Studying with solvers
Solvers have revolutionised poker, and the way we study it. Most pros accept the necessary to study with them nowadays, but not all of them use them optimally in my view. Often they focus too much on specific hands they played, looking to answer the question “Did I play the hand properly?” This is the wrong question in my opinion. While looking at specific hands and spots is a good way to investigate optimal strategy in a concrete way, you really should be focusing on two things:
(1) how should I play my overall range (every hand I could possibly have in this situation) rather than just the specific hand I happened to have on this occasion?
(2) what are the underlying factors and strategic concepts that can be extrapolated from this exact spot to other situations?
To illustrate this I’m going to do something I haven’t before here on the blog, a hand breakdown with a difference. Rather than go through the hand in one go, I’m going to stop at each decision point and give you a chance to think about the spot and what you would do.
A cash game hand
One of my students writes:
“This hand is from a 2/5 live cash game in Vegas. LoJack opens to 15 and I threebet to 45 with black Kings in the hijack. It folds back to him and he calls. Pot is 97 and I have 480 behind.
The flop came 987 two spades one diamond. He checks. My thinking in game was that while I would check this board at a very high frequency, this is one of the hands I want to bet, both for protection and value. On this very wet board, it benefits strongly from folds so I sized up to two thirds pot.
After my opponent folded however, I thought this bet size might be a mistake, as it strengthens my opponents continuing range too much to hands I’m not in great shape against with my kings. What do you think?”
I’ll be back in a few days to give my response, but before I do, ask yourself what you’d do in this situation, and why.
Would you:
(1) Check
(2) Bet small (quarter to third pot)
(3) Bet big (two thirds pot or more)
Assigning ranges
Before considering what we should do on the flop, we should do the following:
(1) Assign ranges to both players based on the preflop action
(2) Evaluate how those ranges interact with the board to see which player (if any) has a range advantage
(3) Based on that evaluation, work out what our strategy should be with our entire range (not just our actual hand)
OK, let's consider the preflop ranges. In these spots, we always know our own range (or at least we should) but we have to guess and make assumptions on our opponent's range. We may know what their range should look like from a game theory perspective, but their actual range may be a lot different in practise.
In this case, my student's range looks like this:
This is a well-constructed balanced polar range consisting of all the strongest hands we want to three bet for value (and don't mind getting 4 bet) mixed in with some suited broadways that flop very well (but can be released if they face a 4 bet), some middling pairs that can flop sets and give us board coverage on boards with no high cards, and a couple of bluffs with blockers that can also flop very well (A5s and A4s). Some of these hands are a mix: the hands with a "1" are always threebet but the others only some of the time. For example 99 is 0.4 meaning we threebet it 40% of the time and flat the other 60%.
OK, now let's look at villain's range. Obviously we can't be sure exactly what the range is, but since villain is a competent reg we can make some assumptions about their range:
(1) It probably doesn't include very strong hands that always 4 bet, likes aces (so to a certain extent the range is capped)
(2) It probably includes other strong hands like KK, AKo and AQo some but not all of the time, as villain will split between sometimes 4 betting these hands and sometimes flatting
(3) It mostly consists of medium strength hands, as well as some suited connectors like 65s that flop very well for board coverage (so it's largely what we refer to as a condensed range: one consisting mainly of medium strength hands)
(4) There are no unsuited weak hands like AJo and down that always get folded to the threebet (as they should) or occasionally gets turned into a light 4 bet if villain thinks we are 3 betting too much, or they're feeling frisky
With that in mind, we get a range that will look something like this:
Evaluating the flop
Now that we have assigned ranges to both players, we can move on to consider whose range hits this flop best. A cursory glance suggests that while the in position 3 bettor has more overpairs, they have less hands that connect with this board than villain, which suggests villain has a range advantage here.
To work out how much of a range advantage, we would need to look at every single hand in both players ranges and work out how often it's likely to be the best hand on this flop, and to stay ahead on every possible turn and river. Luckily, solvers can do this at the click of a button. Loading these ranges into PIO and letting it do the work for us, we learn that in pure equity terms the out of position player has a clear advantage, almost 55/45.
Looking at the equity of both ranges is a good place to start when considering range advantage, but EV is more important. EV represents how much a particular hand will win on average (averaged across every possible run out and opponent holding). Luckily, the solver is also very good at evaluating this for us. In this case, PIO tells us that out of position's overall EV is just over $60, and hero's just under $45. Note that these always sum to $105 (the current size of the pot) but it's actually possible for an individual hand to have an EV greater than the current pot! For example, if we are lucky enough to have threebet 99 and got this flop, our EV is now almost $200, because not only can we expect to almost always win the current pot but sometimes our opponent will put more money into the pot which we can also win. By contrast, while KK is a very strong hand on this board, it's EV is only a little over $75, because not only is it vulnerable but if a lot more money goes into the pot it won't be the best hand as often, and even when it is it's vulnerable to being outdrawn. Note that tens actually has higher EV (over $85) because if it's currently behind it has a better chance to make a really strong hand like a straight, so it plays better in big pots.
Now let's look at it from villain's perspective. Firstly they have slightly more sets on this board. When they do flop a set though, they'll make slightly less on average by virtue of being out of position, We said above that if the 3 bettor flops a set of nines they can expect to make almost $200 on average, but if out of position does, their EV is just over $185. Despite positional disadvantage though, some hands are worth more out of position. For example, we saw if in position player has KK it's worth about $75, but if villain has KK it's worth over $85, by virtue of the fact it's the best hand more often and less likely to be outdrawn. The nuts on this board is JT, and if we look back to the ranges we assigned, we can see in position player never has this hand, but out of position can have JTs. These are the highest EV hands possible on this board, JTs spades (nut straight and flush draw) is worth over $233, JTs diamonds (nut straight and backdoor flush draw) is worth a little over $210, while the other two JTs combos (nut straight but no flush draw) are worth about $208.
We can summarise all this as follows:
(1) Out of position has a clear equity advantage and an even bigger EV advantage on this flop
(2) Out of position also has more really strong hands like straights, sets and two pairs (this is referred to as the nutted advantage)
Because of this, out of position can play this flop very aggressively, and in position has to be a lot more cautious. The first thing the villain (out of position player) should be asking themselves is given how good a flop this is for their range, should they be donking some hands, or should they always check to the raiser?
That's the question I want you to ask yourself now. Imagine you're the villain this case and you have the range illustrated above, what hands if any would you donk, and what sizing would you use?
I'll be back again in a few days to look at what the solvers says the villain's strategy should be, and to explain why.
Donking strategy
To donk or not to donk, that is the question. After I posted part 2, there was some discussion about this on Twitter and Facebook. Many felt they'd check their entire range to the raiser, intending to check raise all their strong hands. That was a very viable strategy a few years ago when preflop aggressors cbet far too often (sometimes 100%), but the problem with that strategy nowadays as Andy Hills pointed out on Facebook is that any competent villain will recognise this is a much better board for out of position, and will just check behind a lot. Because of this, as out of position, we want to go ahead and lead many of our strong hands. This has two advantages: we get more money into the pot when we have a strong hand (not allowing opponent to check behind), and because we have a lot of hands we are leading for value, we get to do a lot of semi-bluffing.If we lead a hand like AT, opponent may fold a stronger hand like AQ, which is obviously great for us.As a general rule the more value bets you have the more other hands you can also lead. I say other hands rather than "bluffs" for a couple of reasons. First we have a lot of strong semi-bluffs (like ATs) that would like opponent to fold but don't hate if he calls either because they can make a very strong hand on a later street. Second, we have some weaker hands (like underpairs to the board) that aren't exactly value bets or bluffs but still benefit when opponent folds. If we donk a hand like 22, we can be pretty happy if opponent folds AQ which is behind but has lots of equity, and betting and forcing him to put more money into the pot to continue is a better result for us than checking and letting him check behind and get a free card.
Now that we have established this is a board out of position should donk (lead) a lot, the next question is bet sizing. When choosing a sizing on flops, there are a number of factors to take into account:
(1) The bigger our range advantage the bigger we go as far as sizing is concerned. In this case out of position has a significant but not massive range advantage: in position has some sets, all the overpairs and therefore has a few hands that can face a lot of heat and a lot more that can call at least one bet. In these circumstances, you generally don't want to bet too big, which has the further disadvantage of being easy to play against (opponent can just continue with strong hands and fold weaker ones)
(2) The bigger we bet, the more we get to bluff. In situations where we have a relatively small number of very strong hands and lots of hands we'd like to bluff, we generally size up (thereby polarizing ourselves). Conversely, in situations where we want to bet a lot of hands not just very strong ones for value (to prevent opponent checking behind) we generally choose a smaller sizing, with less bluffs. The more equity our bluffs have as semibluffs the more of them we can bet at the smaller sizing. This is clearly such a situation, so it's no surprise that even though I set the solver up so that it could bet third pot, half pot, two thirds pot and 1.2x pot (overbet) it only ever really uses the smallest sizing, third pot.
So without further ado, let's look at the solver recommended donking range:
The first thing to note here is just how often the solver advocates donking: roughly 60% of the time.
The next thing to note is that every single hand in its range is what is called a mix, meaning we sometimes check sometimes bet. The reasons for this are as follows:
(1) We don't want to bet all our strong hands all the time. If we do this, what does that it mean when we check? It means our checking range becomes a check fold range. To avoid this, we have to hold some strong hands back that we can check raise (and check call).
(2) If we take the same action every time with a certain type of hand (for example Tx that is openended), then we are exposed on some turn cards when we have taken the other action. For example, if we always bet our open enders, and our opponent picks up on this, then when we check and the turn is a Jack, we can't have the straight.
In the above image, the red bar for each hand on the grid represents how often we bet at equilibrium, and the green bar how often we check. You probably won't be too surprised to see that our strongest hands like straights and top set are bet the most frequently (but not always: as noted earlier we need to be able to check call or check raise them sometimes). Our strong draws are also bet very frequently, because they like when opponent folds (and we win without needing to get there) but don't mind getting called (they can still get there) or even raised (they can call a raise). By contrast, our least frequent leads are hands that don't have much going for them (but may have some showdown) like A3s or KQs.
Ok, now let's flip seats again and asks ourselves how in position should play versus a donk lead. This is something many players struggle with (which is a strong reason to implement don leads into your game). Look back at the preflop range in position has and ask yourself which hands they should fold to a donk lead, which hands they should call, and which hands (if any) should be raised.
I'll be back in a few days with solver results.
Defending against donks
Here's how PIO responds as the in position player facing a donk lead foe half pot:
Overall, PIO advocates folding about 25% of our range, calling 62%, and raising 13%. The first thing to notice is that this is a lot of folding. Because the out of position player is betting half pot, if we fold more than a third of the time, they're making money on the lead even if they never win the pot when we don't fold. Folding 25% is getting up there towards that, so once you add in the times we continue but out of position wins the pot you can see just how profitable the lead is.
Looking at the grid, you can see that most hands are a mix, meaning the solver advocates mixing two or sometimes all three of the possible responses (raise, call or fold). The only hands that have a pure response are 88 (middle set) and 66 (underpair with an open ender) which are only ever called (on a practical note, any time you see a low frequency play of under 5% like the raises for both these hands in the image above, you can ignore them as they usually tend towards zero when we let the solve run for longer, and even if they don't you sacrifice little or no EV by not implementing these low frequency plays and make your strategy much easier to implement).
Some hands mix calling and folding, for example A4s. This is slightly deceptive as it's a pure call when we have a flush draw (A4 spades) and a pure fold otherwise. Note that a backdoor flush is not enough to call in this case (this is a mistake a lot of players make thinking any nut backdoor can't be folded)
By contrast, AQs and KQs are always called when it has a flush draw, but we don't fold the diamond versions that have backdoor flushdraws. So what do they have over A4s that makes them strong enough to continue? The answer is twofold: the queen is an additional overcard to the board that will pull us ahead more often if we hit it, and it also gives us an additional backdoor straight draw. PIO advocates mostly calling (80%) and raising the rest of the time with these hands.
AKs is folded without a backdoor, which is another mistake I see a lot of players make thinking the hand is strong enough to call flop and see what develops on turn. AKs in diamonds are mostly called and occasionally raised, whereas AKs spades doesn't want to fold or reopen the betting so is always just a call.
AKo, AQo and KQo are generally just folded unless they have a spade to give them a backdoor flush draw. When they have a spade they're essentially very similar to AQs diamonds, so it's not surprising the strategy is identical: mostly call, sometimes bluff raise.
AJs, KJs and QJs are stronger hands again since they have a gutshot to ho with their overs (and in some cases a flush draw or a backdoor flush draw). This added equity means we no longer want to raise diamonds as much and risk facing a reraise so we just call, and now the hands with no backdoor flush draws are the ones that are occasionally called but sometimes bluff raised. The spade version of these hands are now strong enough to be played aggressively, and the solver raises AJs 60%, KJs 16% and QJs 9%.
ATs and KTs are open ended. With a flush draw they are strong enough to be played aggressively (ATs spades raises 60% and KTs 30%). Without one they are just called.
The overpairs are all mixes between calling and raising, and the presence or not of a spade in our hand is pretty crucial.We are more inclined to raise aces with a spade (40%) than without (15%). Kings with a spade also raise 40%, but without a spade are raised only 10% of the time. Queens is interesting because it's almost never raised, which is often the case with strong but not super strong made hands with weak backdoors. Jacks on the other hand is a stronger hand on this board with its gutshot. With a spade it's raised 40% and without 15% (the exact same strategy as aces). Tens also raises more frequently with a spade (40%) than without (5%).
That leaves us with the sets. As stated earlier, 88 is a pure call. 99 on the other hand is normally raised: with a diamond (blocking backdoor flush draws and pairs) it's raised 92%, without the diamond only 57%.77 with a spade is raised two thirds of the time, but without one it's never raised.
OK, so that's how we respond if the villain donk leads into us. But to return to the question posed in part one, what do we do with kings (and the rest of our range) if checked to?
Would you:
(1) Check
(2) Bet small (quarter to third pot)
(3) Bet big (two thirds pot or more)
I'll be back in a few days with answers.
Continuation betting and sizing considerations
We have already said that this isn’t a great flop for us so one we need to check behind a lot. Assuming both players are playing GTO ranges our range consists of:
(1) Some sets. We can have all the sets but we would also call 99 88 and 77 some of the time so actually we don’t have many sets. We have no straights so sets are our strongest hands
(2) We have no two pair but all the overpairs which are strong but vulnerable on this flop.
(3) We have no other one pair hands so all that remains are overcards and hands like A5s. With not much Tx and no 6x apart from 66, we don’t have many open ended straight draws. We have some Jx type hands that give us two overs and a gutshot, some flush draws and back door flush draws, plus a lot of hands that have only two overcards
Our opponent on the other hand has:
(1) Made straights (JTs and 65s), all the sets, and some two pairs (98s and 87s)
(2) Not as many overpairs as we have but some pair and a draw hands (T9s, 76s) and more underpairs
(3) A lot less high card only hands
It’s easy to see when we break it down like this that our opponent’s range is stronger. PIO indicates that overall they have a 55/45 equity advantage on this board and because of the nutted advantage an even bigger EV edge (60/45).
This means as discussed in the previous section that the opponent can donk a very high frequency on this board having a lot of strong hands that don’t want to let us check behind, and a lot of good bluff candidates that would like us to fold but can improve to winning hands if we call. PIO donks 60% here and checks only 40% as the out of position player. I was interested to see what it wanted to do in position when checked to. As we both expected, it checks a lot (58% of its range). The interesting thing is that when it does bet, it always uses the small sizing of one third pot. I gave it the options to bet half pot, two thirds pot and 1.2x pot but it never uses this size.
This might seem a little counterintuitive at first when we think about just our value range and how much a lot of it wants to “protect” or deny equity to our opponent. However, when we expand our considerations to our opponent’s range (which contains a lot of very strong hands that can call no matter how much we bet) and our own bluffs, it suddenly becomes clear why the solver prefers the smaller size. We struggle to find a lot of good bluffs on this flop, and the bigger we bet the more of them we need to find. Even when we choose the smaller size of third pot, to get up to the required number of bluffs, PIO ends up having to use some hands that don’t seem like great bluffs. For example, a hand like KQs that has no flush draw front or back door is still bluffed almost three quarters of the time.
That combination of factors makes this a spot where even if we think our big hands want to bet big for protection, a better way to play our overall range is to use a smaller sizing of one third pot.
But does anyone actually donk?
Recent Chip Race guest Callum made the excellent point that while the above analysis is predicated on out of position having an aggressive donking strategy advocated by the solvers, in practise most players don't have a donking strategy, and always "check to the raiser". Callum went on to look what happens if we node lock (fix) the out of position player's strategy to checking entire range, and the result unsurprisingly is that we now check behind a lot more.
As you can see, the only hand in our range that really wants to bet now is 99 (top set), and it balances this with some low frequency betting of other hands. Given that 99 loses less than a dollar in EV by checking rather than betting, we can simplify our strategy to simply checking our entire range behind. This is a very common exploit in position versus opponents who don't donk on boards that clearly favour them: simply check behind, take a street out, and see what they do on the turn.
Callum also points out that even when players do have a donking strategy, it's very hard to be balanced. In practise, most players will lead their strong hands (the ones that really don't want to allow a check behind) but won't find enough bluffs, making the donk leading range too strong, and by extension their checking range too weak. Faced with this strategy, the solver now flips to betting 100% of the time when checked to. This is a point I always try to hammer home to my students: if you bet all your strong hands out of position, that's all well and good when you have a strong hand, but means when you check your range is too weak to take any heat, and makes you very easy to exploit. Conversely, if you're playing against someone who always bets their strong hands, then always pounce on their checks.
On to the turn
Before we look at specific turns. let's ask ourselves whose range is stronger after the flop goes check bet call. As noted before, this is a very good flop for out of position, and when it comes down they have an equity advantage of almost 55/45. Now let's consider what happened on flop, and also what didn't happen, and how this affects the ranges:
(a) Out of position chose to check rather than bet (donk). Given that he will be donking many of his stronger hands (with some weaker ones for balance), this weakens his range somewhat to 52/48
(b) In position elects to bet rather than check. This removes weak hands he's giving up with from his range so he now has a slight equity edge
(c) Out of position elected to call the bet rather than raise or fold. This removes some strong hands that would check raise, and all the weaker hands that just give up, so overall out of position's range is strengthened (to 52/48 again)
OK now let's have a look at the specific turn cards.
3s
This is a scare card but not one that hits either player that much better than the other. Ii's slightly better for in position who has more flushes (11% of the range compared to 9%) so the overall equity advantage for out of position is reduced to a smidgen over 50%. Given that this isn't a good card for out of position and it introduces more nutty hands into in position's range, out of position has no leads on this card, checking 100%. In position conversely gets to do a lot of betting now, and uses a hefty two thirds pot sizing with 62% of its range (comprised mainly of flushes, sets and overpair, balanced with a lot of ace high, king high and air both with and without a spade).It also bets some weaker hands (11% of the range) at a smaller one third pot, and checks the rest of the time. With black kings specifically, the solver always bets again, usually (over 85%) at the bigger sizing.
3c
This bricky looking card is actually very favourable for out of position, as it does nothing to improve in position. Out of position's range advantage swings back up over 56%, and as a result can lead some hands (12% of the range bets one third pot).This is mostly overpairs and better mixed in with some air. Facing a lead, in position mixes raising and calling with kings, mostly raising with the spade and calling without. Facing a check, in position bets less frequently than on 3s. The bigger sizing of two thirds pot is now only used 37% of the time, the smaller sizing 18% and the rest of the time we check behind. With black kings we actually start throwing in some checks (15%), we bet third pot 28% and two thirds the rest of the time.
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