Player Levels

Poker Strategy
63

Level 41: The Aggression Advantage
There’s a simple version of poker that is frequently used in GTO analysis of poker. It illustrates many concepts in poker, from polarization to the power of balance between bluffs and value, but it also seemed to me to show the basic power of aggression, and how it can in theory enable an over realization of equity.

The “test” game is described in many ways in different texts. For simplicity I’ll just make it a game with 3 cards, A, K and Q, and no streets and no community cards. Each player has $1 behind, and there is always a pot of $1, and players are allowed to bet $1 only, or check, and if facing a bet, call or fold. Player 1 will always be dealt either an ace or a queen, while player 2 just always has a king.

It’s pretty easy to see that the equity of player 2’s holding (K) is 50%, as half the time player 1 will be dealt an ace, and the other half a queen. If both players always check, that equity will be realized.

It is also pretty easy to see that player 2 should never bet. Player 1 knows he always has a king, and knows whether or not he is ahead or behind, and will always fold to the $1 bet if he has a queen, and always call if he has an ace. So a bet by player 2 can never win money, but can lose money.

What happens if player one always value bets the ace, and always checks the queen? Well, initially that will allow player 1 to improve his equity above 50% from the times player 2 calls and loses, but if player 2 sees player 1 bet 100 times, and every time has an ace, then eventually player 2 will fold and equity will be back at 50%.

Conversely, if player 1 never bets A, and always bets Q, if player 2 always folds, that will drive player 1’s equity to 100%, but if player 2 shifts to always calling, that would then drive player 1 equity down, as $1 winnings with the ace are offset by $2 losses with the queen.

So both always value betting and never bluffing, and always bluffing but never value betting seem limited in value. While always value betting can lift equity above 50%, especially initially, to the extent your opponent knows you never bluff, your equity again approaches 50%. Always bluffing is considerably worse; while it can again drive your equity initially over 50%, to the extent that your opponent begins to understand that all your bets are bluffs, it can actually drive your equity below 50%.

But what happens if player 1 always bets the ace, and sometimes bets the queen? Well, it turns out that if he bets the queen 50% of the time, there is actually nothing player 2 can do to prevent player 1 from capturing 75% of the pot, and that player 1 is actually indifferent to the calling percentage that player 2 adopts: at every point between 0% calls to 100% calls player 1 still captures 75% of the pot. Player 2 actually has an ideal calling frequency (the MDF from level 30), in that this is the only frequency that restricts player 1 to 75%. Any other frequency would allow a counter exploit (for example, 100% folds from player 2 would allow player 1 to always bet and capture 100% of the pot).

So by betting with a carefully crafted mix of value and bluffs, you can achieve a larger share of the pot than the equity share of your actual holdings that cannot be denied even with perfect play.

1 Like
64

Level 42: Polarization
Yikes… this is something different. It looks like there are spots where it makes sense to mostly only make certain bets with some of your best hands, or some of your worst hands.

I guess I’d been doing this in certain spots, but hadn’t quite realized how pervasively powerful this could be. Somewhere in the middle levels, I’m noticing now that a lot of my betting ranges were becoming increasingly linear/value heavy, and that the correct adjustment against me if I was playing myself and facing a bet was to just fold a lot more, because my betting ranges were quite strong, even if I would often fire some high equity bluffs (semi-bluffs). It’s like that toy game if I only bet with the ace.

The A/Q v K game seems to illustrate how it is possible to derive more value with a mix of bluffs and value than you can with value alone, and so maybe I just need to start thinking of some spots where I should be polarizing, while also thinking of spots where perhaps I don’t need to.

Spots I might not need to polarize (I can keep middle strength hands in my range):

  • where my ranges are broad and with good coverage, I might not need to polarize (normal pre flop open raises, smaller sized continuation bets…)
  • smaller bets in general seem to not require being polarized
  • if I don’t need to defend a calling range, my raises probably don’t need to be as polarized (as a way of including stronger hands in the more passive lines)

Spots I might want to consider polarizing (bet only with strong hands or junk):

  • probably most large, all in bets
  • 3 bets and 4 bets, especially facing someone raising with a wider range, especially when I don’t always want to raise or fold
  • any time I make larger bets, hands in the middle of my range are usually going to be the least happy making the larger sizes

I wonder if I should ever have a polarized calling range…

65

I have no idea what you are saying here. Facing a 3! or 4!? Making a 3! or 4! when you don’t want to raise? Ya lost me there.

66

Not facing… but when you are making a 3 bet or 4 bet. The “facing” part was referring to the width of the opponent’s raise that you are now 3 betting or 4 betting over. I probably could have done a better job writing that initially…

I’m essentially talking about light 3 betting and light 4 betting here, and really following up on the much earlier 3 betting post, and suggesting that one option for expanding your 3 betting (and even 4 betting range) is to fill in the bottom with some junk. Now some junk is going to work better than other junk (blockers tend to be important here)…

Another way I might perhaps have written that bullet as a multi-bullet item:

  • I’ve typically used a linear range to 3 bet and 4 bet, even though my ranges are broader than they used to be, but I think I’m still overly value centric with my 3 bets and 4 bets
  • if my plan is to only raise or fold (3 bet or fold, or 4 bet or fold) in a given spot (such as you might often want to do out of position), then I might want to stay with a linear or merged range (though there are still some reasons you might want to polarize)
  • if I plan on cold calling from that spot some fraction of the time rather than just raise or fold, then polarizing my raising range provides protection to my calling range by allowing me to include stronger hands in my calling range (some of the upper middle hands get demoted from 3! to call, and some junk takes their place)
67

No, I do understand polarizing, but couldn’t figure out what you were trying to say there. Thanks for taking the time to clarify it.

By the way, great thread. I might quibble a bit about the order here and there, but you already addressed that. It’s a difficult thing to organize, you are doing a better job than I could. Well done sir!

68

Level 43: Leverage
Here’s something I hadn’t thought about much before… It appears that when your opponent has a marginal value hand, bets on the turn (or even the flop) can create more pressure than bets on the river. Essentially, to call on the earlier streets, you don’t really know what kind of price you are getting, as you may need to call bets on later streets in order to realize your equity.

I’d often thought of bets I made on the flop and turn as being easier to call, as I’d always imagine there were draws out there that would just call, but it is interesting how someone with a made hand can be in the opposite situation. So bluffs on the turn can have a bit of power, especially with blockers to some of the main draws, and conversely, you can choose to de-leverage by checking on the turn and then betting on the river, to make it easier for your opponent to call a slightly larger bet, especially if you think you only had one street of value left in any case.

1 Like
69

Level 44: Range Elasticity
OK, what does this mean? Hmmm… OK, something is elastic if it stretches, and inelastic if it doesn’t. It seems they are saying that beginners tend to be inelastic in response to bet sizing, and so will make the largest mistakes facing small bets and large bets, as they are not correctly expanding their ranges in the face of smaller bets, or contracting their ranges when calling larger bets.

In general, at high stakes 5k/10k, where I’ve been playing for a while now, if feels like most people are over-folding to every size, (with some obvious exceptions) but I’m not really sure where the biggest mistakes are being made. Do I want to add a lot of small bluffs, or are the mistakes bigger facing larger sizes?

I wonder if I shouldn’t start tracking call rates against 25%, 33%, 50%, 65%, 100% and 175% pot sizes, and then compare those to minimum defense frequencies, and see where people are over-folding by the largest margins? If they are not at MDF, then bluffs will be profitable, and the bigger the gap between call rates and MDF, the larger the profit should be…

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70

Level 45: IP v OOP Raises
So much of the poker literature seems to focus on a button raise versus a big blind defend, and then an early raise versus a big blind defend. It sounds like in stronger games, big blind heads up post flop versus a raise is the most common scenario. I see that happen, but it somehow doesn’t seem very important it is so relatively rare.

Still, I think the range comparison is important. With the button raise, it sounds like you are typically looking at an opening range of between 40% and 60% of hands, with the big blind defending 65% or more (though I think that is quite a bit wider for both seats that what I usually see here), and so the button has both position and the stronger range. With an earlier raise, the big blind is supposed to tighten his defense range, but if anything it seems that the range imbalance gets even more severe.

Given that, in this type of set up the in position aggressor has a significant range advantage on most flops, and as a result will often make continuation bets, as even when they miss they will often be either ahead, or will be behind but with a fair amount of equity.

But that’s when it is the big blind that is calling the pre-flop raise… it sounds like ranges are supposed to be much more evenly matched when you are raising from an earlier seat and get called by the cutoff or button. Even the button will be forced to call with a stronger range: the big blind is last to act, but the button needs to worry that two players behind might have a hand that they want to squeeze with. The cutoff is in an even worse spot, with an extra player behind, and one that will have position on them if they join in the fun. There’s also the simple matter that players outside of the blinds aren’t getting a discount to enter the pot, and so have less incentive to join a raised pot.

So range versus range matchups tend to be much closer when you raise and get called by someone that will have position on you. Both the smaller range advantage and the positional disadvantage create a situation where c-betting as broadly is not as profitable. If you get more than one caller, that just exacerbates the situation.

Adjustments as Out of Position Raiser:

  • move a few more of my strong hands to checks to defend my checking range
  • move a lot of my trash hands out of my c-bet range and into my checking range
  • play fewer small bets, with more bets close to half pot or more (larger bets will create a lower SPR, which will diminish positional disadvantage)
  • take some check raise lines on the flop with both bluffs and value
  • look for some some similar check raise turn lines
  • continue to play aggressively, but be more defensive than when in position
2 Likes
71

@Yorunoame amazing post will read and read again and again - logically written and for to me to want to learn then I would say easy to follow - Tips Hat - Thanks for this

Tiggs xxx

1 Like
72

Level 46: Value to Bluff Ratios

Still trying to digest the implication of that one toy game with the player with A or Q against the player that always has K. There, with what is effectively a pot sized river bet, having 2 value bets for every bluff creates the highest win rate against a perfect defense. I’d always kind of intuitively felt that you can pepper in more bluffs as you make smaller bets, but found it rather surprising that the opposite is true in this simple case: as your bets get smaller, the optimal mix of bluffs gets lower, and as bets get larger and larger relative to the pot, value and bluffs approach a 1 to 1 ratio.

It does seem that there are quite a few differences between this idealized set up and the real situations we face in poker, though:

  • real players do not defend at the optimal frequencies, and any over-folding to a particular bet size will allow an increase in frequency
  • real betting situations are rarely perfectly polarized, where every hand you are betting with is either better than all possible calling hands, or worse than all possible calling hands
  • if my value bets sometimes get called by better hands, then I start to need more value bets relative to bluffs to restore equilibrium
  • if my bluffs have equity, such that they will win some of the time even when called, then I can start having more bluffs than in a situation where all bluffs are zero equity

The interaction of all of these factors further complicates deciding what the perfect mix of value and bluffs is, but we are able to observe that the strongest human players and AI do seem to have a pretty high mix of bluffs.

As a thought experiment, if over betting the pot by a large margin should allow a nearly equal mix of value and bluffs, why can’t I go all in pre-flop with a nearly balanced mix of AA and 72 off (maybe picking exactly 5 combos of 72)?

  • well, you probably could, but I think AA normally over realizes equity by a large margin, and all you are really fighting to capture with this approach is an “optimal” percentage of the starting pot (the blinds in this case), and so you do better than that just playing AA normally
  • you’d also face occasional split pots where AA gets called by AA, reflecting the fact that you don’t hold the absolute nuts
73

Indeed. With the precious AA, you want a caller to your raise, preferably a player with an inferior Ace so that you can win some chips. Just knocking down the blinds is not enough unless perhaps deep into final table of a tournament with very high blinds and you are a large stack already. For this reason if I pick up AA deep into a tournament in the SB and I have a small stack, I will usually limp rather than raise and just have the BB fold. I WANT the BB to make a stab at the pot on the flop, and if he bluff/shoves, so much the better.

In a ring game, just knocking down the blinds with AA is disappointing. But jamming with 72 has inherent risks. Will KK or QQ ever fold? And if you have 72 as opposed to AA, the chances of opponent holding AA or AK are infinitely greater as you have no blockers.

(One of the biggest leaks I see in many people’s game on RP is that they flop a straight, or hit a straight on the turn and immediately jam, causing opponents to fold and so they do not get fully paid off in implied odds. Even worse, they sometimes jam and get called by a higher straight! AA is a hand you want to get paid for.)

74

Yes, I’ve never actually seen anyone shove pre-flop with a polarized and balanced mix of AA and 72 off… mostly just an interesting thought experiment.

In a more realistic situation that reflects the same foundation of theory, any polarized pre-flop raising range is usually doing the same thing, except that the bottom cards selected represent the bottom of the raising range, rather than the absolute bottom of all cards. Likewise, post flop, you are selecting from the bottom of the range of hands that you arrive at the current street with, and selecting a number of combinations of bluffs that balances the value hands you would play the same way.

75

Level 47: More Balance

OK, I’ve finally moved up to 10k/20k, and I’m starting to see players ranked between 100 and 500 quite a lot. I’ve mostly ignored all of this balance stuff, and have just:

  • bet big with most of my best hands, thinking a little about if my opponent has cards in their range that can probably call
  • checked most of my middle strength hands, though I might try to get thin value with them on a street or two with some smaller bets
  • made a variety of bets with hands I thought were mostly behind, but had a lot of outs (though I’d take some free cards a fair amount too)

As I watch things on You Tube (Poker Bank, Jonathan Little, Doug Polk, Alec Torelli, Poker Out Loud), I see these commentators often asking themselves:

  • when I have a strong hand and want to bet, are there bluffs I’d play this way?
  • when I have a weak hand I want to bluff, are there strong hands I’d play this way?

I suspect on low and middle stakes, and even much of high, these questions don’t matter. I remember on middle stakes playing hands in a way that just positively screamed: NUT FLUSH! NUT FLUSH!.. but would get called down for stacks by middle pair anyway. Gradually, I just started waiting for big hands until that became a habit.

But against strong players that are actually paying attention to what I’m doing, just firing with strong value and some occasional semi-bluffs just doesn’t seem to generate calls in the same way. It’s interesting that many of these strong players will say things like: “without bluffs left in my range, it’s pretty hard to make a big bet here and expect to get called… I’m forced to size down to allow my opponent to call.”

I think I need to find more bluffs, but I’m worried about firing them in the wrong spots, and I’m still not too sure what the right mix is…

76

I missed responding to this earlier. It’s true that you usually want nice blockers in selecting the “bluff” portion of a pre-flop jamming range (or 3! or 4! without shoving), but note that it is not strictly necessary. Note that KK or QQ are in a tough spot here. For every 5 times they call and face 72 off (we picked exactly 5 combinations), they will face AA 6 times. Given the ratio of the bet to the pot (200 big blinds deep, with a pot of 1.5 bb), the ideal call ratio with KK or QQ here (assuming you know exactly what your opponent is doing) is very low. Winning 201.5 big blinds 5 times, and losing 6 times doesn’t work out very well. Of course it won’t quite work out that way:

  • AA has roughly 81% equity against KK and QQ (actually better against KK than QQ)
  • KK and QQ have roughly 87% equity against 72 (again it is strange to see QQ do better than KK here)

So 5 combos of 72o might be a little high given that you are a bigger dog with 72 than you are a favorite with AA, not even counting the rare spots where you’ll be against the final combo of AA. Using the 4 combos of 72s might have been a bit closer to balancing things out. But still, AA hugely over realizes it’s base equity, usually realizing somewhere between 250% to 300% of the starting pot (as opposed to the 80% of the pot that its equity share would command). There might be even better reasons why you don’t see strong professionals doing this, but I’m guessing this is at least part of it.

Oh, and you make a great point about AA and AK. Gee, that’s really the reason this isn’t done: opponents left to act pre-flop don’t have capped ranges yet, and so their range is not bluff catchers only, even with AA/72.

77

QQ has more equity than KK in certain situations because it can make more straights than KK.

TT can make even more straights all the way from Broadway down to 6 to 10. Since all straights must have a T or a 5, TT has a blocking effect preventing other hands from making a straight. And if an opponent to TT does make a straight, any pairing of the board gives TT a boat.

78

Let’s modify this with a bit of a complication. Here is the scenario. The BB has JTo. The UTG limps in (a forced bet since the player just joined the table) and so does UTG +2 and UTG +3. The CO open raises 6.5 BB (i.e., 650K into a 450K pot). The CO is a strong player (maybe he is even currently ranked #19) and that UTG +3 is extremely tight, and only open raises with AA. The other players are unknowns. The effective stacks are 50 BB deep.

The small blind folds. What should the BB do? I have a limited understanding of MDF, but if I understand it correctly, JTo is normally a hand that the BB should defend against a RFI from the CO. I would expect that the CO has a wide range in this spot and that the BB would need to call about <41% of the time to avoid being exploited.

But the complication in this situation is that the BB does not close the action – the three limpers have yet to act. I just don’t see how the BB can cold call this bet. How about a 3-bet?

79

I don’t usually see MDF discussed in the context of pre-flop action, but rather in heads up post flop situations, in trying to decide what cards in your range would need to defend to keep pure bluffs from being profitable. Multi-way, MDF doesn’t apply to any single defender, but is the rate that all players combined need to defend against a bet to keep bluffs from being profitable (which starts to get trivially easy, and hence why it is not usually discussed in this context).

Pre-flop is typically multi-way, and there are really no pure bluffs. So it is more a question of range v range. This particular CO will raise over limpers pretty wide, especially when he sees that the initial limp was forced, and so made with a 100% range. What exact cards are in his range probably depends on how much coffee he’s had at that point, but I’d guess it looks something like this (though note that some of the more speculative hands will see a mix of limps and raises):

image

JTo has less than 38% equity against this range, even if you assume all combos of the weaker holdings, and you under-realize equity out of position, so EV will be quite a bit worse than that. If you were last to act pre-flop, it’s probably a clear defend given the discount on your call, but with the limpers behind you, if there are players that take limp 3! lines very often, it becomes a clear fold. Here at Replay, limp 3! lines are pretty rare, and I think that flips it back to a call, as JT does reasonably well multi-way. Let’s assume most of the limpers have ranges like this, and that 2 of the limpers follow if you flat the CO raise:

image

JTo is still in last place, but should be relatively easy to play. I think you can fold or call and expect both roads to be moderately close in value (though the BB positional handicap will again mean that realized equity will look worse than this).

image

1 Like
80

Very informative to see this visualization. I don’t do much off-table work (except for a bit of reading between hands), and am I always appreciative of those that share this in-depth knowledge. This whole thread has a lot to offer.

The larger question when deciding to call or fold in this situation is how to play post-flop, either heads-up or multiway. In other words, if the BB calls here, it must be based a plan of what to do on subsequent streets. This is where poker resembles chess in that every move is linked to future moves in an overall strategy.

As shown, there are a lot of hole cards in CO’s range that have JTo beat outright and plenty of other hands could beat JTo on the flop. If the UTG +3 calls the preflop raise (this player always limps with KK-22 and AK), then JTo is in even worse shape. Not knowing what the other two limpers would be willing to call with added another dimension of uncertainty.

A cold call here seems too much like making a donation to the CO’s stack, since the chances are high that the BB will be forced to fold on the flop. It is not +EV to call a preflop bet if your plan of what to on subsequent streets is likely to fold. A 3-bet, which was briefly considered, may have eliminated the limpers and may have even forced a fold from the CO, given the BB’s nitty table image. This is a bit of a gamble with JTo, though.

Since the CO has a bigger post-flop edge, I don’t see that there is a better option than either to fold or 3-bet this hand. On that note, I would add “self-awareness” as a skill level. That is, understanding your skill level relative to other players’ skill levels is an important factor in good decision-making. I have seen strong players ruthlessly exploit weaker players without such self-awareness. It is about knowing when you can be beaten.

It is better to be exploited a little than a whole lot. Fold JTo and live to fold another day!

81

Sorry, I had assumed that BB defense and MDF were related. Back to Skill Level 1 for me.

83

I think with your image, you could 3! JTo here and show a clear profit, even if you plan on giving up on most flops. Of course table image constantly changes, and there’s only so long that would last, and I think there are better hole cards to pick for the light end of a light 3 betting range.

I think you make a good point about evaluating your post flop chances. If you think the raiser is a fish, or that some of the other likely callers will just bleed chips, then that makes calling a lot more attractive. In all honesty, I’ll usually just toss JTo as the big blind with limpers still to act behind me, and will typically call if I close the action. JTs I would probably split between calls and 3 bets, mostly calling, in both cases, perhaps 3 betting slightly more when I wasn’t last to act.