opening just as wide near the bubble, but fold more tightly because they are aware of bubble factors or give too much respect to the 3bet believing that an opponent wouldn't 3bet shove without a huge hand. Or they raise without anticipating the possible reraise and then once it comes they start worrying about the money so they tighten up.  So I believe there are certain types of opponents where your range won't change that much, and other types in which it will actually WIDEN because of how they play, but for now until I run the math to prove it, that remains speculation.

What I'm very interested in seeing is if an opponent opens up with say 20% of hands and always assumes a certain 3bet range and actually adjusts his calling off range for the bubble, but not his opening range. Since his calling off range will tigthen up as the bubble factor increases, it becomes correct to raise more, if not for the fact that our bubble factor also increases. There is definitely a point in which WIDENING your 3bet range as you get closer to the bubble is correct to certain opponents, but I will be interested to see how great of an adjustments to call off range opponents have to make for that to be the case.

In the future I hope to work some of those numbers.

How To Avoid All Ins To Crush the Early and Middle Stages Of a Tournament

Summary:
Given our assumptions on blind structure and hands per hour, with 40% chips coming from preflop raises plus postflop action, 40% of chips coming from 3bets plus postflop action, and remaining 20% of chips coming from: blind defends, dead money grabs, squeeze plays, blind defends, walks, flat calls, etc. we can theoretically sustain our chipstack for an entire tournament without ever being all in with a preflop raise on 16.16% of hands dealt and a 3bet on 6.5% of hands. Variance and the assumptions not always being consistent with reality will force us to call off most tournaments in reality, but it's a good starting point to understand the value of applying consistent pressure.
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The interesting aspect of tournaments, is that if you have a high enough win rate at low enough variance, you can in theory avoid ever being all in. At a minimum, if you maintain a pretty high winrate at relatively low variance, you can at least prolong the all in for a considerable amount of time... This can result in an enormous boost to your chances of doing well in a tournament. It's not uncommon for me to triple up my starting stack before I'm all in, and at slower structures I could probably go much longer, but I have a tendency to make a very infrequent mistakes that are very, very large in magnitude, as it can be difficult for me to focus for more than 5 hours.

In a few tournaments I have analyzed, you can avoid an all in if you can steal 3 blinds AND antes successfully per rotation. You can get very close with 2 steals and in slower structures, 2 steals may be enough if the speed of play is fast enough (around 80-120 hands per hour rather than 60-80).

The WSOP And Other Big Events are the Exception, not the rule

You should adapt to structure, but the WSOP main event is obviously the exception to the rule. This level has 2 hours before blinds go up and you start with 300 big blinds and the blinds rise at a rate of about 0.40% per hand on average after the first couple levels of the antes. This tournament is such a slow structure that you can for a very long time till have an enormous chipstack relative to the blinds if you just maintain your chips or even if you lose your chips gradually over time. You may not even need to maintain the number of big blinds to get deep, you can just play a few big hands and get a few decent pots every rare once every so often. If you autofold I estimate that you will see 440 hands before you exit the tournament. If you steal one per rotation to maintain your chips until the antes gets involve, then steal 1.3 times per rotation successfully after you get down to 25 big blinds, you will hover above 20 big blinds for the entire tournament until you win if variance doesn't get you. In a tournament like this, coolers and high variance gets in the way and it's sometimes a liability to have good cards in some ways since it keeps you in the hand in an inflated pot that you would prefer to just take down or fold.

If good hands are such a liability, why not just fold if the pot gets too big? Because as much as you'd love to get through an entire tournament without ever being all in, eventually you will. Plus if you don't occasionally show them a big hand your raises may not get as much respect after awhile.

Certainly an average gain equal to 2 steals per rotation plus a few good hands and the very occasional spot where you have a premium hand AND an opportunity to double up is usually enough to have a great ROI. You can still dramatically reduce your chances of elimination, which will translate into a huge win percentage.

Requiring 2 steals per rotation does not mean you should raise 2 times per rotation because your attempts aren't 100% successful. So how many blind steals should you attempt? That really depends on the probability of success.

If you successfully get everyone to fold 80% of the time, expected value of 4.5 big blinds requires that you steal 2.8846 times per rotation or raising about 32% of the time total.

If you get through 70% of the time you need to come up with a steal attempt 46% of the time, and if it works 60% of the time, you need to come up with a steal attempt a whopping 80.65% of the time.

But hold on, This is assuming that your failed steal atempts always lose. If you are called, you will be forced to muck and never flop a good enough hand to stay, and if you are raised, you will never have a good enough hand to stay.

However, I would say that if you KNEW for a fact that you could steal the blinds his frequently, you probably SHOULD never play a big pot, because you can win without conflict. Why take a 30% chance of elimination if you literally never need to. But you never will have that knowledge with certainty, so even if you are 80% confident, you still must have some threshold since there is a chance you are overconfident, and there is still the 20% of the time that you factored in that things won't go your way.

Obviously if opponents recognized this strategy, they'd reraise more actively, your success would decline and it wouldn't be worth folding, and most importantly, you never KNOW that the conditions will remain. In fact, they probably won't... As such, the idea that you can play that perfect tournament 100% of the time and never be all in is probably nothing more than a deluded fantasy. Nevertheless, it's one in which we still should determine correct play IF it were true, so we can adjust from there.

Often times you may get knocked out after getting short and forced to take a less premium hand and regret not calling off with the premium hand earlier. That doesn't mean you shouldn't be aware of the strategy and try to adjust strategically based upon how close or far away from reality you think this is.

It certainly is a worthy calculation to determine how frequently you need to come up with chips to avoid all ins. Since you are all ins, accepting a lower win rate is perfectly acceptable, but not too much lower.

So you certainly should plan on having some threshold where you will fight back if you flop good enough, you certainly will see some rivers, and you certainly will pick up aces and it will be worth playing to a reraise...

One side of the argument is that you will be 3bet, you will be called and faced with calls more often, and opponents will raise ahead of you enough that you couldn't possibly steal that many blinds if you raised when folded to 100% of the time. The other side is that when called or raised, you have more equity than is calculated. You can still maintain relatively low varience chip accumulation with high percentage Cbets, occasional 4bets when facing 3bets, occasionally calling 3bets, occasionally 3betting yourself (plus a Cbet), and occasionally you will see showdowns and have a chance to win, plus implications of the turn and river.

So let's imagine instead you are called 80% of the time instead of 20%, but your Cbet succeeds 50% of the time. Let's imagine there is no raising by your opponent, and that when your Cbet fails you still have 25% equity in the hand. Let's also assume that if your Cbet is called that the turn and river are always checked.

In this scenario, you need to come up with a steal ATTEMPT plus a Cbet attempt about 40.4% of all hands dealt to be able to sustain your umber of big blinds in the ante stages.

But what if only 40% of your chips came from steal+Cbets and another 40% came from 3bets and 20% came from a mixture of walks, flat calling in position and big and small blind defenses and maybe a few turn and river takedowns and showdown wins? That certainly would require us to steal less often.

So let's first try to factor in your 3bets. In a 3bet you are going to be risking 2.5 times the standard raise of we'll say 2.25xBBs for a total of 5.625BBs to win 4.5xBBs when opponent folds. The intention is not to always get a fold, but set up a profitable Cbet in an inflated pot and taking a pot where you have position throughout the hand.

So we will assume your 3bet works more effectively and that you do this with fewer hands, and as such your Cbet is more effective. We will say that you win the hand 25% of the time when your Cbet fials still. If you were to only 3bet/Cbet because table never folds to you, you only need to succeed 16.22% of the time. But there's a HUGE caveat in that you are personally putting over 12 big blinds into the play when you Cbet and in reality probably more. The strategy can no longer REALLY be said to be all that low varience, so there is a definate tradeoff that occurs. Nevertheless, a preflop raise plus a Cbet risks as much as 5.6  big blinds. So if you have 30 big blinds that's still 18.67% of your stack.

The equivalent of risking potentially 18.67% of your stack means that if you 3bet/Cbet at 66 big blinds, it is about the same as a raise+Cbet is at 30 big blinds. At 20 big blinds a PFR+Cbet is about the same as a 3bet/Cbet is at 44 big blinds. Since the 3bet can be done less frequently with a stronger hand and is thus more successful and has with it a slightly bigger edge, in terms of risk/reward you can actually say that it's more preferable to make. As such, it may be about the same risk/reward profile at 30-50 big blinds as a PFR/Cbet is at 20-30BBs.

Since the result of not accumulating chips early on in the ante stages will inevitably result in fewer chips which will force you to raise more liberally anyways, it may make sense to 3bet in the 30-50BBs range before you are forced into a position to take on that amount of risk, and potentially more. You never know if your opponents will fold to you often enough to keep up with big blinds.

So if we get 40% of the 4.5 per rotation we need from PFR+Cbets, 40% from 3bets and 20% from "other", how frequently should we make each play?

Well, we need 1.8 per rotation from each since 40% of 4.5 is 1.8. Based upon some assumptions we made, the EV per PFR is 1.2375, the EV per 3bet/Cbet is  3.0825 per attempt.
1.8/1.2375 means that we need 1.4545 times more than the preflop raise gets per rotation. Or 1.4545 raises per 9 hands or 1.4545/9=~16.16% of hands dealt should be raises. 1.8/3.0825=.583942 means we need only ~58.4% of the EV we get from a steal or we can 3bet 58.4% of all rotations or about .583942/9=~.06488% of all hands dealt should be 3bets.

Since not every hand we get will both be strong enough to 3bet AND not be 4bet before it gets to us we should be willing to raise more than 6.9% of all hands on average, and we should adjust this based upon the position and hand range of the initial raiser. An early position raise indicates more strength, late position raise indicates less such that given optimal opponents we should still 3bet a tighter range vs an early position raiser and a looser range vs a late position raiser.

Also, since it will not fold to us every time we have a raising hand that isn't strong enough to 3bet, we need to widen our preflop range and adjust for position as well.

Overall though, coming up with a preflop raise on 16.16% of hands dealt and a 3bet on 6.5% of hands dealt is definitely attainable. Picking up chips from isolation raise+Cbets or dead money grabs after multiple limbpers and blind defends and flat calls in position of course has to also be done enough and profitably to meet the approximate quota as well.

Additionally the success rate and expected value of each move will change, and the varience caused by a 3bet or large isolation raise may make it less desirable to make as frequently as implied.

So in reality you might 3bet far more frequently when you have over 40 big blinds and raise less often, particularly at a more active table and particular as specific spots come up than implied, but you may make up for it later by not 3betting frequently at all around 20-30 big blinds and then once again you may shift when a 3bet is all in and a PFR gives your opponent the perfect shoving stack such that you have to tighten up your range once again.

Additionally, who knows if your raise will succeed 20% of the time or 80% of the time at a given table, and possibly your Cbet will succeed much more often, or much less often than 50%. Certainly it's possible to set up a spreadsheet or more of these calculations to try to get a better picture of individual situations that favor either one or the other, or where the steal attempt is less profitable. It may also be more accurate to adjust the EV as you increase the frequency of hands since our showdown value and table image will be different and the response should be different.

But ultimately this post is just to get you to understand and respect the fact that if you are profitable enough with just the right amount of activity in the right way at a table, you can in theory dramatically reduce your chances of having to be all in. Certainly if you are more active you can accumulate more chips and as such when you finally get it all in your stealing attempts are so profitable that you have everyone outchipped and you can still recover and accumulate until the next all in you have everyone out chipped and so the next all in you can survive until eventually you win one of those. Or you could get it all in a few times and be much more patient and steal much less often.

The more capable you are of maintaining your big blinds or even staying ahead of them, the more willing you can be to make big laydowns in big pots when you know you have the best of it, when you should fold when you have great pot odds because the pot is inflated to the point where you don't need that varience to advance deeply and when you can continue to apply pressure and at times not care about your two cards as much as the situation.

In reality though, there is ALWAYS varience, and even low varience can force you to take on higher varience or play less active in your 3bets+Cbets or less active in the hands you make a preflop raise with.

Are we optimistic?
Accumulating chips at a high rate has to be done to win a tournament. Two steals per rotation is 2/9 hands which is 22/100 and these aren't big blinds but "M". That translates into 33.33bbs/100 before the antes and 50bbs/100 afterwards. Even though people underplaying the ante stages and overvaluing their survival which allows for consistent accumulation, it PROBABLY isn't enough to maintain that win rate over any very long term period of time with low varience...

That doesn't mean that attempting to do so won't frequently produce enough accumulation plus adding in some all ins with an edge that can give you a pretty darn good shot overall at making a deep run and giving yourself a chance to win.

Unexploitable Bubble Factor Call off for 20 and 30 big blinds

Based upon the pot odds you are getting when facing a shove for your tournament life, the hands you call off with will depend upon your "bubble factor" in a tournament. Bubble factor is basically the multiple of the pot odds you need to call off your tournament life assuming equal skill and a bunch of other assumption the ICM makes. The ICM usually will calculate an ICM close to 1 near tournament starts and it drops somewhat close to one following the money bubble burst when you are far from the final table. It rises to around 1.6 near money bubble and again just above 1.6 at final table bubble, or closer to 2.0 if the money bubble and final table bubble are the same in a top heavy payout.

Utility theory and opportunity cost theory may attempt to adjust these factors based upon the assumptions of there being skill involved. Both try in their own way to determine how to adjust those bubble factors given the skill.

The Utility Theorists believe the skill you have is proportional to the number of chips you have. A larger number of chips creates higher bubble factors for opponents who face you, and thus more fold equity and a greater probability of accumulating more chips without as great of risk on your tournament life, and greater probability of forcing other people to risk their tournament life.

The opportunity cost theorists believe that players do not respect ICM and thus will knock each other out sooner as well as provide much more "future value" which means survival is more important than expected value. An all in not only risks your tournament life which is more valuable if you do have skill, but that it also costs you the opportunity to see more profitable spots in the future plus all the hands you would have been dealt if you were not eliminated. If you still have plenty of room to accumulate chips, all in becomes unnecessary, particularly since it's possible you may not have to be all in to win a tournament.

Also, as blinds and antes rise proportional to everyone's chips, not only do you stand a better chance of getting called, possibly getting called twice and tripling up.... You also have the additional fold equity from someone raising another player out of the pot and the ability to wait for a dominant hand. Youalso build up a tight image that becomes relevant if/when you win a double up that increases your chances of additional blind steals when blinds are much, much higher than whatever they were when you passed up an edge. It also increases your probability of survival deep, and requires fewer steals or more hands before you have to double up before you can get back to where you were before you started blinding off.

I believe both have some merit. Between 15-30 big blinds there's an argument for taking slightly more risks than the ICM suggest to gain utility and make up for the edge you may give up by stolen pots in the future. There's also the argument for taking some risks vs some good players and loose aggressive players to your immediate left that otherwise might hinder your ability to accumulate chips.

Between 5-25 there's the argument that because of the antes rising, and blinds rising, you can win around 20% more on a blind steal a level later which allows for a tighter strategy to still be profitable, but also that ensuring your survival  and increasing the probability of a double up and triple up when you do get that hand.

Because of theses bubble factors dictating a different "call off" range, it should change the opening range if you want to avoid being exploitable to a raise.

So the first step is determining a few guidelines for the "call off" percentages. These can be determined by a formula that relates the pot odds and bubble factors to the win percentages. Once we have these, we can assume that opponents 3bet such that you have a 50% chance of being 3bet by one opponent. We then need to calculate which hands have the equity to call off. When we have the "call off" range we can extrapolate the raise percentage.

The raise percentage must not only be unexploitable, but also compensate for bubble factors in which the chips we lose or may lose are worth more than the chips we gain.

The assumption is that when opponents are faced with a raise where the iniaial raiser had this many players left to act, they 3bet with the following:

8 8.3% 88+, AJ+,ATs+,KTs+,QJs
7 9.43% 88+, AJ+, KQ, A9s+,KTs+,QJs,JTs
6 10.91% 66+, AJ+, KQ, A9s+,KTs+,QTs+,JTs,T9s
5 12.95% 66+, AJ+, KQ, A9s+,K9s+,Q9s+,J9s+,T8s+,98s,87s,76s,65s,54s
4 15.91% 66+, AJ+, KQ,QJ,JT, A8s+,KTs+,QTs+,J9s+,T8s+,97s+,87s,76s,65s,54s
3 20.63% 55+, AT+,KT+,QT+,JT,A7s+,KTs+,Q9s+,J9s+,T8s+,97s+,87s,76s,65s,54s
2 29.29% 22+, A9+,KT+,QT+,JT,T9,98,87,76,65,A2s+,K9s+,Q9s+,J9s+,T8s+,97s+,87s,76s,65s,54s
1 50% 22+,A2+,K4+, Q8+, J9+,T8+,98,A2s+,K2s+,Q2s+,J5s+,T6s+,96s+,85s+,75s,64s+,54s

I only ran the calculations for 8 players, 5 players, and 2 players since I think having optimal call off vs 8.3% 3bet range, 12.95% range and 29.3% range is enough to get an idea. I may also add in a tighter range like top 4% later just because tighter 3bets are certainly common, especially with regards to shoves.

Keep in mind this is just as a means to calculate our opening strategy though so we aren't easily exploitable relative to our call off. I believe our call off range should actually be tighter because they 3bet less often than they should and because they are exploitable enough with raises that exploiting with calling 3bets is less necessary.

So with 30 BBs and a 0.8 bubble factor we need to be 36.69% to win (raise 2.25x BBs with antes)
So with 30 BBs and a 0.8 bubble factor we need to be 39.15% to win
So with 20 BBs and a 1.0 bubble factor we need to be 42.01% to win
So with 30 BBs and a 1.0 bubble factor we need to be 44.58% to win
So with 20 BBs and a 1.2 bubble factor we need to be 46.51% to win
So with 30 BBs and a 1.2 bubble factor we need to be 49.11% to win
So with 20 BBs and a 1.4 bubble factor we need to be 50.35% to win
So with 30 BBs and a 1.4 bubble factor we need to be 52.97% to win
So with 20 BBs and a 1.6 bubble factor we need to be 53.69% to win
So with 30 BBs and a 1.6 bubble factor we need to be 56.27% to win

So for example, if we raise UTG with 30 BBs and are shoved on and call with QQ+ and assume our opponents have a range of 88+,AJ+,ATs+,KTs+,QJs, then we are collectively are 72.568% to win vs that range. So 72.568% of the time we win say 32.5 and the remaining 27.432% we lose 48 (adjusting for bubble factors of 1.6 the chips we lose are worth 1.6 times more). Overall we can multiply .72568*32.5 plus .27432*-48. This is about 10.4.

What that means is that we can raise and fold X times and break even without being exploited. The X is what we have to solve for. Keep in mind that when we are raising, although we are losing 2.25 BBs when reraised, we actually are losing 1.6x more accounting for bubble factors. So we are risking in effect 3.6BBs on our raises and when we call, we gain an average of 10.40. So this means we can raise fold with 10.4/3.6=~2.89 times and call on the next one. Or we need to have QQ+ 1/3.89 times that we raise in this spot. Or we can raise 3.89 times more than we are willing to call off with, or with about 5.28% of hands

That translates into an opening hand range UTG 9 handed on the stone cold bubble with 30 big blinds of about 5.3% (roughly 88+,AQ). The actual call off range should try to account for the opponent because we need to be 56.27% to win a 30BB shove, but only 53.69% vs a 20BB shove. If a shorter stack shoves the bubble factors might even change and also, the opponent's hand range are most likely tighter than "cash game optimal" and a raise is probably profitable all of which means we should play tighter when calling off (and perhaps slightly looser when raising).

Nevertheless this is all work to establish a reasonable BASELINE to give us a better feel of approximately correct play.

What you have to understand about the bubble is IF opponents adjust and tighten up their 3bet range, you can profitably open much wider and call off very tight to the point if you want to exploit opponents. Even to the point where  you are very exploitable such as raising 70% of hands and only calling off with KK+. Obviously if opponents even 3bet you once you should dramatically reduce your raise percentage and reevaluate after a couple more raises on whether or not you should tighten or loosen it up. If on the other hand you are seeing opponents 3bet shove every single hand, you should call off much looser (remember a 30BB shove you should be 56.27% to win on the bubble, so 57% over a random hand vs any two shover ), but basically only open with a hand that you are willing to call off with.

30BBs opening range and call off
(before separator are hands you are willing to call off a shove with)
0.80 bubble factor
8 players left to act (vs 8.3% reraise shove range)
99+,AJ+,KQ+|28.13%
4 players left to act (vs 15.91% reraise shove range)
77+,A3+,A2s+,K9o+,K6s+,QTo+,Q8s+,JTo+,J9s+,T9s|71.25%
2 players left to act (vs 29.29% reraise shove range)
22+,A2+,K2+,Q3+,Q2s+,J6o+,J3s+,T8o+,T6s+,98o+,96s+,86s+,76s|100%
1.0 bubble factor
8 players left to act (vs 8.3% reraise shove range) 
TT+,AQ+,AJs+|17.65%
4 players left to act (vs 15.91% reraise shove range)
88+,ATo+,A8s+,KJo+,KTs+,QJs|33.33%
2 players left to act (vs 29.29% reraise shove range)
44+,A3o+,A2s+,K7o+,K3s+,Q9o+,Q7s+,JTo,J9s+,T9s|67.36%

1.2 bubble factor
8 players left to act (vs 8.3% reraise shove range)
JJ+,AQ+ | 11.18%
4 players left to act (vs 15.91% reraise shove range)
99+,AQ+,ATs+,KQs|19.23%
2 players left to act (vs 29.29% reraise shove range)
66+,A9o+,A5s+,KTo+,K9s+,QTo+,QTs+,JTs|31.45%
1.4 bubble factor
8 players left to act (vs 8.3% reraise shove range) 
JJ+ | 6.89%
4 players left to act (vs 15.91% reraise shove range)
TT+,AKs | 10.56%
2 players left to act (vs 29.29% reraise shove range)
77+, ATo+,A9s+,KQo+,KTs+,QJs |20.49%

1.6 bubble factor
8 players left to act (vs 8.3% reraise shove range)
QQ+ | 88-JJ,AQ+
4 players left to act (vs 15.91% reraise shove range)
TT+|77-99,AQ+A9s+,KQs,87s,54s
2 players left to act (vs 29.29% reraise shove range)
88+,AQ+,ATs+ | open with 13.7% of hands

20 BB opening range and call off
0.80 bubble factor
8 players left to act (vs 8.3% reraise shove range)
99+,AJo+,ATs+,KQo+,KTs+|23.11%
4 players left to act (vs 15.91% reraise shove range)
66+,A2o+,K7o+,K2s+,Q9o+,Q2s+,J9o+,J9s+,T8s |100%
2 players left to act (vs 29.29% reraise shove range)
22+,A2+,K2+,Q2+,J3o+,J2s+,T6o+,T2s+,96o+,94s+,87o+,85s+,75s+,65s,54s|100%

1.0 bubble factor
8 players left to act (vs 8.3% reraise shove range) 
TT+,AJ+,KQs|14.05%
4 players left to act (vs 15.91% reraise shove range)
88+,A8o+,A2s+,KTo+,K9s+,QJo+,Q3s+,J8o+,J7s+,T8s+,98s|46.21%
2 players left to act (vs 29.29% reraise shove range)
33+,A2+,K4+,K2s+,Q6o+,Q3s+,J8o+,J7s+,T8s+,98s|41.64% (note: I used propoker tools to determine the call off range, and pokerstove to do the remaining math.... In propoker tools J9s means J9s,Jts,Q9s,QTs,QJs,K9s,KTs,KJs,KQs where as I've always interpreted it as only J9s,JTs. As a result, whenever low suited one gap connectors or low offsuit cards like 97o+ were in the calculation it include more than I intended like Q7o so it messed with some of the results like this one. In most cases, it didn't change the results very much if at all unless facing a wide 3bet range.)

1.2 bubble factor
8 players left to act (vs 8.3% reraise shove range)
TT+,AQ+|9.88%
4 players left to act (vs 15.91% reraise shove range)
88+,AT+,A9s+,KQo,KTs|17.94%
2 players left to act (vs 29.29% reraise shove range)
44+,A7o+,A2s+,KTo+,K9s+,QJo,QTs+,JTs|32.77%

1.4 bubble factor
8 players left to act (vs 8.3% reraise shove range) 
JJ+,AK,AQs+|6.93%
4 players left to act (vs 15.91% reraise shove range)
99+,AQo+,AJs+,KQs| 11.51%
2 players left to act (vs 29.29% reraise shove range)
66+,A9o+,A7s+,KTo+,KTs+QJo,QTs+|18.94%

1.6 bubble factor
8 players left to act (vs 8.3% reraise shove range)
JJ+|4.42%
4 players left to act (vs 15.91% reraise shove range)
TT+,AKs|6.77%
2 players left to act (vs 29.29% reraise shove range)
77+,ATo+,ATs,KQo,KTs+|11.62%
===========================================================================
I believe I have provided enough of a context for you to get a good idea of how to adjust based upon everywhere between these numbers or even outside of these ranges as a good approximation. But a 3bet range of TT+,AK,AQs may be more common from early or middle position. So

Early and middle position strategy for different bubble factors.
20 BBs

0.8 bubble factor
TT+,AK,AQs|12.41%
1.0 bubble factor
JJ+,AK|7.94%
1.2 bubble factor
JJ+|5.33%
1.4 bubble factor
QQ+|4.18%
1.6 bubble factor
QQ+|3.35%

30BBs
0.8 bubble factor
TT+,AK|15.55%
1.0 bubble factor
JJ+,AKs|9.54%
1.2 bubble factor
QQ+|6.82%
1.4 bubble factor
QQ+|5.22%
1.6 bubble factor
QQ+|4.02%

Note:Raising more often will probably still be profitable since opponents 3bet range is tighter, but because of complications about them potentially just calling in position, it's probably not that great.

The "optimal 3 bet range" we used was one in which opponents deny you the odds to profitably steal with any two, and thus it's a very wide range, particularly in bubble situations, and probably looser than is realistic. Thus in real life you probably can raise wider to exploit and fold tighter and assume opponents aren't trying to exploit. IF you get 3bet, you might reduce the raise percentage and increase the call off percentage and reevaluate later. On the other hand, It probably is correct on the bubble to 3bet much wider IF opponent's open with the same range and fold very tightly.

I think 88+,AJ+ is a common 3bet range vs a late position raiser plus maybe some rare bluffs we won't include because they aren't common enough and enough people will shove tighter than this. I will add this calculation in later.

Complete Tournament Preflop Strategy Guide

Gameplan:
The goal early in a tournament is to survive to the antes with a competitive stack. The primary goal is to protect chips when you have enough worth protecting. You should still seek out weaker opponents by making larger raises with strong hands and larger bets on the flop, turn and river, but more checking to protect your strong hands from reverse implied odds. If you don't catch any big hands or win pots with reasonable hands here, you will have to eventually shift gears and loosen up your standards. Raise with a big hand, protect your big hands by checking more often to get to showdown, and betting larger when you do bet. Follow this tapering guide.

With over 50 big blinds pre antes we raise with the following hands or tighter.

No Antes:
8: JJ+,AK || 77-TT A9s+,KTs+,QTs+,[67s-KQs]
7: JJ+,AK,Aqs || 55-TT, A9s+,KTs+,QTs+,[67s-KQs]
6: JJ+,AQ+ || 33-TT, A4s+,KTs+,QTs+,[45s-KQs]
5: TT+,AQ+ || 22+,A4s+,K9s+,Q9s, [45s-KQs]
4: 99+,AQ+,AJs,KQs || 22-88, A2s+,K9s+,Q9s+, [64s-J9s]+,45s
3: 88+,AJ+,ATs,KQs || 22-77, KQ, A2s+,K9s+,Q9s+, [64s-J9s]+,45s
2: 66+,AT+,KQ,A9s,KJs+ || 22-77, [78-QJ], A2s+,K2s+,Q8s+, [53s-J9s]+
1: 44+,A7+,KT+,QJ,A2s+,K9s+,QTs+ || 22,33,A2-A6,k2-K9,Q8-QT,J7+,T7+,97+,[45o-87o], Q2s+,J7s+,T6s+,95s+,85s+,74s+,64s+,54s,43s

The separator shows the hands which have some value over "optimal" 3 betting strategy vs those that don't and are effectively a semibluff. Just because a hand has "value" doesn't mean you should raise. If your opponent 3bets with 20% of hands, 10% of hands have value, but if you raise, your opponent might only call with the top 5% of hands. That would mean that only the top 5% of hands could be raised while the 5-10% range should be called... with some bluff raises with hands not in the top 10%and slow play calls of the top 5% mixed in to balance your range. Bluff 3 betting hands might be a single suited connector or two single suited connectors like 65s and 78s.

The 3betting strategy of opponents is calculated such that they have a 50% chance of having the best hand if they had acted with the information you had when they raised, and a 2/3 chance of being able to make you fold if you don't mix in bluffs or implied odds. But the real optimal solution would be more complicated and contain more calls and some bluff raises. But if they call or raise more often than this, in general play fewer hands for bluffs, and slightly more of your "bluffs" become value.

If they 3bet wider but fold to 4bets, you can mix in some more 3 bet bluffs and also call with the worst part of your range. If they rarely 3bet and often call, you should fold to a 3bet but raise more often since you will have steal opportunities and showdown value by seeing more flops, turns, and rivers.

The postflop value of position means you probably should play a lot tighter in early and middle position than this and looser in later position. Also, a lot of the suited connector types out of position are "semibluffs" as said, and as such are probably not profitable, and not necessary since opponents probably won't adapt to your game the way that they should.

Against people who reraise and call more liberally, we can get rid of a lot of the "semibluff" hands when raising, and can widen the range which gets value. Many people are the opposite and don't 3bet very often, particularly early, and as such we can open slightly wider but should be much tighter about continuing vs a 3bet.

With over 50 big blinds after the antes:

Antes:
8: TT+,AQ+,Ajs | 22-88, A9s+,[45s-KQs]
7: TT+,AJ+,Ats+,KQs | 22-88, {A9s-J9s}+,[45s-KQs]
6: 88+,AJ+,Ats+,KQs | 22-88, A4s+,{A9s-J9s}+,[45s-KQs]
5: 77+,AT+,KQ,A7s+,KTs+ | 22+,[KQo-T9o],A2s+,K9s+,Q9s+,[64s-J9s]+,54s
4: 77+,A8+,KT+,A2s+,K9s+,QTs |22+,[45o-KQo],K7s+,Q8s+,J8s+,[53s-J9s]+,43s
3: 66+,A8+,KT+,A2s+,K9s+,QTs+ |22+,[45o-KQo],K4s+,Q8s+,J8s+,[53s-J9s]+,43s
2: 22+,A2+,k2+,Q2+,J5+,T7+,97+,86+,Q2s+,J2s+,T2s+,94s+,84s+,74s+,64s+,54s,43s
1: A2+,K2+,Q2+,J2+,T5+,96+,85+,75+,64+,54,T2s+,93s+,84s+,74s+,63s+,53s+,43s

Although you should start tight early, if you don't pick up a good spot or a good hand, eventually you may have to increase your aggressiveness to pick up some chips. So 2-4 levels before the antes with a  30 to 50 big blinds you still have the utility left to make a move so you give yourself a chance to go into the antes with a decent stack. So you are going to have to try one of 3 moves. The dead money grab, the resteal, or the position play (flat call on the button without much of a hand). A squeeze play shove also isn't a bad idea but is probably the riskiest play and hopefully isn't necessary. Alternatively you can just open up your range on the button, cutoff and hijack to virtually any two at tighter tables to try to give yourself a chance at having a good stack entering the ante stages and use the postflop position and Cbets.

In the ante stages you get a much better price on your steals, but most of the hands that have enough equity when called will lose enough from reverse implied odds that they won't be worth it. The weak kicker types are emoted and replaced with lower connected cards that won't be trouble and that can be semibluffed. Due to position over the blinds and tightness of other players, usually opening this range or wider even is fine. However, you need to avoid drawing too much attention to yourself by trying to avoid raising twice in a row when you don't have a good hand the 2nd time and avoid 3 hands in a row when you don't have a hand that can withstand a reraise like JJ+,AK. Avoid raising the short stacks blinds because they have nothing to lose. Also, you want to target the blinds of players with more chips to remove their utility and get them in a position when they are playing to min ash so that they aren't a threat to you.

3bet strategy NO ANTES between 30 and 50 BBs (or more)
When facing tight raise (sometimes flat to trap with QQ+, sometimes 3bet jam with AK)
3bet: AA,KK,QQ,[AK]A5s,{T8s}
flat: 22-JJ,[QQ+],AK,Ats+,Kts+,Qts+,{JTs,T9s,98s,87s,76s}
{only flat the suited connectors in position and only raise T8s in position}
When facing loose raise
3bet: QQ+,AK,A2s-A5s,{T9s,87s}
flat: 22-JJ,AQ,A6s+,KTs+,QTs+,JTs,{98s,76s}
{in position only, otherwise fold}
Antes 3bet strategy 30-50 BBs (or more)
3bet or flat:
when facing tight raise
3bet: QQ+,AK,A2s-A5s,T9s,87s
flat: 22-JJ,AQ,A6s+,K9s+,Q9s+,J9s+,T8s+,98s,76s,65s,54s
when facing loose raise
3bet: TT+,AQ,AJs,A2s-A5s,T9s,98s,87s,76s,65s,54s
flat: 22-TT,AJ,KT+,QT+,JT,T9,98,87,76,65,54,A6s+,K9s+,Q9s+,J8s+,T8s,97s,86+,75s

2 levels before the antes and 2 levels into the antes, Utility Theory comes into full effect where we are approaching this tournament as if the bubble factor is 0.80 which means the chips you gain at this stage is worth more than the chips you risk due to the future return you get from having a larger stack. You have to make it to the antes with a competitive stack or once you get to the antes you have to try to make some moves to get a stack.

This means we should "gamble" to gain a stack even when we don't have the pot odds.

0.80 bubble factors
When you make 2.25xBB and are faced with a 20BB shove
3bet range of KK+ it still requires AA to call
3bet range of JJ+ requires QQ+
JJ+,AK 3bet requires JJ+,AK
TT+,AK,AQs requires JJ+,AK as well but TT and AQs are very close.
88+,AQ+,AJs,KQs requires 88+,AQ+,AJs+
77+,ATs+,AJo+ requires 22+,AQ+,AJs+,KTs+
77+,A9s+,KTs+,QTs+,ATo+,KQo requires 22+,A9o+,A2s+,KQo,KTs+
22+,A9s+,KTs+,QTs+,ATo+,KQo requires 22+,A5o,A7o+,A2s+,KTo+,K6s+QJo,Q8s+,J8s+,T8s+,97s+,86s+,76s
About 25% of hands or
66+,A2s+,K7s+,Q9s+,J9s+,T8s+,97s+,87s,76s,65s,A8o+,KTo+,QTo+,JTo
Requires 22+,A2+,K2+,Q9o+,Q2s+,J9o+,J7s+,T9o,T7s+,96s+,86s+,75s+,65s,54s
About 50% of hands requires 22+,A2+,K2+,Q2+,J5o+,J2s+,T7o+,T2s+,97o+,95s+,87o,84s+,
74s+,64s,53s+,43s
Any two and you can call pretty much any two suited cards and all but 72o,62o,52o,42o,43o,23o.

Flush draw or open ended straight draw call off on the flop for twice the pot raise or smaller, or make an up to 3.5x pot shove yourself or smaller. Call off your tournament life with a low pair and back door straight flush draw for a pot sized bet or smaller and make a 2.5x shove or smaller.

Approximate Open Shoving for 20BBs with 0.80 bubble factor Without Antes:
8 99+AK,AQs
7 88+,AQ+,AJs
6 88+,AQo+,ATs+,KQs
5 66+,AJo+,A9s+,KTs+,QJs
4 55+,ATo+,A8s+,K9s+,Q9s,J9s,T9s
3 33+,ATo+,KQo+A7s+,K9s+,Q9s+,J9s,T8s+,97s+
2 33+,A9o+,A5s+,KTo+,K9s+,Q9s+,J9s+,T9s,97s+,87s
1 22+,A8o,K8o+,K2s+,Q8o+,Q5s+,J8o+,J7s+,T8o+,T6s+,97o+,96s+,86s+,75s+,65s,54s

Approximate open shoving strategy with 0.80 bubble factor and 20 big blinds WITH antes:8 88+,AQ+,AJs+,KQs+
7 66+,AQ+,ATs+,KJs+,QJs
6 55+,AT+,A9s+,KQo,KTs+,QTs+,JTs,T9s
5 55+,A9o+,A7s+,KJo,K8s+,Q8s+,J9s+,T8s+,98s
4 22+,A8+,A4s+,KTo+,K8s+,QJ,Q8s+,JT,J8s+,T8s+,97s+,87s
3 22+,A6+,A2s+,KT+,K7s+,Q9+,Q8s+,J7s+,T7s+,97s+,86s+,75s+,65s
2 22+,AT+,K8+,K2s+,Q9+,Q7s+,J8+,J7s+,T6s+,96s+,86s+,75s+,65s
1 22+,A2+,K2+,Q7+,Q2s+,J7+,J2s+,T7+,T5s+,98o,95s+,87o,85s+,75s+,64s+,53s+

We want to stay above 30 big blinds in the first few levels of the antes so we can preserve the ability to maintain full utility and decent raise sizes with our full range of hands. Staying in the 20 to 30 big blind range is still adequate due to tapering.

Once you get past the first couple levels of the antes, the utility theory declines and the opportunity cost for surviving rises. Treat the 3rd to 5th level into the antes like a cash game which will still be slightly lower than the ICM suggests.

1.0 bubble factors or cash games: calling off 6BBs 1.375 to 1 not in blinds*
Need to be 42.1% to win
8 44+,AT+,A8s+,KQs
7 33+,A9+,A7s+,KQ,KJs+
6 22+,A8+,A5s+,KQ,KTs+,QJs
5 22+,A7+,A2s+,KJ+,KTs+,QJs
4 22+,A4+,A2s+,KT+,K9s+,QTs+
3 22+,A2+,KT+,K8s+,QJ,QTs+,JTs
from blinds*
SB button push:22+,A2+,K5+,K2s+,Q9+,Q6s+,JT,J8s+,T8s+,98s
BB button push:22+,A2+,K2+,K2s+,Q7+,Q2s+,J8,J4s+,T8+,T6s+,98,96s+,85s+,75s+,64s+,54s
BB w SB push 22+,A2+,K2+,Q2+,J4+,J2s+,t6+,T2s+,97o+,95s+,87o,85s+,75s+,65s,54s

Calling off 10BB shoves
With 1.225 to 1 Odds, Need to be 44.94% to win
8 77+,AJ+,ATs+
7 66+,AJ+,ATs
6 55+,AT+,A9s+,KQ,KJs+
5 33+,A9+,A5s+,KQ,KTs+
4 22+,A7+,A2s+,KQ+,KJs+
3 22+,A4+,A2s+,KJ+,KTs+,QJs
from blinds*
SB button push: 22+,A2+,K7+,K5s+,QT+,Q9s+,J9s
BB button push: 22+,A2+,K8+,K2s+,Q9+,Q8s+,J9s+,
BB w SB push: 22+,A2+,K2+,Q6+,Q2s+,J8+,J4s+,T9+,T7s+,97s+,87s

Otherwise use shovebot charts.
Depending on when the money pays and your skill, you could potentially quickly shift to where your survival is worth much more. You really want to survive to the bubble with a competitive stack so that means not gambling more than the ICM suggests, so a major inflection takes place.

You should quickly approach the bubble factor as if you are actually on the bubble before you get there, so you can maximize your probability of getting there. Once you are there you want to push as if the bubble doesn't matter, but call off your chips as if it does.

Bubble Factor 1.6: Facing 6BB cash game optimal shoves when we have 6BBs left.
need to be 53.78% to call off 6BB shove vs these ranges on bubble.

8 88+,AQo+,AJs+
7 88+,AJ+
6 88+,AJo+, ATs+
5 77+,AT+
4 77+,AT+,A9s+
(did not adjust pot odds for being in the blinds)
3 66+,A9+,A8s+,KQs
2 55+,A8+,A7s,KQ,KTs+
1 44+,A3+,A2s+K8+,K5s+,QT+,Q9s+,JTs

Facing 10xBB shove vs cash game optimal shoves with 1.6 bubble factor:
We need to win 56.64% to call vs these ranges
8 TT+,AK
7 99+,AK+
6 99+,AQs+
5 99+,AQ+,AJs
4 99+,AQ+,AJs
(did not adjust the pot odds for being in the blinds)
3 88+,AJ+,ATs+
2 88+,AT+,A9s+
BB 66+, A8o+,A7s+,KJo+,KTs+

Facing 10x open shove with 1.6 bubble factor:
Call off with KK+ only vs a TT+,AK pushing range
Call off with QQ+ vs 99+,AQ
Call off with QQ+ vs 8% range
Call off with JJ+ vs 10% range
Call off with TT+,AK vs 15% range.
Call off with 99+,AK,AQs+ vs top 20% range.
Call off with 99+,AQ,AJs+ vs top 25% range.
Call off with 77+,AJ+,ATs+ vs vs top 50% range.
Call off with 55+,A7+,A2s+,KJ+,K9s+,QTs+ vs any two shover.
note: We should be slightly tighter if we still have people left to act.

20BBs call off facing 3bet that puts you all in or commits more than 25% of your stack:
Call off with KK+ vs TT+,AK pushing range.
Call off with QQ+ vs 99+,AQ.
Call off with JJ+,AKs vs 8%+10% range.
Call off with TT+,AK,AQs+ vs 15% range.
Call off with 99+,AQ+,AJs+ vs top 20% range.
Call off with 88+,AJ+ vs top 25% range.
Call off with 55+,A8+,A7s+,KQo,KTs+ vs top 50% range.
Call off with 44+,A4+,A2s+,KT+,K7s+,QJ, Q9s+,J9s+ vs any two card shover.
Never raise more than 65% of the time with 20BBs and 1.6 bubble factors or you are very exploitable to a reshove in any and all situations.

Unexploitable opening 2.25x hand range with 1.6 bubble factors (hands before the | signifies hands that you can call off a "cash game optimal" shove with.)
8    JJ+ | 99+,AQ+ATs+,KQs,QJs
7    JJ+,AK,Aqs| 66+,AJ+,Ats+,KTs+,QTs+,JTs
6    TT+,AK+,AQs| 55+,AJ+,A9s+,KQo,KTs+,QTs+,JTs
5    99+,AQ+,AJs+,KQs| 22+,AT+,A5s+,KJo+,K9s+,Q9s+,J9s+,T9s,
4    99+,AT+,ATs+,KQs| 22+,A8+,KT+,QJ,JT,A2s+K9s+,Q9s+,J9s+,T8s+,98s
3    99+,AJ+,ATs+,KQs| 22+,A8+,KT+,QJ,JT,A2s+K9s+,Q9s+,J9s+,T8s+,98s
2    77+,AT+,A9s+,KQ,KTs+| 22+,A5+,KT+,QJ,JT,A2s+K9s+,Q9s+,J8s+,T8s+,98s

Note:If opponents respect bubble factors and don't think you do (won't 3bet liberally) you can exploitatively raise wider to much wider than this. If opponents do not respect bubble factors OR are utility theorists OR assume you do and thus 3bet liberally, there is less need to raise with hands that you will not call off vs shoves which thus effectively are in some ways "bluffs" or at least semibluffs. Similarly, you can shove much wider than the shovebot charts or squeeze play math suggests but should call much tighter.

On and near the actual bubble you can raise with virtually any two cards. You want the image of someone willing to call off your tournament life without actually wanting to. That way people shouldn't reraise you if they care about their tournament life unless they have a really powerful hand.
As the bubble bursts everyone else who survived is going to start getting it all in MUCH looser and begin racing again. Hopefully you have built up the chipstack to survive the all in fest because you still want low varience accumulation. So after the bubble I would probably open much tighter as if I was on the actual bubble, but be willing to call off a little bit looser and not require that great of edge.

As you approach the final table bubble it's different. Even though you probably could exploit and steal by raising more aggressively, it becomes too costly. Moving all in becomes too risky, and you've already accumulated a huge percentage of all the chips in play. You don't have much farther to go. You should instead be playing to triple up rather than double up which means you should be comfortable as a short stack.

Playing to survive carries with it some value, but if everyone is playing to survive you can with very low risk, thrive off of low risk steal attempts. I think the mistake that most people make is they are initially planning on folding their way into the final table, but then at some point they change their mind or think they are more desperate than they are so they jam with A9 and get called by AJ or something.

If you are going to adopt a max patience strategy, you need to know just how patient you can be. You also will want to know what the equivalent of that patience is with regards to the odds you have the best hand preflop so that you can take an adjusted max patience strategy, such that when it folds to you, given the number of players you have the equivalent hand of a max patience strategy given the number of players left. You also probably want to consider raising about 3 times as often as you are willing to call a shove with such that you have a max patience raise strategy as a baseline. This gives you a chance to buy a little bit more time without being exploitable. You also might jam 1.5 to 2.5 times the hand range you are willing to call off with if you have less than 15 big blinds.

You should make a few larger raises that make it look like opponents have zero fold equity such as 3-4 big blinds when you have 12-15 big blinds or up to 5x when you have nearly 20BBs. It's odd, pot committing raises that you are actually willing to fold that are exploitable, but who's going to call it unless they are willing to call a shove? And what looks stronger? Probably a 3-4x raise. You shouldn't mind making these with mid pair and AK and AQ as well and if it fails and someone shoves and you fold, you probably should think about doing it with QQ+ happy to steal the blinds, but not concerned if someone shoves.

You also might be willing to give up a higher rate of gain for a higher rate of survival as you approach the final table such as limping in with your raising range with under 25 big blinds, and reraise shoving your hands that have value vs an optimal 3bet. Or such as checking the river rather than thin value betting in position. Or checking the FLOP for pot control rather than just the turn and potentially ALSO the turn depending on what card comes with top pair and better hands that are vulnerable to monster draws. You might use FLAT betting rather than a percentage of the pot size such that you might bet the same amount on every street.

Another key principal is to steal from the other big stacks as you approach the final table and especially 5 handed on in, while giving the shorter stacks a pass. You eventually want EVERYONE playing for second place so that requires taking a few more chances against those with more chips or as much or nearly as much as you. If people are trying to fold up the money and you keep this situation around as long as possible, it will benefit you. Make big folds as long as you are able to hang in there and get your fair share of the blinds, even short handed, but don't be afraid to push your stack in the middle and put the opponent to the tough decision if they are capable of folding.
Technically the final table bubble pops and the ICM is in slow decline to 1 from the final table on. I believe people underestimate how easy it is to be incredibly "sticky" by playing what seems like way too tight. Playing to "survive" seems like a loser's strategy, but when you are close to the end of the tournament, surviving long enough equals a second place finish or even a win.

A Maximum patience strategy may more accurately assess the blinds, the probability you get dealt a better hand, and how to play given the structure and your chips stack than an ICM based strategy with a high bubble factor might. It is at this point where there are probably a fixed amount of double ups before someone wins the tournaments plus or minus a few steals. If you only need 3 double ups to win the tournament, it doesn't really make sense to double up sooner and win the hand earlier, which is what happens if you take a slightly higher EV strategy, but still get money in with 3 double ups to win. Granted having an extra big blind or two has a slight possibility of having implications in that if your opponent who you have covered doubles up, you may have a few more chips from which to double up back or if they have you covered and you win, you are a little bit closer to the win. But the chances of a few big blinds making the difference is probably smaller than the advantage you get from being more patient, particularly due to the equity rise of waiting until other people knock themselves out. While a ICM of 1.6 would say that you can call off a 10BB shove with 99+,AK,AQs+ vs top 20% shoving range. That's pretty tight, but if you were in a tournament where the blinds rise faster, there may be more value in calling off looser because you are unlikely to find a better spot, and the equity jump of moving up a few spots isn't necessarily worth folding. Conversely, if you have about 40 hands left, you have well over a 50% chance of getting JJ+. If it folds to you with 4 players left to act and you have JJ, you have a 94.7% chance of a remaining opponent not having QQ+ and an 85.75% chance of no opponent having QQ+,AQ+.

If you really wanted to, you could determine a call range, a value derived in BBs from when opponents fold, and an approximate value they call based upon the probability you get called and the range of hand you get called with and compare JJ+ to 99+,AK,AQs and see which provides the greater chance of survival. Of course that isn't enough information since you could also call with 99+,AK,AQs now and occasionally survive to get that JJ+. And the blinds may go up since then and the possibility of you having to widen your range to include AKs or TT, AQs or AKo exists. Plus you may not get JJ+ in middle position and folds to you, you may get a raise from a stronger range of hands or it may fold to you in the big blind.

But there are probably complicated methods and monte carlo simulations that could be done to try to assess the value of a call off in final table spots vs the probably alternatives.

When you are ridiculously short stacked, people will call you, and you may get called twice, or better yet called once and then reraised shove isolated to where the first caller folds. The antes rising can actually improve the value of patience as well. So the more patient you are, the more value there is in being more patient to a certain extent. The blinds and antes are always increasing insuring the price you get on your "double up" or "triple up" is constantly improving. While just maintaining your chipstack isn't going to buy you any additional hands, it's only a matter of time before the bigger stacks run the AQ vs AK hands and think it's good. So even surviving caries with it a huge payout in moving up the money. More importantly, if/when you do double/triple up if you can show a big hand, people are much much more likely to fold to your raises and shoves.

People tend not to notice your style as often as you might think, but the combination of not seeing you in any hand and then finally seeing you turn over a monster will probably get enough attention immediately after to make them second guess calling.

The overall goal is to think about the total odds of winning every all in until you win the tournament and try to both reduce the number of all ins required and increase the odds when all in, plus add enough uncontested steal and add tremendous amount of value for strategies that help moving up in the money. The final table and near it is where you see Phil Helmuth do some very odd things like make what seems like a lot committing raise UTG as a really really short stacked with just about any two and then fold to a raise. Granted, this is with tournaments with much slower structures. But he is using his tight image to make that raise, and the value of survival being disproportional to try to pick up the next round of rotation at the lowest risk possible. If he doesn't get it, he still can get great pot odds to call off an all in from the big blind, or even survive another rotations worth before the next big blind he is basically calling no matter what. A lot can happen in two rotations and if the steal succeeds he can probably fold for another rotation or two before he needs to make steal attempt and if it fails he again probably has to go with whatever hand he has in the big blind.

The fold is about chance of survival. There is very little difference between doubling up to 16 big blinds or 13 big blinds in regards to your Overall equity and probability of finishing in each place. This is true such that it's okay to fold with good pot odds or even great pot odds and make a steal attempt that in normal scenarios should be a shove. But buying another rotation plus the ability to make another steal later can be invaluable as a short stack as you can easily see two people get knocked out in that time and still manage to either pick up a monster hand or another steal and buy more time. If it fails, a double up can still buy you a lot of time, particularly if the blinds rise or there's more dead money in the pot due to a raise and a call in front of you or a few limpers and you shove and a few limpers fold. For all you know you may even get a walk in the big blind if the late position players are worried about the small blind having a hand and the small blind has 26o.

Be ridiculously tight until you catch that double up hand and probably get action. You constantly adjust the number of "hands left". So your actual odds of catching a hand before you blind down to whatever line in the sand you draw are much higher than 50% overall. Since you are gladly trading some EV in chips for greater probability of survival, you are waiting to close to the best spot/hand you can reasonably expect to get. The goal is to play to survive until short handed with the tightest image possible, and use whatever recent attention you have drawn about being tight to your advantage. Shift gears to become much more aggressive.

Utility Theory

The concept of Tapering as illustrated in this post and shown in the series below introduces an understanding of "utility" as it relates to having fold equity on every street.

We can see that when we get down to under 40 big blinds the bet sizes become small enough where we don't really have the same effectiveness on our bets, even though we maintain the ability to bet on every street. Below 25 big blinds in order to preserve utility on all 3 streets to have a full pot sized bullet on the river, we have to reduce our bet size to less than the preflop raise.

If we choose larger bets to maintain the effectiveness and profitability that a half pot bet has, we are committing ourselves a lot more by the river, and thus our river bet is losing some utility. We also lose the ability to raise preflop and continue on all 3 streets. We lose some ability to 3bet the "standard" raises which are generally smaller than the tapering strategy raise and are generally under 3 big blinds at most stages. We also lose the ability to raise on other streets and maintain the full aresenel of moves on every additional street as we decrease our stack size relative to the blinds.

Tapering is a concept that EXTENDS utility longer on in the tournament. Those with an edge that can play larger pots will have more chips for longer allowing them to keep playing slightly larger bets until their opponent's effective stack force them to play for less.Even then, as the blinds rise, tapering maintains full utility even as the blinds rise to where everyone becomes shorter stacked as even with smaller raises, the implied odds aren't there for opponents to call, and the risk management and stack preservation makes it difficult to even call a smaller bet.

However, tapering doesn't extend utility forever. Eventually there is a tradeoff regardless of the choice you make. 20-35 big blinds is really where that tradeoff occurs.

Option 1 is limp in to preserve utility on the flop. Limping adds the "last one in" advantage with 20-40 big blinds and under where you can shove rather than get shoved on and have no fold equity and only the equity of your hand, and where folding to an all in shove over your limp would represent a very small percentage of your stack. Limping strong invites multiple callers and isolation raises which requires you to have a range that contains enough drawing hands that play well multiway in your range, and enough strong hands that you are willing to 3bet shove for value or as bluff. Your strong limps offer opponents with small pairs the implied odds to setmine when you are strong. However, you maintain the ability to setmine vs someone who limped first or only made a small raise or when there are multiway pots and play drawing "implied odds" type of hands like suited connectors and suited aces that otherwise would lose their implied odds to be playable. It requires a very different logic to playing out of position such as frequently leading out small on draws as well as strong bets with a smaller than normal bet. You want to lead out weak often enough with strength as well so if you get played back you can shove, and so you aren't too predictable, but you also want people to just call often enough that they allow you to create the pot odds such that if they bet it, you would get the right price to call. You can use board texture to overbet the pot on later streets, and use information on wet, boards with say ten or jack high to follow up with a bigger bluff on the turn since they probably would have raised to prevent a difficult decision on the turn.

The real drawback becomes "forking your range" or "forking your strength/weakness". This means that your opponents can typically tell by how you acted preflop (raise or a limp) what you have. This forces you to always limp, or be slightly more predictable about what you have, or to complicate things by SOMETIMES limping with hands you would raise and vise versa. If you are able to limp raise often enough, you probably cannot raise during that period without making it easier for opponents to read your hands, so you'd have to pretty much always limp which certainly loses a lot of opportunity preflop and on the flop when you get multiple opponents as you cannot as easily buy pots with aggression. So limping usually switches mentality to playing for value rather than playing to steal, unless you can limp raise often enough to where you are restealing. The limp raise doesn't have to occur nearly as often for it to be more profitable since it's like a 3bet in that it wins more chips.

A limp in the middle and later stages effectively plays like a raise and a few callers in the earlier stages in that you need stronger hands to continue or draws to strong hands, and good implied odds hands in your range as opposed to the AJ,KT,QT,J9 and Qx suited type of hands.

Option 2 is to just bet less than half the pot and taper down to a minraise. A minraise is okay even though they will defend more, but betting less than half the pot is certainly questionable.

Option 3 is to remove the all in river pot sized bet and instead just have stack sizes where a shove might only be half the pot on the river, and all 3bet pots will be all in or effectively playing all in on the flop.

Option 4 is just to accept the lost utility, mostly play as normal, but plan to get the money in on earlier streets and readjust your bet sizes and hand strength accordingly. Since you are trying to get the money in earlier, slightly larger bet sizes now might also become acceptable again, and the hand ranges should reflect this.

So there could be a phase where you REVERSE taper and instead inflate the bet size again to make it more challenging to players who are unprepared for this style and also to take a higher risk/higher reward line since the bigger risk will soon become not taking one.

If you are going to play draws they will probably be the all in semibluffs as described in the nutball supersystem. This also means you will not be floating on marginal hands or checking to induce bets very often. You are reraising to get the money in or folding, or checking to give up if you don't improve. If you float it's only on small bet sizes and you'll probably be shoving in on the turn if you detect weakness.

Option 5 is to be anticipatory. Rather than WAIT until this moment comes, prior to the blinds rising, more liberally seek out all in situations and more aggressive plays in anticipation of losing that utility such that you maintain the ability to steal at a lower risk for longer if your aggressive stage is successful. So in the 25-35 or even 30-40 big blind range, you are willing to flip or even take the worst of it with pot odds or if you are a utility theorist, you might not even need a +EV decision to get it all in, you just need to be close enough to where you can make up for it later due to your extra utility when you otherwise would have been in the 10-20 big blind range that may last for a few levels.

Option 6 is to just completely change styles entirely. For example potentially open shove or check raise shove flops for large overbets as somewhat alluded to in the option 4. You might have a mixture of semibluffs and made hands and maybe more pure bluffs an extremely low percentage of the time.  Then continuing for larger bets with the intention of shoving the turn with some top pair hands and possibly some gutshot draws that improve to 15 outs by picking up a flush and double gutshot on the turn and perhaps some bottom or middle pair hands that improve or see a scarecard on the turn. Your hand range may also change dramatically and possibly your bet sizing.

Option 7 is to change hand selection and criteria for a "stacking off hand" since you won't have that extra river to worry about or draw to so draws decrease in value if semibluffing is not profitable. Medium strength hands decrease in value unless opponents are making too many "hero calls" and medium strength hands aren't really worth stacking off except for top pair weaker kicker and top pair now goes up in strength. To really understand how your hand values should change, you need to use the rule of 3 and 6 and 5 and 10 and/or understand stack to pot ratios as described in Professional No Limit Hold Em Volume I and how commitment threshold of hands should change against different types of opponents based upon SPR..
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I believe the theoretically equilibrium shoving bluff to balance your nut hands is if you have a made hand 2% of the time, you might overbet shove for 5x pot 0.33% of all flops (1/pot size +1 times the made hand). And your semibluffs should on average break even to force opponent to widen his calling range. Or you might just include some of the more longshot semibluff hands on very rare frequency. This system was partially explained in nutball supersystem. I haven't really used it, but maybe I should. It requires some serious planning if you want to get the bet sizing and math right and definately is a high varience strategy that could get you back into the "full utility" zone. But in tournaments the ICM is really weird since if opponents obey the ICM they should really make big folds, which makes it profitable to more liberally push. But if they don't do as they should and you know it, you shouldn't be so aggressive and should have a much higher made hand to semibluff shove rate and eliminate a lot of the worst semibluffs and instead play those passively/cautiously.

You could argue you no longer need that full pot sized bullet all in on the river, but it's nice to have the flexibility to bet for value. Also, opponents have the power to eliminate all your utility by 3betting and effectively committing you to the all in.

Aside from this, being able to get the full value of 100 big blinds is usually going to result in a higher BB per hour rate if you are a skilled player, as well as present situations where you can play with greater implied odds if you choose to never have a preflop opening raise over 3.5 big blinds. With these implied odds more hands become playable for longer and there's added value early that gets lost as your chipstack declines.

It's not real easy to draw a clear line in the sand of where the benefits of getting more chips are large enough that it's worth accepting a lower bubble factor or even gambling with the worst of it in order to preserve the utility over multiple blind levels and parlay it into more chips in the future. I don't know what option is best and at what stage. You can also argue that the gains of utility are not sufficient enough to outway the opportunity to find a very profitable spot in the future while maintaining a large probability of survival and chance to make money by folding, as well as the possible opportunity in the future to get isolated heads up for more than a triple up after loose action and a big isolation reraise which may have a better probability of getting you back in it with more chips anyways.

Nevertheless, we can see that "utility theory" has a clear argument for consideration.

If I had to describe where I think Utility Theory takes place, it's where you have 25-35 big blinds, and as low as 15 if the situation dictates it. The odds to eliminate a very skilled opponent seated to your left, or a very loose, aggressive opponent to your left at a table where they are very passive and loose preflop and tight postflop do not have to be very high for you to be able to make up for it in future steal opportunities. Your chances of advancing deep will have to go through that player as you will probably not be able to outchip him by waiting for a hand, and if he won't let you steal your fair share of pots, you will be forced to bet a parlay with MULTIPLE all ins.

However, if you go after this guy and can cripple his utility or go bust, you immediately end the need to make multiple all ins as you have a high probability of being able to just accumulate chips and keep up with the rising blinds without a lot of risk to your tournament life aside from the initial all in.

Exceptions aside, anywhere from 5-15 and possibly as high as 25, I think "opportunity cost of better opportunities" is greater than utility gained by doubling up. So really in the 15-25 range there is a pretty big overlap between two schools of thought. I hope to cover "opportunity cost of survival" in a future post.

Hand Analysis: Dead Money Grab Vs the Stop and Go

The Scenario: You are playing an online $10 buy in $1250 guaranteed tournament. With 40 big blinds after several limpers, you limped in late position with suited connectors figuring to have good enough implied odds. 3time bracelet winner Dutch Boyd on the button has made an isolation raise of around 1/4th his stack, and you no longer are getting good implied odds, but are getting 2.7 to 1 pot odds to call.

The Problem: Without knowing Dutch's exact cards or exact range, is it possible to play 87s profitably at this point?

The Set Up: Leading up to the hand dutch had been powering the table quite well and shortly after started to pick up AA, 44 and AK a few times and was on a pretty nice run of cards. So his image is more wild than usual. Normally he preaches being the bully without looking like the bully. However, sometimes you catch a run of cards and you end up looking like one more than you intended. I think that's the case here.

I believe that may turn out to be important from the perspective of chefward. Since Dutch's image is more wild than usual, chefward can more easily diagnose Dutch's hand range as wide, and determine that he won't have a hand that can call the flop quite as often as he might otherwise have assumed vs your typical player. Even though dutch hadn't made a lot of these big isolation raises, it'd be hard for chefward to give dutch credit for only premium hands here and he'd at least have to include some broadway hands and suited aces and maybe suited connectors and lower pairs in his range. That will translate into dutch finding it slightly more difficult to hit the flop enough to be able to call.

Dutch decides to from the button to isolate his standard $3300 at this level PLUS 1x per limper. The raise size actually should be 8100 if you want to be picky. Aside from a philosophical discussion on whether or not the "standard" should be normal raise plus 1x or 0.8x or 1.2x, or if he should make a multiple of the pot when he's going for the "dead money grab" I am entirely fine with this raise size and don't have any qualms about it.

I analyzed the raise over several limpers for 1.5x and 2x multiples of the pot in this "dead money grab" post. It really depends on the limpers limp range, and their ability to trap. But even with much larger raise size than the 8/9ths of what's in the middle, there's a mathematical argument for raising any two. The argument is entirely mathematical but requires assumptions about opponent's limping range that are speculative. Since the risk of 8/9ths is far less than the 1.5x and since the positional player usually has an advantage, the dead money grab with the J9o is perfectly fine.

I say no problem at all making this play as the raiser, you have to do something to keep up with the rising blinds, and waiting for aces probably isn't going to cut it.

BUT... what about chefward2 who's sitting there with a suited connector? I don't think he can play such that he can make it unprofitable for Dutch, but that doesn't mean he can't fight for his share of the dead money in the pot.

So say you're sitting with 87s after several limps and a seemingly LAG player raising behind you. You're getting 17,000 to 6400 or 2.656 to 1. Certainly if you knew the hands were always checked to the river you'd only need to be 27% to win. But it will almost never check to the river. Especially out of position with shallow stacks, you can't make the usual assumptions. You can't assume postflop that "bets break" postflop so you cannot just assume if you are 27% equity it's a good call. Instead you will be committed on the flop to either calling or folding.

While you may get some implied odds, You most certainly are NOT getting the implied odds to hit 2 pair or a straight or flush. You do have some fold equity on a jam, but probably not enough to jam any flop here. Even if you include 8+outs in addition to any two pair or better hand you are only going to be hitting around 25% of the time. With that range of hands, depending on Dutch's range, you'd probably need to be getting much closer to 3 to one pot odds because you'd probably only be slightly ahead of Dutch's calling range.

So the next question is can chefward just shove every single flop profitably? I say the answers also no.

You're not getting 1:1 on your steal attempts because you're just calling preflop. You're actually risking over 30k to only get the 17k that's in the flop before your decision. You gotta be right close to 66% of the time to break even on your pure bluffs or make enough from your shoves when you do hit to make up for the difference. Probably not happening if you always shove.

We certainly could try to come up with a more precise answer to this in terms of what sort of range our Dutch has to have and has to raise with and call a flop shove with to make a shove with any two profitable, but it probably will involve folding some pairs and when you are getting 2 to 1 it's hard to fold a pair since you will only flop a pair 1/3rd of the time and you probably will have some outs... I just think Chefward's opponent (in this case Dutch) will have a pocket pair or flop a pair or combo draw too often for an autoshove to be correct. And if that's the case, where does that leave you? Does that mean you should fold?

Not so fast.

In order for a call preflop to be profitable for chefward, he has to be creative. The stop and go play with any flop is too reckless and the fit or fold style is too tight. So we have to determine if there is somewhere between where it may work.

How much equity is it going to cost chefward to also push with any pair on no facecard boards and pair with backdoor draws such as 3 to a straight with 87 on a 76J or 89A flop for example? Well, on a board like 56J with a single spade he's still going to be 27.37% to win vs KK. That's not bad at all. Sure he's only 8.25% vs a set but that's a small percentage of his Dutch's range.

While I think it's probably more ideal to take an exploitative solution of checking our made hands far more often, and open shoving our semibluffs and pair with backdoor draws, it's far easier to look at this as more of an all in or fold calculation, and it's already pretty complicated with shove or fold, without adding in a mixture of checks folds and check shoves and shoves.

Chefward may even be able to shove with only a backdoor draw. Sometimes he's going to be against AK, get called, and actually be ahead. If he shoves with a 5 and a spade or a 9 and a spade and no pair he is 10.40% to win against say KK and 7.37% to win vs a set, but he is about 30.6% vs two overcards. Certainly that's not terribly ideal, but when you have a lot of fold equity and also are planning on shoving with straights, flushes, two pair, trips, and 8+ out draws the collection of all your shoves gives you pretty good equity when called.

So let's just look at the equity vs a few basic flops, estimate the probability of a similar flop, and try to estimate the overall equity of the decision to shove certain flops. Then we can look at the dutch's approximate raising range and approximate the probability he is going to find a call there.

So let's start with the pure double backdoor draws.
On a 6s Kh2d flop I don't know that we'll get called as often as we should because of that King, but just so I try to approximate the averages of the same flop with a T,J or Q flop I'm going to include AJ+, 99+, 22,66 plus KT+. When I do this, we have about 15% equity. Certainly could be a lot worse, especially since on these types of flop Dutch will probably have a decent chance of folding. I have a feeling shoving with ALL backdoor draws is too much and will tend to make the AVERAGE equity when called too low, but I think including some of these shoves may end up being right just because we aren't going to flop made hands and strong draws often enough not to include some bluff shoves, nor can we shove on the AK2 type of flops where we have nothing at all and no backdoor draws.

What about a 92J flop. This represents backdoor flush draw with the "idiot end" of the gutshot. If we have 78s and are facing a calling range of 22, 88+, A9, T8,T9,QT, any Jack in his range, some ace kings and some Ace queens (because that's a hand that he may fold, he may not so by only including some we weight for the possibility that he doesn't) Then I come up with about 22.9% equity with 78s. Certainly that's not the best flop, but once we add the fold equity in on a flop like that, it may be okay to shove as part of our entire shoving range since our overall range will be much stronger than the lowest 25% of our shoving range.

If we change the flop to 98K with backdoor flush draw AND now we have a pair and backdoor straight draw, now the equity is up to about 37% given the hand range I come up with. Pretty good.

Now if we add in open ended straight draw with backdoor flush draw we shoot up from the 15% equity we had with only double backdoor draws to 38% equity.

If we have a flushdraw and a backdoor straight draw our equity is nearly 40%. Actually, I forgot to add flush draws in Dutch's range.

If we crush the flop with 2 pair, we are probably far better than 75% up to around 80% or as bad as around 70% depending on how much credit Dutch gives us and depending on whether or not there are flush draws for Dutch or backdoor flush draws for us.

If we flop a straight we are probably above 85% vs Dutch's calling range.
If we flop a flush we are probably about 85% as well since many hands that hit the flop will also have a redraw to either a fullhouse or higher flush.

The difficulty in the math is now subtracting the overlaps. For example, there may be a 16% chance of hitting a gutshot draw on the flop, but that includes when you hit gusthot as part of a combo draw including gutshots, double gutshots, gutshot with flushdraw and/or overcards and when you do the math of backdoor flush draw it is going to include backdoor flush draws plus gutshots or open ender or whatever.

You are not looking at the probability of a double backdoor draw, but the odds that a double backdoor draw isn't also a gutshot or a straight draw, or otherwise you end up counting them twice. I'm not going to do all of the math here, instead I'm going to do a really rough approximation. If anyone knows any tricks or software that can help, I'd like to hear it. I think flopzilla may be the tool I need if I want to be really accurate in this stuff but there are others too, I'm sure.

Based upon my calculations, assuming the equity stays as calculated if Dutch's calling range represents 60% of his holdings, it's not profitable to include double backdoor draws and gutshot draws and if we do, overall the strategy is unprofitable by over 300 chips. If Dutch calls 50% of the time, it's not profitable to shove with double backdoor draws, but it is profitable to shove with gutshots and overall the strategy is profitable. If Dutch calls 40% of the time, it's profitable to shove with double backdoor draws, but probably the worst of flops are unprofitable to shove.

Overall, I think if Dutch's range preflop is wide enough chefward could probably make the call profitable if he was very calculated about which flops he shoved. I also think that if many of the calling hands would be a very difficult call to make, we can justify the call.

Since we are calling 6400, the pot will be 23400 on the flop. The effective stack signified by the shorter stack (in this case Dutch) is 24387 behind. So if you are to shove the flop, you are putting in BOTH the 6400 to call PLUS 24387 or a total of 30787 to add 17000 to your stack from this stage in the decision tree. If you shove and are drawing DEAD when called, you have to get folds nearly 2/3rds of the time. But as we know from the analysis we just did, you are rarely drawing dead, and probably can avoid shoving the worst of the flops.

In Conclusion, chefward almost certainly cannot make 8000 worth of profits which is what's needed to make Dutch's decision unprofitable on his own, but it seems he can probably make a selective stop and go profitable to fight for a small share of the dead money AT the cost of great volatility to his own chipstack, that given ICM factors may not be worth the chips he gains. After working the math I can conclude that my intuition that calling CAN be marginally profitable for chefward2 is correct, but the margins are very thin and seem more thin than I would have thought, and given ICM impact that chips you gain are worth less than the chips you lose, it may be better for chefward to fold anyways. It's one of those really razor thin decisions where either one could be correct. Certainly it's not terrible to gamble a little bit to potentially knock out a talented player, particularly if there's enough dead money that it's positive chip EV and go down swinging at who you perceive to be the biggest threat at the table to you taking the title.

But you can also argue that there's no reason to fight so hard in a go big or go home situation when you have almost 40 big blinds and there are better spots.

With that being said, I'm pretty confident it's possible to get more precise about which flops you can shove with given a lot of in depth analysis of ranges. I very much doubt that chefward has done the kind of work to get there... but who knows, maybe he has. Some flops will be hard enough for your opponent to call, or will miss opponent that even if he does that it will be profitable to bluff shove a lot more, where other spots you will have a decent draw and still  have to fold because opponent's range crushes yours. Like if you had T9 and flop was KQJ, a Ten brings you a straight but probably is almost never good when you get called and hit.

I also still think that you could milk a little bit more out of calling preflop if you check the flops you hit hard enough to be a favorite vs calling range(12+ outs or 2pair+), you will get paid off more, and the result is that you probably will not have to semibluff shove quite as light because of the added equity you get most of the time from checking. You also may be able to get your money in with one pair hands this way and end up ahead if the flop is exactly right (like all 3 flop cards below ten or 3 uncoordinated cards and only one of them being a non ace face card).

I think Dutch probably can find a call 55% of the time chefward shoves. If he starts unpaired, he'll pair up 1/3rd of the time, plus he'll start with a premium pocket pair often enough and hit a big draw himself often enough or have overcards like AK and a chance at having the best hand often enough given the pot odds to make a call. I think it might be closer to 50% in Dutch's case and closer to 60% in the typical opponent's case who doesn't punish the limpers as liberally as they should, but I think 55% is a fair, conservative assumption that chefward should be comfortable making.

Even though it's unprofitable to shove double backdoor draws... if we don't have these in our range we lose too much from check/folding too often such that it is more unprofitable if we don't include them than if we do. So we have to include all gutshots too. The ideal solution to make the most profit is going to only include some of the double backdoor draws, and probably include some more one pair hands without double backdoor draws when the flop misses opponent's range, but I am not going to try to even come close to solving this since I already took some liberties in approximating my assumptions and it would be a nightmare to try to solve every scenario.

So I think if you were VERY VERY precise you could make a selective stop and go play in this spot profitable to the point where it might even be worth it after factoring in ICM, but that would require virtually a photographic memory and a lot of hours of work that are best suited working on core concepts that can help develop better "feel" instead of memorizing precise decisions for exact situations that will almost never arise again.

Nevertheless, it's a very interesting topic for consideration. It would be interesting to look at the stop and go with suited connectors in the same spot when we have a lot more chips behind and thus have better implied odds to hit the better flops. It also would be interesting to compare it where you are even shorter stacked such that you can jam any pair and you get more out of your semibluffs due to not laying as big of a price vs hands that can call you. If we could get a few spots like that, we could develop a much better "feel" for making moves like this profitably and then repeat that with a ragged ace or two offsuited broadway cards or low and medium pairs so that overall we have a good "feel" for when to make a more flop texture based stop and go with around 10 big blinds, 15 big blinds, 20, 25 and the check raise shove variety of the blind defend vs someone who always Cbets as well as vs a more selective Cbetter that Cbets 60% of the time.  If you never limp, you don't probably need to concern yourself too much with the stop and go to defend against a potential dead money grab, but there may still be some spots to use it from the small or big blind.