1. Introduction
This is Part 12 of the article series "PLO From Scratch". The target audience is micro and low limit players with some experience from limit or no-limit Hold'em, but little or no PLO experience. My goal with this series is to teach basic PLO strategy in a systematic and structured manner.

First, let's talk about the status for this article series. There's been a long break between Part 11 (June 2010) and Part 12 (January 2011). The reason for this is mainly that I spent a lot of time working on NLHE theory during the summer/autumn 2010, and I used this work as basis for an NLHE article series. The PLO series was put on hold while this was going on, but "PLO From Scratch" is very much alive. The plan is to publish several articles throughout the spring of 2011, and also finish the bankroll building project for the series (building a bankroll from 50 BI =$250 at $5PLO, and stop the project when we have 50+10 BI ($5000+$2000 =$7000) at $100PLO, ready for a 10 BI shot at $200PLO).

The practical part was also put on ice while I was working on NLHE theory. This grinding project is done on my spare time, and the last months have been busy for me. But it's important for us to finish the grinding project, and the plan is to kick the grind up a notch and log more sessions in 2011. Status for the project as of Jan 01 2011 is that we're done with the first limit ($5PLO), and most of the next limit ($10PLO). When the bankroll has grown to 50+10 BI ($500+$250 =$750) for $10PLO, ready for a 10 BI shot at $25PLO, I'll update the article series blog and discuss the results there, before we move on to $25PLO. I hope to get this done in January/February 2011.

Before we define the topics for Part 12, let's summarize what the previous articles have covered:

  • Part 1: Introduction
  • Part2, Part 3, Part 4, Part 5, and Part 6: Preflop play
  • Part 7: About online low limit PLO
  • Part 8, Part 9, Part 10, and Part 11: Postflop play

I consider our treatment of preflop play as finished, and the remaining articles in the series will be about postflop play, plus one final article where we summarize the series and discuss the results from the bankroll building project $5PLO to $200PLO. I am not 100% sure about how many articles remain, but I expect there will be at least 4 more before we end this series.

What have we covered so far in the area of postflop play?
  • Part 8 and Part 9: General discussion about postflop planning, and a simple postflop planning model (assess the situation based on the number of opponents/position/SPR/equity) to help us with this
  • Part 10 and Part 11: We discussed c-betting, with many examples of c-bet decisions in singly raised, 3-bet and 4-bet pots

These are important topics, since these scenarios occur frequently. Our approach in Part 8 - Part 11 was generally practical. We have talked about some fundamental principles for postflop planning and c-betting, and then we illustrated them with many thorough examples. The examples served both to illustrate general concepts, and to provide practical guidelines for how to play postflop in typical and frequently occurring scenarios.

We'll continue our discussion of postflop play in Part 12 and future articles, but we start by taking a step back from the specific and towards the general. In parts 10-11 we looked specifically at c-betting, and illustrated the discussion with many examples. In the articles to come our discussion will be more about postflop betting in general, and we won't distinguish sharply between different types of scenarios (e.g. whether we're c-betting as the preflop raiser, donk-betting into the preflop raiser, or betting out in a limped pot). We'll focus on learning sound PLO thought processes that tie together the postflop play over all streets from flop to river. And then we'll pause here and there and look more into the specific details of various common and important postflop scenarios.

Part 12 will be more theoretical than the 4 previous postflop articles, but there will of course be examples along the way. Hopefully, we'll find a good balance between theory and examples that will help you understand why PLO postflop betting strategies are the way they are. As a part of our discussion of postflop betting in this and future articles, we'll also return to the important concept planning over multiple streets (that we began discussing in Part 8 and Part 9).

To avoid biting over too much at once, we'll mostly talk about heads-up scenarios to illustrate how theoretical concepts are put to use. But we'll also generalize to multiway scenarios along the way. Below are some important big picture ideas that we'll talk about in the articles to come:
  • How the concept "the strength principle" is used in PLO (different from NLHE)
  • Why the concept ""way ahead/way behind" (and the corresponding WA/WB postflop line) is less useful in PLO than in NLHE, and the consequences this has for PLO postflop betting
  • Bet-folding versus passive play when we have a medium/weak hand that is often best (a consequence of the two topics above)
  • The concept "the betting machine", and why this is a good mindset to have when we're starting a postflop betting process in PLO
  • Simple hand reading based on relative strength (which is simpler than you might think, especially when you have position)
  • Looking ahead, and having a plan for the next street and the different cards that can come

In these topics there will also be included plenty of discussion about the transition from flop to turn, and from turn to river. Instead of talking about turn play/river play in isolation, we'll view these topics more as a continuation of flop play and the postflop planning process we began on the flop. Just like good preflop play sets us up for profitable postflop scenarios, so will good flop play set us up for profitable turn and river situations. But we will of course also talk about some turn/river specifics (for example, how to play the river with a bluffcatcher heads-up and out of position).

So we choose to discuss play on the different streets in a unified way, and be very aware that the scenarios we find ourselves in on one street always are consequences of choices made earlier in the hand. The strategy for the street we're at right now is therefore connected both with the plan we started out with earlier in the hand, as well as all possible scenarios we can find ourselves in on the next street (and we have to be prepared for all of those).

We'll complement this "holistic" way if thinking about betting by also discussing some concrete postflop scenarios in some detail. Some specific postflop topics we'll talk about are:
  • Checking behind on the flop as the preflop raiser instead of c-betting. Why do we do it? Which hands do we bet and which hands do we check?
  • Situations for bet-folding
  • 2- and 3-barreling (bluffing over multiple streets)
  • Donk-betting on the flop (betting into the preflop raiser)
  • Checkraising
  • Playing medium/weak hands in position and out of position. When do we play them passively, and when do we play them aggressively? When should we fold them, when should we take them to showdown, and when should we turn them into bluffs?

To summarize: The plan for this and the future postflop articles is to talk about postflop betting in general. We'll use theoretical concepts and "big picture ideas", while also giving practical guidelines for various specific scenarios that occur frequently. But we'll try to avoid getting bogged down in details. We first and foremost want to learn general betting concepts that apply to a wide variety of situations.

We start this process in Part 12 by discussing a theoretical model of betting heads-up in position ("the AKQ game"). From this model, we'll extract some important betting concepts, and then we'll use these concepts in the scenario where we're the preflop raiser in position in a heads-up pot, having to make a c-bet/check decision after our opponent has checked to us.

This is a c-bet/check decision that occur frequently (for example, after we openraise on the button and get one caller in the blinds), and we'll look at the c-bet- and check-behind ranges for this situation in Part 13. In Part 12 we'll warm up by looking a t a special case of this scenario, namely the way ahead/way behind (WA/WB) scenario. We'll compare the WA/WB scenario in PLO with the WA/WB scenario in NLHE to illustrate an important difference between the two games: In PLO we're more often happy to win the pot right there, without extracting more value from weaker hands, and so we more often prefer that they fold when we bet.

Part 12 contains some theory and discussion that at first glance might seem unrelated to PLO. But the point of this article is to build a theoretical foundation for future discussion, so please be patient. In Part 13 and future articles we'll use the concepts from Part 12, and we begin in Part 13 with a discussion of c-bet-range and check-behind-range when we c-bet heads-up and in position. When we have understood this scenario well, we'll have learned a lot about good PLO thought processes for betting the flop. Then we'll apply this knowledge to future articles where we'll look at c-betting out of position, planning 2- and 3-barrel bluffs, donk-betting, checkraising, and various other topics.

We begin with a presentation of the general strength principle for poker betting:

2. The Strength Principle
The Strength Principle is a macro concept for betting in all forms of poker:

- Bet and raise with your strongest hands
- Check and call with your medium strong hands
- Check and fold, and occasionally bluff, with your weakest hands

Defaulting to betting with our strongest hands is obviously correct, since we want to profit from getting called by weaker hands. And with our very weakest hands, we usually give up the pot postflop, but sometimes we bluff. Note that bluffing with our weakest hands makes things easy for us when our bluff doesn't succeed. For example, if we bet a worthless hand and get raised, we can fold without having to worry about folding the best hand.

So the class of hands that gives us the most difficult decisions is the class of medium strong hands. These are hands that are sometimes best, sometimes behind, and where a bet often doesn't work well neither as a valuebet (since few weaker hands call) or a bluff (since few better hands fold). With this class of hands it intuitively makes more sense to keep the pot small, and try to get cheaply to a showdown and see if we win.

But as we shall see in this and future articles, this way of thinking often breaks down in PLO. So in PLO we'll often bet medium strong hands simply because we want to give the opposition a chance to fold weaker hands (since they usually have decent chance of drawing out on our hand, if we allow them to stay in the pot). Only getting called by better hands is not a big problem for us if our bet forces our opponents to fold sufficiently many weaker hands that have decent equity against us. This is often the case in PLO.

And we also haven't got any problems turning a medium strong hand (for example, top pair with nothing else) into a bluff against better hands when we suspect our opponents aren't very strong. It's hard to call big bets in PLO when you don't have the nuts, since the nuts is often in the hands of someone else. So the assumption that few better hands fold when you bet a medium strong hand doesn't work as well in PLO as it does in NLHE. More about these concepts later.

The next concept in this article is "way ahead/way behind" in an NLHE-setting. Then we'll study a game theory model of betting in position (the AKQ game) to build a theoretical foundation for understanding the strength principle and the way ahead/way behind-scenario. When we have understood the strength principle and the way ahead/way behind scenario in a NLHE setting, we'll turn to PLO and see how these concepts work there.

I have chosen to take the route from general poker theory via NLHE to PLO to build an general understanding of these concepts since most new PLO players already know basic NLHE strategy. By taking these steps, it will be easy for us to understand principles of PLO betting, based on our understanding of well-known NLHE betting principles, plus an understanding of how the structure of PLO (4-card starting hands and pot-limit betting) differs from the structure of NLHE (2-card starting hands and no-limit betting).

3. The way ahead/way-behind scenario
In NLHE it often makes sense to use the concept "way ahead/way behind" (WA/WB) and the corresponding WA/WB postflop line (play passively, and try to get to showdown in a small pot). These are scenarios where we have a medium strong hand that is either far ahead or far behind, typically on dry flops with no draws. In these situations we don't expect worse hands to call or better hands to fold if we bet. The logic is then that a bet does not work well as a value bet, and it doesn't work well as a bluff, so it's best to check the flop, planning to get to showdown in a small pot and see who wins. If our opponent bets the turn after we check the flop, we are prepared to call a lot, since we expect our check to have induced bluffs from our opponents. But our goal is still to get to showdown without building the pot big.

Below is an example to illustrate the WA/WB postflop line in NLHE:

Example 3.1 A way ahead/way behind situation in NLHE

Preflop:
$100NLHE

Hero ($100) raises to $3.5 with K K on the button, big blind ($100) calls. Hero expects big blind to play the top 15% against a button raise, and that he would have 3-bet {99+,AQ}, which is the top 5% of hands. So our starting assumption is that the big blind's preflop calling range is the top 5-15% of hands (top 15% minus top 5%).

Flop: A 9 2 ($7.5)
Big blind ($3.5) checks. Should Hero c-bet or check behind? What should his overall postflop plan be?

We flop the best possible underpair on an ace high and very dry flop. We're assuming that Villain has a medium strong range with many Broadway hands (ATs, KQ, JTs, etc). Since the flop is without draws, and since we have at most 2 outs when behind, we can conclude that:

- We're either ahead with very good equity
- Or we're behind with very poor equity

Thus, we're either way ahead or way behind. One assumption we made about Villains range is that he would have 3-bet any pocket pair AA-99. So if he has a pocket pair now, it will be 88 or lower. If he doesn't have a pair, he has two cards that can't beat us by spiking a pair on the turn. It seems reasonable to assume these low pocket pairs and undercards will fold on the flop if we c-bet. It's also more or less irrelevant to us whether or not these hands get to see a free showdown, since they almost never draw out on us anyway.

Those times Villain has us beat, he as at least top pair, and then our equity is very poor (we have at most 2 outs). It's also obvious that Villain won't fold these hands if we bet. So we see that a c-bet doesn't do anything good for us, regardless of whether Villain has a better hand (he never folds), or a worse hand (he will fold, but he won't make a mistake by doing so, and he would rarely have drawn out on us anyway with these hands).

We can formulate our problem like this: The small equity we gain from folding out Villain's weak hands, thus denying them their tiny chance of drawing out, does not compensate for our big loss those times we bet into Villain's better hands. So intuitively, a flop check seems better than a c-bet. We'll now do some simple modeling to show that this is indeed the case.

Modeling the EV of c-betting and checking
We'll assume the following:
  • Villain's range is the top 5-15% (top 15% minus top 5%), based on the ProPokerTools' hand rankings
  • He will fold any hand worse than top pair if we bet (and remember that he doesn't have QQ-99) in his range, since he would have 3-bet them preflop)
  • He will call down with all hands top pair or better if we c-bet the flop. In other words, he will check all streets, planning to call if we bet, and he will never bet himself
  • Our postflop strategy is to either a) c-bet pot (7.5 bb), planning to check down if we get called, or b) check the flop, planning to check the hand down

Note that we're now working with a model of reality to estimate whether c-betting or checking has the highest EV. And we're making some simplifying assumptions in order to get a model that is easy to work with. Note that it's reasonable for Villain to check-call the flop with the hands he doesn't fold, since these are mostly medium strong hands. He would have 3-bet AA/AK/AQ/99 preflop, and he can't have 22 (bottom set), A2 (top and bottom two pair), or 92 (bottom two pair) with a top 5-15% range.

Now we turn to our poker software toolbox (I have used EVPlusPlus.com) and compute equities and range distributions. Below is a summary (our hand/Villain's range to the left, equities in the middle, and the number of combos in Villain's range to the right):


KsKh 51.6%
Top 5-15% 48.4% 96 combos

KsKh 8.6%
Top 5-15% & (A*,99,22,92) 91.4% 47 combos (49.0%)

KsKh 93.5%
Top 5-15% ! (A*,99,22,92) 6.5% 49 combos (51.0%)

EV (c-bet and check down when called)
=EV(villain check-folds) + EV(villain check-calls)
=0.51(7.5 bb) + 0.49{0.086(22.5 bb) - (7.5 bb)}
=(3.83 bb) + (-2.73 bb)
=+1.10 bb

EV (check down)
=0.516(7.5 bb)
=+3.87 bb


About the notation:

Top 5-15% & (A*,99,22,92) means "the hands in the top 5-15% range of the type A*, 99, 22, 92", while Top 5-15% ! (A*,99,22,92) means "the hands in the top 5-15%-range that are not of the type A*, 99, 22, 92").

Our calculations show that:

- We have 51.6% equity against Villain's total range
- We have 8.6% equity against the range he calls down with (49% of his total range)
- We have 93.5% equity against the range he folds (51% of his total range)

We're 90+% favorite around half the time, and the other half Villain is a 90+% favorite, so this is an extreme way ahead/way behind scenario. In general, the best postflop line for these scenarios is to get to showdown in a small pot, and our model illustrates this clearly.

C-betting and checking down when called is +EV in isolation (+1.10 bb), but checking the flop and then checking down is more than 3 times as profitable (+3.87 bb). We see from the calculations that the EV for c-betting can be written as a sum of two components: What we win when we pick up the pot on the flop (+3.83 b), and what we win when our c-bet gets called (and here we lose -2.73 bb). Since the hands that fold almost never would have drawn out on us anyway, a c-bet is simply a lot of extra risk for very little extra reward, and checking is clearly the best option.

Now we have seen a practical example of a WA/WB scenario in NLHE. The next step in our theory study is to model c-betting heads-up in position using a "toy game" that we can solve exactly using game theory. We will then see the same as in the example above, but this time using a mathematical and exactly solvable model.

Then we'll end this article by connecting the concepts "the strength principle" and "way ahead/way behind" with PLO and use an example to illustrate how they work there for a c-bet decision heads-up in position. We will then discover a fundamental difference between NLHE and PLO with regards to postflop betting. We will continue with our discovery in Part 13, where we'll discuss the consequences this has for our postflop betting lines in PLO.

4. The AKQ game
"The AKQ game" is an example of a "toy game" that models important aspects of real poker, but in such a way that we can solve the game exactly. Solving the game means finding the optimal strategies for the players involved. "Optimal strategy" here means the best strategy against opponents who always exploit our strategy maximally, whatever our strategy is. In other words, we're playing opponents who will exploit maximally all mistakes we make. By defining, solving and studying the solutions for such toy games, we can gain insight into real poker strategies.

The AKQ game is a so-called "half-street game", defined as:
  • There are two players, Alice (out of position) and Bob (in position)
  • The game uses a deck with 3 cards: A, K and Q
  • The games begins with both players putting a 1 bb ante into the pot
  • Each player is then dealt a card from the deck
  • Alice checks "in the dark" (a forced check)
  • Bob can now bet 1 bb, or check behind and let the hand go to showdown
  • Those times Bob bets, Alice can fold, or call and let the hand go to showdown
  • Those times the hand goes to showdown, the highest card wins

The classification of this game as a "half-street game" stems from the rules that force Alice to always check to Bob. Thus, she does not have any decisions to make unless Bob chooses to bet, and she is not allowed to play her half of the street (i.e. she's not allowed to force Bob into a decision by betting into him). This model is similar to the scenario where we openraise on the button, get called by a player in the blinds, and he starts the postflop play by always checking to us. Not entirely the same situation, since the player out of position can bet into us if he want to (so we can never be sure his flop checking range is identical to his preflop calling range). Also, on the flop the hand values are not static like in the AKQ game (the turn and river cards will change hand values drastically). Nevertheless, we can use this model to extract qualitative guidelines for c-bet/check decisions heads-up and in position on the flop.

We'll now solve the AQK game. Solving the game means finding the answers to the following questions:
  • What is Bob's optimal strategy for betting/checking behind?
  • What is Alice's optimal strategy for calling/folding those times Bob bets?

"Optimal" in this context means finding the most profitable strategy against an opponent who uses the most profitable strategy against our strategy. If one player makes a systematic mistake, the other player will adjust his strategy so that he exploits this mistake. Then the first player can realize this, fix his mistake, and adjust to the opponents' adjusted strategy, and so on and so forth. So we can imagine a continuous process of adjustments and counter-adjustments that converges towards a pair of strategies that can not be improved further. When the two players have reached this point, neither of them can change their strategy without giving their opponent an opportunity to exploit them, and the process ends.

Alice and Bob then end up with an optimal strategy pair, and this strategy pair is the solution to the AKQ game. It's not a certainty that any of them can make money from an optimal strategy, but this result is acceptable to both of them. A game theory optimal strategy is first and foremost a defensive strategy. We want to make money by exploiting opponent mistakes, but since our opponents are trying to exploit us at the same time, we also have to think abut not giving them openings they can attack profitably. But an optimal strategy pair can also allow one player to win (and we'll see in a minute that Bob wins in the AKQ game because of his positional advantage). But in many game theory optimal strategy pairs, both players break even when they play perfectly against each other.

What is Bob's optimal strategy for betting/checking?
It's obvious that Bob's optimal strategy must have the following characteristics:

- Always bet an A (nuts)
- Always check behind a K (medium)
- Sometimes bluff with a Q (air)

Because when Bob as an A, Alice has a K or a Q. So Bob can never lose with an A, so he bets for value every time, hoping Alice calls. Similarly, when Bob has a K, Alice must have an A or a Q. Therefore, it's impossible for Alice to make a mistake if Bob should choose to bet a K. If she has an A, she has the nuts and she calls. If she has a Q, she can not beat any of Bob's hands, and she folds. So Bob can not gain value by betting a K, and he checks behind and takes a free showdown.

We see that the principle "don't bet when no worse hands will call, and no better hands will fold" rears its head again. This game is of course only a simplistic model of real poker but the principle is the same for this toy game and for the way ahead/way behind scenario we studied in the NLHE example previously in this article.

Finally, Bob knows that he has to sometimes bluff with a Q to gain value from betting his A hands. The reason is that Alice (who always adjusts to Bob's strategy) will fold K every time when she discovers that Bob is only betting his A nut hands. So if Bob only bets A and checks down K and Q, he is giving Alice an opportunity to exploit him by never paying off his value bets when she has a bluffcatcher (the K). The question now becomes, how often should Bob bluff to ensure himself the best possible minimum profit from betting?

This is a pot odds question. When Bob bets 1 bb into a 2 bb pot, Alice get pot odds 3 : 1 on a call. She of course calls every time with an A and folds every time with a Q, but when she has the bluffcatcher K (loses to Bob's best hands and wins against his bluffs), she has a decision to make. She can't call every time, because then she is paying off Bob's A every time, and Bob will simply stop bluffing his Q's (Bob will stop bluffing and only valuebet when Alice always calls with a K, because then his bluffs never succeed).

For example, if Bob chooses to bluff 10% of his Q's, the A : Q ratio in his betting range is 100% : 10% =10 : 1 (1 bluff for every 11 bets). Since Alice is only getting pot odds 3 : 1, she has to fold her K's every time even if she knows Bob is sometimes bluffing (but he is not bluffing enough for her to call profitably with 3 : 1 pot odds). But then Bob makes more profit than if he only bets his A hands, since he now gets to sneak in one successful bluff for every 11 bets.

Reversely, if Bob should bluff with, say, 50% of his Q's, the value : bluff ratio for his betting range becomes 100 : 50 =2 : 1. Now Alice can call profitably every time with a K, since the pot odds 3 : 1 are better than the odds 2 : 1 against Bob bluffing. She will lose 1 bb 2 times, and win a 3 bb pot 1 time for a 3 - 2 =1 bb net profit for every 3 bets Bob makes. So her call with the bluffcatcher K has an expected value of +1/3 bb per call.

We see that Bob can get away with a certain amount of bluffing (for example, 10%) that Alice can't defend against (she loses to Bob when she folds every time, but she loses more if she calls every time). So Bob can ensure himself a steadily increasing guaranteed profit by starting to bluff, and then bluffing more and more. But there is a certain threshold that he can't cross (and we saw that this threshold must be below 50%), because then Alice can switch from the necessary always-fold-strategy (since she's not getting he pot-odds to call) to a profitable always-call strategy, so that Bob begins losing money on his bluffs. Next step is to find this threshold, and from the calculations above we know that it must be somewhere between 10% and 50%.

What if Bob uses a value/bluff ratio identical to the pot odds Alice is getting? When Alice has the bluffcatcher K, she's getting 3 : 1 to call and snap off a possible bluff, but when the odds against Bob bluffing is also 3 : 1, it doesn't matter for Alice whether she's calling or folding. If she calls with her K every time, she loses 1 bb 3 times to Bob's A's and wins a 3 bb pot 1 time against Bob's Q's. So her net profit from calling is 3(-1 bb) + 3 bb =0.

So Alice becomes indifferent to calling or folding with her bluffcatcher when Bob employs this optimal bluffing strategy. Therefore, when we calculate what bluffing does for the EV of Bob's bet, we can simply assume she folds every time with a K and only calls with an A. When Bob bets 8 times with a 3 : 1 value/bluff ratio (6 times with an A and 2 times with a Q), Alice folds every time he has an A (because then she has either a K or a Q), and Bob picks up 6 x 2 bb pots =12 bb. When Bob bets his 2 Q's, she calls half the time (when she has the A) and folds half the time (when she has the K), so Bob picks up 1 x 2 bb pot once and loses 1 bb once. Therefore, on these 8 bets, Bob's gain is 6 x 2 + 2 - 1 =13 bb.

This is 1 bb more than if he only bets his 6 A's and checks the 2 Q's, because then he wins 6 x 2 bb pots when Alice folds to his A value bets, and he loses 2 showdowns against Alice's A's and K's when he checks behind with his Q's. So Bob's optimally balanced valuebet/bluff strategy ensures him an extra 1 bb for every 8 bets he makes, and there is nothing Alice can do to prevent this. It's 3 : 1 against Bob bluffing, and Alice's getting pot-odds 3 : 1 when she calls to snap off a bluff, so calling with her bluffcatchers becomes break even. She can play her K's however she wants, and Bob will still make 1 bb more (compared to never bluffing) per 8 bets.

Bob's total optimal strategy that guarantees him a 1/8 extra profit per bet then becomes:

- Always bet an A
- Always check behind with a K
- Bet 1/3 of the time with a Q

Because then the value/bluff ratio becomes 1 : 1/3 =3 : 1, which is what Bob wants. Now we move on to Alice, and find her optimal call/fold strategy to use against Bob's betting strategy:

What is Alice's optimal call/fold strategy?
It's obvious that Alice's optimal strategy must have the following characteristics:

- Always call with an A (nuts)
- Sometimes call with a K (bluffcatcher)
- Always fold a Q (air)

Calling with the best hand is automatic, and the same is true for folding her air hands (Q) that can't beat anything that Bob bets. So what we have to do now is to find Alice's optimal calling frequency with her K's, so that she can keep Bob's guaranteed profit to a minimum. She can't call every time, because then Bob stops bluffing, and exploits her by always getting his value bets with an A paid off. Now Bob's bets win a 3 bb pot half the time (when she calls with a K) and a 2 bb pot half the time (when she folds a Q), so Bob's gains 3 + 2 bb =5 bb for every two bets, or 2.5 bb/bet. This is an extra 1/2 bb/bet compared to never getting paid off getting called, which is better than the extra 1/8 bb/bet he gains from his optimal strategy.

But Alice can't fold every time either, because then Bob will bluff his Q's every time and rob her blind. When Bob has a Q, there's a 50-50 chance that Alice has an A or a K. So Bob risks 1 bb to steal 2 bb, and he succeeds half the time if Alice always folds her K's. So when he has bluffed 2 times, he has stolen 1 x 2 bb pot once and lost 1 bb once for a net profit of 1 bb (=1/2 bb per bluff). So Alice loses a lot to Bob's adjusted strategy when she always folds as well.

Keep in mind that Bob can guarantee himself an extra 1/8 profit per bet by following the optimal valuebet/bluff strategy outlined above, and his profit is the same no matter whether Alice always folds or calls with her K's. But if Alice should choose one of these extremes, Bob can win even more by moving away from his optimal strategy (he can choose to never bluff or always bluff). So Alice's goal is to deny him this opportunity, and keep him down to a 1/8 bb per bet minimum.

What if Alice calls and folds in a ratio identical to the pot odds Bob is getting on his bluffs (2 : 1. since he's risking 1 bb to win 2 bb). When Bob bluffs a Q 3 times, he will get called twice (and lose 1 bb each time) and succeed once (and steal a 2 bb pot) for a net profit of 2 x (-1) + 2 =0 bb, which is 0 bb per bluff. So if Alice chooses to call and fold in this ratio, she guarantees that Bob can't profit from bluffing his Q's every time. She will still lose minimum 1/8 bb per bet to Bob's overall betting range when he sticks with the optimal valuebet/bluff strategy, but he keeps his profit down to this amount, and she doesn't give him an opportunity to profit even more by always bluffing or never bluffing.

So Alice needs to call 2 out of 3 times (66.67%) when Bob bets. She can get to 50% by always calling with her A's (since A's make up 50% of the 2 hands A and K she can have when Bob bets), and she needs to call enough with K's to get to 66.67% total call percentage. She can get there by calling with 1/3 of her K's (since 50%/3 =16.67%). Alice's total optimal strategy then becomes:

- Always call with an A
- Call 1/3 of the time with a K
- Always fold with a Q

The reader can easily verify that Bob now can't gain by switching from his optimal strategy to an always-bluff or a never-bluff strategy.

What is Bob's win rate in the AKQ-game against Alice?
First, if Bob never bets, but checks down all his A's, K's and Q's, the game is symmetrical and break even for both players:

- Bob has an A =he wins against K and Q
- Bob has a K =he wins against Q and loses against A
- Bob has a Q =he loses against A and K

So Bob wins 3 pots and loses 3 pots, and all pots are 2 bb in antes since there is no betting. So Alice and Bob on average get their 1 bb ante back. The same will happen if Bob never bluffs. He now checks all K's and Q's and only bets his A's. But then Alice can fold her K's every times against Bob's valuebet, so the result becomes the same as if he had checked down his A's. Therefore, Bob can not make money with a strategy where he always checks, or where he only value bets his best hands and checks everything else.

But when Bob employs his optimal valuebet/bluff strategy with a balanced mix of value bets and bluffs, he is guaranteed to win 1 bb per 8 bets. Alice responds with her optimal strategy that keeps Bob's profit to this minimum. If Alice and Bob plays the AKQ game 18 times, Bob will on average have A, K and Q 6 times each. The distribution of outcomes becomes:
  • Bob value bets A 6 times and bluffs Q 2 times (=3 : 1 value/bluff ratio). As explained previously, he picks up 13 bb by doing so, regardless of Alice's strategy. So his net gain is 5 bb (13 bb won minus the 8 x 1 bb he paid in antes for those 8 hands)
  • Bob checks K 6 times. He wins 2 bb 3 times (against Alice's Q) and wins nothing 3 times (against Alice's A). His net gain is 0 bb (6 bb in won pots minus 6 bb in antes)
  • Bob checks Q 4 times. He loses all of these pots to Alice's A and K, so his net gain is -4 bb (0 bb won minus 4 bb in antes)

So Bob's net gain on these 18 hands is 5 bb + 0 bb - 4 bb =+1 bb.

Conclusion: Bob's guaranteed profit from the AKQ game is 1/18 bb per hand =0.056 bb per hand, or 5.6 bb/100.

Not bad! And all of this profit stems from the fact that Bob has position on a player who never bets into him. Note that the game is symmetrical with respect to the player's hands (each player is equally likely to get dealt an A, K or Q), so without any betting both players break even. Thus the only profit source is the betting round after the cards are dealt. In other words, this result can be interpreted as a measure of the value of position against a passive player.

What we can learn from the AKQ game
We'll use insights from the AKQ-game in future articles, where we'll discuss the value of position and what the player out of position can do to counter this (for example, by leading into the raiser and checkraising him). What we'll use from this solved toy game right now is the insight into why Bob shouldn't bet his medium strong hands (his K's), and how this is related to real poker games.

The reason it was "forbidden" for Bob to bet the medium part of his range was that hand strengths were well-defined and completely static. When Bob had a K, Alice either had a hand that was always better (A) or a hand that was always worse (Q), and Alice always knew whether she was ahead or behind with these hands. There were no future streets that could change relative hand strengths, and their relative ordering stayed constant.

Now we ask:

Why are we using the rule of thumb "don't bet when no worse hands call and no better hands fold" in NLHE postflop betting decisions? It's correct in the AQK game, but does this rule apply to a real poker game?

We can of course always use this rule at the river, where hand strengths are completely static (there are no more cards to come), but it often makes sense to use the same line of thinking in earlier streets in NLHE as well. One reason for this is:

The structure of Hold'em results in a game where relative hand strengths change less from street to street than in many other games. Especially after the flop. Thus, Hold'em after the flop is more similar to the structure of the AKQ game than many other games.

For example:

Alice has 6 6 , Bob has A K . Alice has the best hand before the flop. The probability that Bob outflops Alice with a better pair, two pair, trips, straight, or flush is 32%. So Alice is a big favorite (68%) to also have the best hand on the flop.

Let's say the flop comes 8 3 2 . Alice is still ahead, and Bob now has 6 outs to a better hand among the 45 unseen cards. So on this flop, there is a 39/45 =87% chance that Alice will still have the best hand on the turn.

If the turn comes 8 3 2 Q , Bob is still behind with 6 outs among the now 44 unseen cards. So there is a 38/44 =86% chance that Alice will also have the best hand on the river.

One reason for this stability in the relative strengths of hands from street to street is the use of community cards in Hold'em (the board cards that all players have to share). For example, if you have two opponents who are both drawing to a flush to beat your straight, you will either be beat by both of them on the next card, or by none of them. But if you had played 7-card stud, each opponent would have gotten a separate card, so that both could have beaten you, or one of them, or none of them. So you would have gotten drawn out on more frequently in stud.

Another reason for this stability, compared to PLO (where we also use community cards) is the starting hand structure. We use 2-card hands in Hold'em, and 4-card hands in PLO. It's much harder to hit the board with only two cards in your hand, which has given rise to the expression "most hands miss most flops" in Hold'em (which isn't the case in PLO).

Another important reason why we won't like to bet in NLHE when worse hands won't call and better hands won't fold has to do with the betting structure:

In a "big bet" game (no-limit or pot-limit) it's expensive to bet when you shouldn't

If you think no worse hands will call, and no better hands will fold, you risk a bet comparable to the pot size to win only what's in the pot. If the hands you beat rarely will draw out on you if you allow them to get to showdown, and if you rarely draw out on the hands that beat you, you might be in a situation where the risk/reward ratio for a bet becomes so poor that a bet can't make you money. The cost of paying off the better hands might be too high compared to the pots you pick up when you are ahead. Then it might be best to check the hand down, or at least pay the minimum to get to showdown. The way ahead/way behind example earlier in this article illustrated this.

5. An illustration of how NLHE betting strategies can break down in PLO
We have now reached a very important point in the theoretical work done in this article, namely that postflop planning and betting strategies that we carry over from NLHE, aren't always applicable in PLO. We have discussed the strength principle and way ahead/way behind scenarios (and the corresponding way ahead/way behind postflop line) in a NLHE context. Then we did a mathematical model study (the AKQ game) which told us something about why the WA/WB line works well in NLHE, and why we in NLHE don't like to bet when worse hands don't call and better hands don't fold.

The gist of it is that the relative strengths between hands have a statical nature in NLHE. Medium strong hands therefore often find themselves in situations where they don't make much money from betting.

We'll now go through a tandem example that illustrates how it can be correct to check down a Hold'em hand in a WA/WB scenario, while in a similar PLO-scenario it's better to bet and be happy to win the pot right there on the flop. Even if no worse hands call or no better hands fold.

5.1 Comparing postflop betting lines for NLHE and PLO in similar situations and under equivalent assumptions
We raise pot (3.5 bb) with AA/AA7s on the button in NLHE and PLO, respectively:

- NLHE: A A
- PLO: A A 7 2

The big blind calls, and the flop comes: J J 9

In both cases we have a naked overpair without draws on a paired and coordinated flop against a Villain with a medium strong calling range. We're assuming (as in the previous WA/WB example) that the big blind plays the top 15% of hands preflop, and that he would have 3-bet with the top 5% of hands. So his flatting range preflop is top 5-15% of hands. Big blind checks to us on the flop, and we have a c-bet/check decision to make.

We'll now calculate the EV for c-betting and checking in NLHE and PLO. We'll use a model with assumptions similar to the ones in the previous way ahead/way behind example:
  • Villain's range is Top 5-15% (Top 15% minus top 5%) in both NLHE and PLO, based on the hand rankings from ProPokerTools
  • He folds all hands worse than trips if we bet. This is a very reasonable assumption in the PLO case. And since Villain doesn't have overpairs or good underpairs in his range (he would have 3-bet {AA-99,AQ+} =5% preflop) it's also a reasonable assumption in the NLHE case.
  • He calls down with all hands trips or better if we c-bet. In other words, he checks every street, planning to call if we bet, and he never bets himself. This is an assumption we make to get a simple model that is easy to work with.
  • Our strategy on the flop is to either a) c-bet pot (7.5 bb), planning to check the hand down if called, or b) check the flop, planning to check the hand down


EV for c-betting and checking i the NLHE case

AsAc 70.8%
Top 5-15% 29.2% 85 combos

AsAc 7.7%
Top 5-15% & (J*,99) 92.3% 15 combos (17.6%)

AsAc 84.4%
Top 5-15% ! (J*,99) 15.6% 70 combos (82.4%)

EV (c-bet and check down when called)
=EV(villain check-folds) + EV(villain check-calls)
=0.824(7.5 bb) + 0.176{0.077(22.5 bb) - (7.5 bb)}
=(6.18 bb) + (-1.02 bb)
=+5.16 bb

EV (check down)
=0.708(7.5 bb)
=+5.31 bb


The results are similar to the previous WA/WB-example:
  • We're a big favorite (84%) against 1/6 of Villain's range, and a big underdog (7.7%) against the rest of his hands
  • We're not getting called by worse hands, and we're not folding out better hands (per assumption)
  • As a result, checking the hand down (+5.31 bb) is better than betting (+5.16 bb)

EV for c-betting and checking in the PLO case

AsAc7d2h 52.8%
Top 5-15% 47.2% 12973 combos

AsAc7d2h 8.2%
Top 5-15% & (J*,99) 91.8% 3086 combos (23.8%)

AsAc7d2h 67.0%
Top 5-15% ! (J*,99) 33.0% 9887 combos (76.2%)

EV (c-bet and check down when called)
=EV(villain check-folds) + EV(villain check-calls)
=0.762(7.5 bb) + 0.238{0.082(22.5 bb) - (7.5 bb)}
=(5.72 bb) + (-1.35 bb)
=+4.37 bb

EV (check down)
=0.528(7.5 bb)
=+4.00 bb


The situation in the PLO case is similar to the NLHE case:
  • we're a big favorite against 1/5 of Villain's range, and a big underdog against the rest of his range. However, when we're ahead, we're less of a favorite than in the NLHE case (67% in PLO versus 84% in NLHE).
  • We don't get called by worse hands, and we don't fold out better hands (per assumption)
  • But it's still better to bet (+4.37 bb) than to check the hand down (+4.00 bb)

Also, note that in the NLHE case we're a 70.8% favorite against Villain's total flop range, but we can't bet (within our model). In the PLO case we're basically a coinflip with 52.8% against Villain's total flop range, but now a bet is obligatory.

The mathematical explanation for this curious result is that we make so much money (+5.72 bb) from betting and folding out weaker PLO hands (that nevertheless have decent equity against our hand) that we can afford to lose a bit (-1.35 bb) those times Villain calls. The reason is that the hands Villain folds have much better equity against us in the PLO case (33%) than in the NLHE case (15.6%). So it costs us much more to give Villains weaker hands a free showdown in the PLO case. But in NLHE, betting to protect ourselves against Villains weak hands isn't worth it, since they draw out so rarely.

Conclusion:
Checking down a medium strength hand that is often best on the flop, but will only get called by better hands, doesn't necessarily do anything good for us in PLO. In NLHE it's often best to take these hands to showdown in a small pot. In PLO it's often best to bet them on the flop and hope to win the pot right there.

Consequence:
In PLO we will bet-fold medium strength hands on the flop more often than in NLHE. In NLHE we will more often turn medium strength hands in to bluffcatchers (checking the flop, planning to call if Villain bets on a later street).

We have now illustrated, with some general poker theory and a couple of detours into NLHE strategy land, an important concept for PLO postflop betting. We will often find ourselves on the flop with a hand that is probably best at the moment, but not really strong enough to bet for value (if we define "for value" as betting and being willing to get stacks in if we get checkraised). For example, if we raise Q Q 9 2 on the button, get called by the big blind, and the flop comes J 6 4 . We're often ahead on the flop, but it's clear that we have to fold to a checkraise if we bet. So a bet is probably not for value against the range Villain continues with.

So is a bet a bluff then? Yes and no. We fold to a checkraise (like we would with a pure bluff), but it's possible that Villain will call with some hands we beat, and we can win some showdowns if this happens. So a bet does have a value component too.

But the main point is that we can argue that our overpair is too strong to check and give up, but too weak to check behind and use as a bluffcatcher on future streets. Because there aren't really many good turn cards for us. There are 10 diamonds that will make a flush possible, all 2, 3, 5, 7, 8 will make a straight possible, all hearts put another flushdraw on board (which we don't have), and all A and K might give Villain a better pair. So the list over turn cards we'd like to see is short, and we will rarely find ourselves on the turn with the desire to call a bet, should we check the flop, planning to call the turn bluffs we then sometimes induce.

Therefore, since we're probably ahead on the flop, with the possibility of getting called by some hands that we beat, and since we will often win the pot right there, it seems best to simply bet and hope Villain folds. Even if our hand is too weak to continue against a checkraise, or if Villain should call the flop and then bet the turn. Note that if Villain only calls the flop, he has revealed weakness. Then we can use the turn and river to make his life very difficult, even if his weak hand is better than our weak hand. Sometimes we will win a showdown after checking the turn and river, and other times we will turn our hand into a bluff and 2- or 3-barrel to make Villain fold a better hand (it's hard to call down in PLO with a hand of moderate strength).

More about these things in future articles, but you should already now see that getting our flop bet called doesn't necessarily make our medium hand difficult to play on future street. We have position, Villain has told us that his range is weak, and we'll have all the good options available on later streets if he continues to check to us.

6. Summary
You can view this article as a "theoretical interlude" on our journey through PLO postflop theory. The main point of the article was to introduce some important betting concepts. A concepts we will use a lot in future articles is:

In PLO it's often better to bet a marginal hand and hope our opponent(s) fold (if there aren't too many of them) than to check and try to sneak our hand to showdown in a small pot.

This concept will be the starting point for lots of aggressive postflop betting, especially when we are heads-up in position and we have the opportunity to valuebet and bluff as we please against a player who has told us he has a weak range out of position.

We will put this concept to work immediately in Part 13, where we'll begin with a discussion of c-betting/checking behind as the preflop raiser heads-up and in position on the flop. After this we'll talk about 2- and 3-barreling (bluffing over multiple streets), which is also a very profitable and enjoyable activity.

The day a new and inexperienced PLO player understands how barreling works, it's like "the light goes on" in his head, and a new world of profitable opportunities is laid before his feet. He now realizes how much of the game is about position and reading the situation, and how little his cards often mean. So we'll spend plenty of time on this subject.

Part 13 will be a practical article with lots of specific strategy talk. I expect to have it ready in a few weeks, and then I plan to publish about 1 article per month until the "PLO From Scratch" article series is done.

Good luck!
Bugs