See Optimal Preflop Poker 1 and Optimal Poker 2 first.
Once the antes are involved, our raises tend to only be around the size of blinds plus antes (2.25x) rather than twice the blinds (3x). As a result, collectively opponents should raise us 50% of the time to prevent us from raising with any two and force our steal attempts to break even. As such, we need to have value over these hands.
How much value? Really, we only may need to call a small raise or 3bet a small amount proportional to the pot to take it down. Even if our opponent puts in 3 times our bet for 7.5 big blinds, we will only have to call 5 more big blinds into a pot that will be inflated to 17.5 after both our opponents and us call. So we only need 28.6% equity for a call to be better than a fold, once we are raised.
If we are in raise or fold mode, 3x our bet is about 1.5 times what's in the pot. To prevent our opponents from breaking even we must be able to raise 40% of the time. In other words we can multiply our "value" hands by 2.5 to form our initial steal range. Because of the antes and the equity both us and our opponent gain on the backs of those that fold, we actually can increase this more. However, we will not try to calculate this just yet.
If we are getting 1:1, our opponents need to raise us 50% of the time to force us to break even on our steal attempts.
Note:If they just call, they are forced to call even more often or make a raise at some point in the future more often or mix in a wider range of call with some mixture of raises this amount or tighter.
That translates to the following hand range given our raise started with X players remaining.
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.
So now what hands have equity over them? If you look at the pot odds after a reraise and you call, you are putting in 7.5 to win 17.5 and that means you need 43% equity assuming it's checked to the river. If you look at hands that have 33% equity over that range since you are getting 2:1 on your calls, then you only need 33% equity vs the calling range. Although once you put in the raise there are hands with more than 28% equity but less than 33% equity that would profit more than folding on calls, that is only attempting to salvage a portion of what already was risked with the steal attempt and it would be more ideal to have never raised at all with those hands. As such, if anything that should only allow you to widen your steal range because when raised you don't always lose the full 2.25 you risk even when your hand may not have value.
Since we don't know exactly what the optimal percentage of raising vs just calling is and how that manifests in equity postflop we cannot know the exact equity one should have since it's impossible to know just how often opponent should call or raise and how often we should call or raise when opponent raises. However, I am very confident the ideal range for when we have "value" vs a raiser is somewhere between 33% and 45% equity.
These numbers need to be much higher with "trouble hands" or "reverse implied odds" hands such as high cards with no drawing potential or weak kickers and drawing potential in an aggressive game, while they retain more of their value in passive games. Suited connectors and draw heavy hands can actually play much stronger when you are the aggressor and have implied odds. In other words, tend to play a lot more suited connector type hands and fewer weak kings and queens and weak aces type hands than the "optimal play" actually suggests.
We needed a way to be completely objective AND approximate optimal play, so we didn't account for actions beyond the flop and assumed all bets break even and are checked to the river. In reality, you can ignore that last bet unless it's break even or better when you are betting on a draw, where you can get paid off when you hit, and you tend to lose that last bet with the one pair, weak kicker hand and get in trouble with opponent has you outkicked or plays to beat one pair.
But I digress.
Opponents may call and defend blinds often enough to minimize the amount gained from our steal attempts in addition to raising occasionally. This probably requires us to have closer to 45% equity and as a result, a tighter strategy is better. However many times both the players in the pot win at the expense of the folders, which favors us forcing the action more often which somewhat counterbalances the need to be tighter to compensate for optimal blind defenders, with allowing looser action to still be profitable. Having positional advantage and postflop edge may allow for more hands, but it simultaneously doesn't require you to play as many to do well and in some cases in tournaments chip accumulation at lower or no risk is better than more chips at higher risk. Since I haven't calculated the optimal rate of defending the blinds, I can't factor everything in perfectly.
I do know that the optimal solution really will allow blinds to defend very liberally (especially for only a 2.25x big blind bet) by calling often. This means our steal attempts won't win the full pot but instead a fraction of a slightly larger pot which translates into slightly less than the full steal vs optimal opponents. However, the optimal solution also will require them to defend pretty liberally after the flop which may allow us to gain quite a bit of chips a high percentage of the time when they don't
So I'm not sure but since you never need to play "optimal" preflop and opponents will never be "equally skilled postflop" plus position does matter this may not be that far off. You also should in reality really consider the opponents and postflop actions and other variables. This is just a baseline for thinking about decision making and understanding how the antes effects play.
So let's provide two guidelines with the optimal play likely somewhere between:
These hands have 33% equity over initial hand range of opponent. (Every single hand in this range is individually profitable vs opponent's range if opponent just calls and checks to the river)
vs range of 3bet defender
8.3% range: 44+,A9+,KT+,K9s,Q9s+
9.5% range: 22+,A2+,KT+,QJ,K4s,Q9s+,JTs
10.91% range:22+,A2+,K9+,Q9+,JT,T9,K2s+,Q7s+,J8s+,T8s+,98s
12.95% range:22+,A2+,K2,Q6+,J8+,T8+,98,Q2s+,J2s+,T4s+,96s+,86s+,76s,
15.91% range:22+,A2+,K2+,Q2+,J8+,T8+,97+,J2s+,T5s,95s+,85s+,75s+,65s,54s
20.63% range:22+,A2+,K2+,Q2+,J7,T8,97+,87,J2s+,T6s+,95s+,84s+,74s+,64s+,53s+
29.29% range:22+,A2+,K2+,Q2+,J2+,T2+,95+,85+,75+,65,54,92s+,82s+,73s+,63s+,52s+,43s
50% range:22+,A2+,K2+,Q2+,J2s,T3+,95+,85+,75+,64+,54,92s+,83s+,73s+,62s+,52s+,43s
These hands have 45% equity over 3bet range. (Will be profitable if opponent occasionally raises and we call and hands are checked to the river even if steal attempts from bluffs are break even).
45% equity:
8% range: TT+,AQ+,AJs (5% of hands)
9.5% range:TT+,AJ+,ATs,KQs (6.5% of hands)
11% range:88+,AJ+,ATs+,KQs (7.4% of hands)
13% range:77+,AT+,KQ,A7s+,KTs+ (11.2% of hands)
16% range:77+,A8+,KT+,A2s+,K9s+,QTs+ (16.3% of hands)
20% range:66+,A8+,KT+,A2s+,K9s+,QTs+ (16.75% of hands)
30% range:44+,A4+,K8+,Q9+,JT,A2s+,K5s+,Q7s+,J8s+,T9s (30% of hands)
50% range:33+,A2+,K7+,Q9+,JT,K4s+,Q8s+,J9s+,T9s (33% of hands)
In either case, we can widen our range considerably compared to non ante stages. That's not even including the "bluff raises" as our steal attempts should break even vs optimal opponent.
If all these hands have "value" vs a hand opponent plays, we should add in bluffs such that we have a "value" hand 40% of the time. That means multiply the hand range by 2.5.
For example, take the 33% equity.
Total hand range:
8:39.59%
7:69.76%
6:88.61%
5 or less players remaining: over 100% or Any Two Cards.
This is approximately:
8:22+,A2+,K8+,QT+,JT,T8+,98,K3s+,Q7s+,J7s+,T7s+,96s+,85s+,76s,64s+,54s
7:A2+,K2+,Q2+,J4+,T6+,96+,86+,J2s+,T2s+,93s+,84s+,74s+,64s+,54s+,43s
6:22+,A2+,K2+,Q2+,J2+,T4+,92+,84+,74+,64+,54,T2s+,92s+,82s+,72s+,62s+,52s+,42s+,32s
5 or less: any two.
OR
5:22+,A2+,K2+,Q2+,J2+,T2+,92+,82+,72+,62+,52+,42+,32
4:22+,A2+,K2+,Q2+,J2+,T2+,92+,82+,72+,62+,52+,42+,32
3:22+,A2+,K2+,Q2+,J2+,T2+,92+,82+,72+,62+,52+,42+,32
2:22+,A2+,K2+,Q2+,J2+,T2+,92+,82+,72+,62+,52+,42+,32
1:22+,A2+,K2+,Q2+,J2+,T2+,92+,82+,72+,62+,52+,42+,32
Button, cutoff, hijack, and even middle position raises any two.
I only multiplied the initial 45% equity range by 2 since I assumed this strategy was probably for a tighter strategy against larger reraises or opponent who bluffs more often and wanted the so called "conservative estimate" in which we want the true optimal strategy to fit somewhere between these two. Also, opponents who 3bet may 3bet with odds to keep you in, but if you choose to fold, you need a 4bet hand more often. So this leads to the following hands as the "tighter end" of correct play during the ante stages.
8: 66+,AJ+,ATs+,KTs+,QTs+,JTs (10%)
7: 44+,AJ+,KQ,A9s+,K9s+,Q9s+,J9s+,T9s (13%)
6: 33+,AT+,KQ,A8s+,K9s+,Q9s+,J9s+,T9s (14.8%)
5: 22+,A8+,KJ+,QJ,JT,A2s+,K9s+,Q9s+,J9s+,T8s+,98s (22.3%)
4: 22+,A2+,K9+,QT+,JT,a2S+,k5s+,Q8s+,J8s+,T7s+,97s+,86s+,76s (32.6%)
3: 22+,A2+,K9+,QT+,JT,a2S+,k5s+,Q8s+,J8s+,T7s+,97s+,86s+,76s,65s (33.5%)
2: 22+,A2+,k2+,Q6+,J7+,T7+,97+,87,Q2s+,J3s+,T5s+,95s+,85s+,74s+,64s+,54s,43s (60.03%)
1: 22+,A2+,k2+,Q2+,J5+,T7+,97+,86+,Q2s+,J2s+,T2s+,94s+,84s+,74s+,64s+,54s,43s (65.8%)
If multiplied by 2.5 instead:
8 12.44%
7 16.21%
6 18.48%
5 27.90%
4 40.72%
3 41.86%
2 75.04%
1 82.20%
That translates into roughly
8 55+,AT+,KQ,A9s+,K9s,Q9s+,J9s+,T9s
7 33+,AT+,KJ+, A5s+,K9s,Q9s+,J9s+,T9s,98s
6 22+,A9+,KJ+, A4s+,K9s,Q9s+,J9s+,T9s,98s
5 22+,A4+,KT+,QJ,JT A2s+,K8s,Q9s+,J8s+,T8s+,97s+,87s
4 22+,A2+,K9+,QT+,JT,T8+,98,87,K3s+Q6s+,J6s+,T6s+,97s+,85s,75s,64s+
3 22+,A2+,K9+,QT+,JT,T8+,98,87,76,K3s+Q6s+,J6s+,T6s+,97s+,85s,75s,64s+
2 A2+,K2+,Q2+,J2+,T5+,96+,85+,75+,64+,54,T2s+,93s+,84s+,74s+,63s+,53s+,43s,
1 A2+,K2+,Q2+,J2+,T3+,94+,84+,74+,63+,53,43, any suited
Because the blinds are correct to defend more liberally and they will
slightly eat into your profit, you can probably raise less often.
However, from an exploitative perspective, and because we don't know quite how often opponents should raise and they sometimes should call, if opponents just call, rather than raise, you can widen the ranges.
If you prefer not to 4bet light, and you don't want to be quite as much of a target, you probably can get away with raising much less often, particularly if opponents fold and you will still be able to accumulate chips. That will allow you to fold a lot more to aggressive 3bets and 4bets and avoid all ins. You may be able to win a tournament without ever having to show down a hand or ever being all in which supports cutting back on your profitability and loose play in exchange for lower variance and less risk, and also not being quite as much as a target by stealing less often.
In either case, we can dramatically widen our raise range but as a result our opponents should have a pretty wide 3bet range and we in turn will need a pretty wide 4bet range.
One problem with this is it assumes stacks are infinitely deep meaning there is no all in move. Since there is, this strategy plays too many hands when a 3 or 5bet all in is perfect stack size for shoving and too few when a 4bet or 6bet or 2bet shove is perfect stack size. Eventually we no longer have equity over opponent and eventually we have to either call and gain whatever equity remains or fold.
Also, because of post flop decisions and it being very unlikely the hand will play to the river, we need to start out much stronger if we want to continue to the river a high percentage of the time that we play.
The 33% equity "value" range also works for flat calling raises from the button. Since we may be squeezed this could be slightly tighter, particularly if a squeeze play is likely going to be all in, but we could potentially play very loose on the button vs raises because we only need 33% equity vs the initial raiser and we have position.
Another decision we may make is facing a 2x BB raise. Although game theory solution is to 3bet often enough to force opponent to break even, we can also look at when we have equity to call assuming the blinds fold.
33% equity to flat on button when facing X% raising range:
8.3% range: 44+,A9+,KT+,K9s,Q9s+
9.5% range: 22+,A2+,KT+,QJ,K4s,Q9s+,JTs
10.91% range:22+,A2+,K9+,Q9+,JT,T9,K2s+,Q7s+,J8s+,T8s+,98s
12.95% range:22+,A2+,K2,Q6+,J8+,T8+,98,Q2s+,J2s+,T4s+,96s+,86s+,76s,
15.91% range:22+,A2+,K2+,Q2+,J8+,T8+,97+,J2s+,T5s,95s+,85s+,75s+,65s,54s
20.63% range:22+,A2+,K2+,Q2+,J7,T8,97+,87,J2s+,T6s+,95s+,84s+,74s+,64s+,53s+
29.29% range:22+,A2+,K2+,Q2+,J2+,T2+,95+,85+,75+,65,54,92s+,82s+,73s+,63s+,52s+,43s
50% range:22+,A2+,K2+,Q2+,J2s,T3+,95+,85+,75+,64+,54,92s+,83s+,73s+,62s+,52s+,43s
We can see that even against tight opponents, it's correct to have a very wide range when calling. However, this range should be tighter because blinds will not always fold, players will not check until the river, and we occasionally will get "squeezed" by the blinds and have to fold. Also, our opponent's hand ranges are not "known", and there are "reverse implied odds" for a lot of these weak kicker hands.
Nevertheless, the concept of occasionally calling on the button can never be that bad, and if we are likely to get squeezed, that favors just calling with premium hands occasionally too. Even though we may let our opponents outflop us or get money in and outdraw us sometimes giving up that risk for value and balance and for the potential to resqueeze shove with aces or getting opponent's entire stack when he hits top pair or decides to continue with a low pair or ace king on low card flops may be worth it.
Since it's mathematically such an advantage to play lots of hands and call, and since opponents do not always adjust, the ante stages is THE MOST profitable stage of the tournament. We want to both maximize our chances to get to the ante stages while also trying to get there with enough chips to be able to make a handful of failed raises and still have over 20 big blinds.
There may be some tradeoff between giving up some probability of getting to the ante stages, and ensuring that when you do get there you have enough chips. Every decision to play or not play a hand may cost us chips. Costing us chips may force us into future situations where we are more desperate which could cost us more chips. So any decision could hurt our chances of getting there. usuaully the decision of playing a hand decreases our chances of getting there more than not playing a hand, but usually not playing a hand decreases our chances of having well over 30 big blinds when we do actually get there. But having more than 30 big blinds is not really necessary since you can clearly get a lot more chips from 30 big blinds, and still be able to raise preflop, Cbet the flop and still have chips left over to do it again. I'm not convinced that having more than 30 big blinds really adds a lot of utility, beyond giving you a few more shots at a particularly level of utility and perhaps sustaining it for maybe one more level of play which you may be able to parlay into two more levels of play. However, you can parlay a tight image into additional value in future levels of play as well and tend to get there more often, so those effects are not really that significant if you know how to use a tight image.
Certainly there is a lot of missed opportunity by not having more than 20 big blinds in the ante stages. However, even with 15 big blinds it's probably only a matter of time before you double up. You can increase your stack significantly with a move in or two to maintain or even grow your chips. Once the double up happens you probably have a really tight image, a lot more chips, and the blinds may be much higher so it probably still is worth quite a lot and can nearly be made up for particularly AFTER a double up with a monster hand when opponents are now forced to recognize how tight we've been, then we can go to work with much less resistance. Even if you are still shortstacked after playing this patiently, you can shove a few times in a row after the double up then scale back for a rotation or two and then do it again. Then you can play ultra tight again and before you know it you pick up a big hand again and no longer have the tight image until you're called again. You win 2 of these spots where you are probably on average of 60-80% to win and now you have the ability even in a fast structure to win with only a few all ins all tournament and in a slow structure, you may be able to avoid ever being all in.
So being all in twice and called with 60-80% to win is not really all that different from being all in once with the 40-65% chance to win earlier on and parlaying it into a lot of chips in smaller pots early.
Either way, we have to recognize that it's profitable like no other to play a much crazier strategy than you'd ever play in a cash game. If there tends to be a lot of pots that make it to the turn and river with lots of bet, a lot of the preflop edge may be neutralized and the equity will have to be higher and a bit closer to the no ante stages. But when opponents start playing more hands, we then can profitably be aggressive to attempt the limp raise or squeze play which is another story to cover later.
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