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%
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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.
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