In a tournament, there's a "correct" way to play on the bubble given certain assumptions. This "correct" way to play is determined by what's known as the Independent Chip Model or "ICM". It assumes that skill is equal and skill is basically determined by theoretical odds of finishing in a given place if players were to coinflip for their lives until the tournament's over. Thus anyone still around in a tournament still has some chance of making money by other players getting knocked out first and thus folding always provides them with SOME chance to make the money and so folding is always +EV.
If 5 players were playing where 4 players win a $100 seat for $400 prize pool and even in chips, each person's tournament life is worth $400/5=$80. Since by winning the tournament you only add $20 of value, but risk $80 of value, you need to be a huge favorite or get enormous pot odds to justify a call that risks your tournament life. If other players are certain to bust themselves before you are at risk then there is absolutely no need whatsoever to even play a single hand. But the assumption made in the ICM model is that everyone is equal in skill so you cannot wait for them to bust and at some point you must call.
As such, it is much less correct to call off your chips since folding an all in gives you a chance of making money by other players knocking each others out. Except when one player is an overwhelming favorite, all in confrontations hurts both the initial raiser and the person who pushed while those not in the hand gain. How big of favorite?
In the example, the "bubble factor" is risk/reward of calling off stack. $80/$20=4. This means we need 4 times the pot odds or 4/1 on coinflips or better to call. Alternatively we can take the pot odds in a given spot divided by the bubble factor. That's the win ratio to 1 needed. So if we have 1.2 to 1 pot odds we take 1.2/4=0.30 then we need to win 1/(.30+1) or 1/1.3=~0.7692=76.92% with 1.2 to 1 pot odds to call off our tournament life. Mostly, optimal opponents with high bubble factors will be doing a lot of raising and folding, and very little calling. But the bubble factors will rarely be this high. Usually they will be around 1.2 and rise to 1.6 near the bubble as well as the final table bubble. Following the bubble bursting they drop sharply, but also individual bubble factors exist when facing opponents with different sized chipstacks.
The more chips one has, the greater their opponent's bubble factors and the greater hand or pot odds that an opponent needs to be able to risk their tournament life. The calculation gets complicated with varying chip stacks and a lot more players left with a payout that isn't flat, but the concept remains the same as explained.
Opponents are SUPPOSED to avoid races with those who have them outchipped and require a more significant edge or pot odds to call off their tournament life and the closer to the bubble and the flatter the payout, the more these bubble factors grow. Ironically, because of this, having more chips in turn has MORE value than just the equity assumed by the ICM model against opponents who respect this fact.
Thus, the assumptions that all things are equal is somewhat invalidated. Opponents do NOT always have an equal skill edge in reality. But even if they did, if all opponents are "optimal" and we know they will respect the threat of a player with more chips, then we can see how getting more chips would allow for one to continuously have profitable steal opportunities over the course of dozens, or hundreds of hands due to a slight chip advantage that one could parlay into a greater chip edge, which would lead to larger bubble factors of opponents as the tournament gets closer to the actual bubble and the rate of profitability would grow.
If opponents respect chip utility, having chips is an advantage which leads to additional fear equity which creates an implied advantage not considered by the ICM. If you were to factor that in, players of equal skill actually should somewhat loosen up their standards in order to accumulate chips, since having more chips would more frequently lead to additional steals and an eventual win more often than implied by the ICM model. Players should be more liberally willing to risk their own survival early for chip utility and fear equity as a result of the ICM players tightening up vs the chip leader.
However, the assumptions made with ICM are close enough to be a baseline for decisions from which we can still attempt to handicap "utility". If we want to adjust, we can simply lower the bubble factors so we "gamble" more than our opponents in order to get more chips with the theory that we can parlay that into enough chips to compensate for the equity we gave up by being slightly more reckless than our opponents. We also may not want to be so quick to adjust without looking at the other side of the coin.
On the opposite side of the spectrum if opponents all played based upon chip utility, or played as they did in a cash game and didn't consider bubble factors at all, they would bust themselves much more quickly which would result in a much larger probability of someone playing very patiently making the money. There is also "opportunity costs" you give up when you risk it all not related to just the equity you preserve by folding. For example, on the bubble in the first example with enough big blinds against weak fields, opponents will probably play poorly enough that we never should risk calling off all of our chips.
Opportunity cost of survival takes place early as well. If you can with low variance triple your chip stack just by playing in small pots, by the time you are all in you will have many more chips, and you may not ever need to be all in at all. As such, with this kind of skill edge, any all in carries with it the possibility to destroy your chances of all that future chip accumulation. ICM only assumes equal skill and assumes there is a chance of making money without risking your tournament life. Opportunity costs theorizes that not only do you have the possibility of other opponents taking on unnecessary risk to bust themselves, but you also have the ability to avoid all ins and accumulate chips due to your edge. The slower the structure and the larger your edge and less aggressive the opponents, the more likely it is that you can avoid all ins and thus the more value in passing up high variance situations.
This equity is derived from the ability to accumulate +EV spots even after considering the ICM. It is derived from the profitable skill edge that the player can exploit through both the knowledge of ICM AND the overall advantage over opponents as well as preserving the probability of finding a better hand/spot in the future while also surviving in a condition where opponents probably play the ICM model suboptimal as well, providing more equity than the ICM implies due to their mistakes.
The concept of willingly accepting a less correct than the ICM implies, with the intention of gaining back more in the future is known as "utility theory".
Both opportunity cost and "utility theory" are concepts of knowingly taking the worst of it (or passing up profit opportunity) to get the best of it later. It is like a pool shark who is correct to intentionally lose a $10bet if there is a greater than 50% chance that his opponent will up the stakes to $100 next game and he knows he has a skill edge. If you KNOW opponents will tend to treat the bubble and other situations such that it allows you to steal and resteal profitably far more often over the course of the tournament, than the sum of those future chips gained as a result could easily make up for the decision that was slightly -EV. If these conditions could only be brought about by taking the -EV decision, then it certainly may be correct to take.
The other argument for "utility theory" aside from just the fear equity, is that in reality the skill advantage is largely influenced by the number of chips. The best deep stack cash game players usually have a better BB/100 hand win rate than the best short stackers even when in some instances the deep stackers may be slightly handicapped when playing with other short stacks.
This is because with 100BBs you can still force 20BBs opponents all in, but you can also call with the implied odds to hit and open more hands vs other players with 100BBs. You can float flops, minraise and raise flops, utilize positional advantages over all 3 streets and get paid off. The deep stacker CAN be handicapped by the short stacker as the deep stacker will have to open less often with short stacker to his left as the short stacker has the "last in" all in 3bet advantage that eliminates the option to gain fold equity with a 4bet. The short stacker can neutralize much of the deep stackers edge while having a slight edge himself, but the edge that a good deep stack player has over a bad deep stack player is always going to be greater than the edge that short stackers have over a bad short stacker.
As such, good players with more chips are worth more than good players with less, and any player with an edge still has greater equity than the average in this situation if he is still around.
So Utility Theory and Opportunity Costs are often in opposition with each other. Depending on which concept is more dominant at a given stage and set of conditions will dictate which play is correct. Unfortunately, these concepts are very abstract and impossible to calculate with certainty, so the default analysis is to start with the ICM and determine what "correct play" is assuming optimal opponents and then try to deviate from there. You can approximate or guess which is more dominant and adjust the bubble factors and decisions as a result but that is still speculative and requires an intuitive skill to understand creative, abstract thinking
For example, at a given table if a maniac player to your left is neutralizing your skill edge, while other players are bad enough to continue to let him accumulate chips, the only thing you can do is wait for cards or wait for the opponent to hopefully bust out. If you wait for cards you will have to survive multiple all ins. As such, it may be worth far more taking the worst of it while you still have opponent outchipped if in exchange you gain the ability to steal small pots left and right. Particularly if the conditions are not going to remain such as near the bubble or 3 really bad players that are just putting tons of chips in preflop and on the flop, then folding the turn every hand. If 45% of the time you survive and knock the maniac out you go on to steal thousands of additional chips, the additional gain from those conditions may justify the -EV spot in the first place.
With that being said, I personally think that someone with a skill edge should in general place a premium on their tournament life. They may have to work a little harder to get chips, but they always have the possibility for greater utility in the future anyways, and it may not even take them an all in to win the tournament if the table draws and cards run well.
Neverthless, you cannot apply this knowledge without understanding what the ICM model suggests so we will investigate bubble play and unexploitable GTO strategy given equal stack sizes and bubble factors of 1.6.
Tournament odds divided by bubble factor equals your tournament odds.
Bubble factor of 1.6
pot odds of 1.3125
Tournament odds of 1.3125/1.6=.8203125
With tournament odds of .8203125
Note: I initially calculated calling 20BBs with 46.25 chips in play including your 20 after a reshove over 3x raise from opponent but that ignores the initial raiser. In reality I meant to calculate for calling 17 after raising 3x and a 20x shove but the differences are minor enough to not matter. If you are raising 2.25 this is actually correct to call a reshove when you started with around 16.65BBs left.
You can evaluate the tournament odds in a few key situations to approximate based upon your opponent with a bubble factor of 1.6 which is fairly common on or near most bubbles.
20BBs facing resteal:
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.
Knowing we can only call with 44+,A4+,A2s+,KT+,K7s+,QJ, Q9s+,J9s+ vs any two, we need to raise as often as possible without opponent being able to hurt us by autoshoving any two.
We must get 1.6 more than we risk overall. So for example if we ONLY raised with 44+,A4+,A2s+,KT+,K7s+,QJ,J8s hand range,we'd have 64.84% equity over any two.
So if we called off with this range on average we'd be 64.84% to win or we'd expect to yield about 7.557 in chips.
But we'd win 22.5 64.84% of the time for 14.589 (but the chips we win are worth 1.6 less than the chips we lose)
We'd lose -20 the remainder. After multiplying the chips we lose by 1.6 we get a tournament odds adjusted 3.338 EV when we call on average.
So that means we can afford to lose 3.3378 on our folds when we are pushed so when we raise we have to have the goods 37.47% of the time.
This translates into needing to raise 2.6688 times the percentage of hands we can call off with.
The opponents will tend to 3bet a much tighter range if your raise comes from early position. It's really hard to say what's "correct" because that depends on whether or not opponents assume that you adjust to the bubble factors and whether or not your opponents adjust by 3betting more liberally or less liberally. If you knew your opponent would adapt to bubble factors, it'd be correct to 3bet him far more liberally. If he didn't, you'd actually have to only 3bet with a hand with enough equity where you want to get the money in, plus a few more because you still have some fold equity.
You need to have "the goods" often enough when you raise to be able to "call off" 37.47% of the time and "the goods" is defined as being xxx% to win.
Knowing we can only call with 44+,A4+,A2s+,KT+,K7s+,QJ, Q9s+,J9s+ vs any two, we need to raise as often as possible without opponent being able to hurt us by autoshoving any two.
That establishes a limit to how loose we can raise preflop. So multiply that range by 2.67 and you get about 65% of hands.
Since there are multiple opponents in most situations they can shove a lot tighter and collectively force you to fold more often than the calling range allows since you will have to have a much stronger hand vs multiple opponents that shove say half the time.
So I think I will just assume opponent's 3bet you as they're supposed to which is such that you will be reraised 50% of the time. For reasons that opponents are shoving and we cannot gain any fold equity on them and bubble factors, they should be looser. But if so, we will just get our money in much better when we call so it's not much of a sacrifice.
Opponents should force us to break even on our steals. Bubble factors may change that as if they know we will properly adjust they can 3b more loosely
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.
If everyone has 20 BBs and we make the raise, we need to call off such that we are 53.69% to win I believe. As such the hands that theoretically have value vs "cash game optimal opponents"** given bubble factors
can be calculated and from there we can multiply the value range such that we have a hand often enough to call off adjusted for tournament odds to break even.
So here is value over the range.
53.69% to win vs these ranges with:
8 JJ+
7 JJ+,AK,Aqs
6 TT+,AK+,AQs
5 99+,AQ+,AJs+,KQs
4 99+,AT+,ATs+,KQs
3 99+,AJ+,ATs+,KQs
2 77+,AT+,A9s+,KQ,KTs+
We then look at the EV of each entire range vs our opponents entire 3bet range.
We adjust for tournament factors and then raise often enough where we can fold and break even.
we are 63.13% when we call shove with 2 opponents remaining. But chips we lose are worth 1.6 times as much and as such we need to have value calling shove hands on 42.323% of the hands we raise with. If we raise with 2.362781 times the number of value hands total, 42.323% will be value.
As such we take the value range times 2.362781 to determine how often to raise on the button.
UTG 9 handed we are 68.42% so we can raise 3.52 times the range of hands we are willing to call off with.
Even though we were always looking for a 53.69% chance of winning, sometimes there were very few hands that were just barely in that range and more hands that were way above that amount so the overall equity of our entire range may vary.
Either way, we can either calculate for every single spot how often to raise, or we can just approximate and multiply by some arbitrary amount for the remaining.
Because there are so many other factors, and often there will be opponents with fewer chips that we will be able to call wider with left to act that could push, and opponents should actually adjust and 3bet much lighter due to bubble factors, I am comfortable just applying a 3x multiplier on the hand range for all hands except those we already solved for and also being aware that raising wider and folding tighter, while exploitable to some degree, can be adjusted quickly the moment we have shown we are capable of folding to a 3bet shove.
Additionally, because I think opponents tend not to 3bet nearly as much as they should, you can be a bit more cautious. Opponents have to be aware that if they are called too loosely it hurts both the caller and themselves. As such, they might push a lot closer to the hand range they are willing to call off with vs loose players.
So here's a rough approximation of how our opening range might look like with 20BBs on the bubble with bubble factor of 1.6.
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
**How they should play against you if survival weren't a factor and they were just playing tournaments... Meaning they raise such that they force you to break even on bluffs or 50% of the time. The actual optimal solution involves more calling and the tournament odds factors in.
Although normally I would look for bluff hands that are not likely to be dominated, there also is some consideration to the probability that an opponent has a hand that he can call or shove with (if you hold an ace it's less likely opponent has one) and also there's a chance that a shorter stack will have you committed to calling off your stack. You don't want to have 45s in these circumstances and would rather have a weak ace which at least has a 33% chance vs a pair, and may dominate a few desperate players weak aces. If your pot odds are good enough, you can call after factoring in the tournament/bubble factor.
The bubble factor is much lower when a player has much fewer chips than you do and you already will have some chips in the middle so you will get a discount to call the shove which means it's not that unlikely that you still may call off with the hands that don't have value vs a 20xBB shover, but might over a 14xBB shover.
As such, I just went by pushing hand rank according to the charts at the "Kill Everyone" book by Lee Nelson, Tysen Streib and others.
Facing a 10BB push we will not in the blinds assuming blinds fold we are getting 1.225 to 1.
We have to get 1.6 times better than this to call with the same odds or 1.96 to 1.
Or with the same pot odds we can take 1.225/1.6=.765625 to 1 or needing to be 1/1.765625=56.64% to win.
Pot odds over bubble factor in this case is .765
Facing 10x open shove:
Call off with KK+ vs TT+,AK pushing range
Call off with QQ+ vs 99+,AQ
Call off with QQ+ vs 8%
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.
Let's look at the 'equilibrium" shoving strategies FAR from the money to represent the people who use that autoshove chart without considering bubble implications and see how often we should call.
We will then find out that if opponents play like this, it's correct to shove more often.
Always note that having more chips means bubble factor allows for shoving more vs smaller stacks, less vs stacks that can hurt you and more vs stacks you can hurt that should fold a lot.
In general the big stacks and medium stacks shouldn't really mess with each other, but because they may occasionally raise it's correct to shove a lot wider.
But we are just looking at everyone having the same stack for now and you can try to adjust for these factors by "feel".
Calling off vs opponents who do not adapt to the bubble.
10BB shoving according to pushfoldcharts.com
10 Big Blinds Left
Players With Antes
8 33+ A8s+ A5s AJo+ K9s+ KQo QTs+ JTs T9s
7 22+ A8s+ A5s ATo+ K9s+ KQo Q9s+ J9s+ T9s
6 22+ A8s+ A5s-A4s ATo+ K9s+ KJo+ Q9s+ QJo J9s+ T9s
5 22+ A2s+ A9o+ K8s+ KJo+ Q9s+ QJo J8s+ JTo T8s+ 98s
4 22+ A2s+ A5o+ K7s+ KTo+ Q8s+ QTo+ J8s+ JTo T8s+ 98s 87s
3 22+ Ax+ K6s+ KTo+ Q8s+ QTo+ J8s+ JTo T7s+ 97s+ 87s 76s
2 22+ Ax+ K2s+ K8o+ Q6s+ Q9o+ J7s+ J9o+ T7s+ T9o 96s+ 86s+ 75s+ 65s
sb 22+ Qx+ J2s+ J6o+ T2s+ T7o+ 94s+ 97o+ 84s+ 86o+ 74s+ 76o 63s+ 53s+ 43s
Calling off with bubble factor of 1.6 from the button assuming blinds are on autofold:
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+
6 Big Blinds Left with antes
"shoving equilibrium far from money"
Players
8 22+ A3s+ A9o+ K9s+ KJo+ Q9s+ QJo J9s+ T9s 98s
7 22+ A2s+ A8o+ K8s+ KTo+ Q9s+ QJo J9s+ T8s+ 98s
6 22+ A2s+ A7o+ A5o K7s+ KTo+ Q9s+ QJo J8s+ T8s+ 98s 87s
5 22+ A2s+ A3o+ K6s+ KTo+ Q8s+ QTo+ J8s+ JTo T8s+ 98s 87s
4 22+ Ax+ K5s+ K9o+ Q8s+ QTo+ J8s+ JTo T8s+ 97s+ 87s 76s
3 22+ Ax+ K2s+ K7o+ Q6s+ Q9o+ J8s+ JTo T7s+ 97s+ 87s 76s
2 22+ Kx+ Q2s+ Q8o+ J6s+ J8o+ T7s+ T9o 97s+ 86s+ 76s
sb 22+ Tx+ 92s+ 94o+ 82s+ 85o+ 73s+ 75o+ 62s+ 65o 52s+ 54o 43s
6 big blinds left with antes
need to be 53.78% to call off 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+
What's interesting if your opponents are folding this tight, you can widen your 3bet all in range, which gives the aggressor the advantage and the person with more chips usually has the advantage. If a chip leader has 50 big blinds and everyone else has 20, it's an advantage to push in more widely as a bigger stack because the bubble factors of opponents are larger and thus they must fold far too often such that it's profitable to raise, 3bet or shove with far more hands. Conversely, if there are other opponents with 50 or more big blinds, that represents added risk, which forces you to tighten up. Shorter stacks is an advantage provided there are other equally big stacks at the table that limit the pressure they can apply.
When opportunity cost is greater than utility theory, you effectively are just playing with a slightly larger "bubble factor" than you actually have according to ICM dependent upon how large of skill edge you have. When utility theory suggests future opportunity of having more chips outweighs opportunity cost, you are playing with a lower "bubble factor" than the ICM suggests.
I hope to go back in and fill in some more details and solve for spots such as with 30BBs and also for bubble factors of 1.2 and 0.8 (to get a feel of when the utility and future opportunity of having chips slightly outweighs cost of survival far from the money).
See part 2
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