The 3rd Reason For Betting In Poker

Why do you bet in poker? When does it make sense to do it? 

If you’re not a beginner anymore (and I imagine you aren’t, since you’re subscribed to a relatively advanced strategy newsletter) then your immediate answer to this question is really obvious.

The dichotomy value and bluffs.

That’s how we learn about poker, and that’s how 99% of professional players perceive the incentives for betting.

These incentives do exist. That’s a fact. If you’re on the river with a very strong hand, you want to bet because you can extract chips from your opponent through getting called by worse hands. You can’t only bet value hands, however, because then your opponent would never call you (assuming he has information on your actual strategy). Therefore bluffs emerge as a potential threat to your opponent – if he doesn’t call you, you make money by betting your weak hands. You force your opponent into calling you by constructing a range that contains enough weak hands to benefit from an overfolding mistake. Your opponent’s best response then is to call you enough of the time to neutralize the threat of your bluffs – he makes your bluffs indifferent between betting and checking by calling at the famous 1 – alfa frequency. 

While those concepts and rationalizations are certainly useful, they are not sufficient to explain everything that happens in poker. 

What if I told you that there is another reason for betting – one in which you are never getting called by a worse hand, and also never getting a better hand to fold… but you should still bet.

To see this very clearly and understand the mechanics of it we need to study the AKQ Toy Game.

A toy game is a simplified version of the full game of poker. Poker is extremely complex and big, so in order to properly study it we can create toy games to observe and study the behavior of specific variables of the game. The findings can then be extrapolated to the full game, helping us build better strategies.

The AKQ toy game is very simple: two players are on the river. Their ranges are completely symmetric and each of them can only have 3 unique hands – aces, kings or queens. Blocker effects exist here, so when the OOP player holds aces, then the IP player can only be holding queens or kings. When the OOP player holds kings, the IP player can only be holding aces or queens; and so on. 

The particular AKQ toy game we are going to solve contains these parameters:

• Pot size of 4 bbs;

• Each player can only bet or raise 1 or 2 bbs at their turn;

• The board is set up in a way that Aces beats Kings and Kings beats Queens.

What do you think is the solution to this game? 

If you’re like most people, your rationalization will be something like this: “Well, each player has the nuts, a bluffcatcher and a bluff. We should take our aces and bet the biggest amount allowed, 2 chips. We can’t only bet aces though, so we also bet queens sometimes to balance our value bets. And then we check with kings as those are just middling hands”.

This thinking is absolutely reasonable – in fact it’s completely congruent with everything we’re usually taught about poker: take your nuts and bet as big as possible, throw some bluffs in so that your opponent calls you sometimes. And then with a hand that can’t really get value from worse, you just check. 

Take a look however into what’s the actual solution to this game:

OOP PLAYER

IP PLAYER

While the IP player’s strategy follows the typical logic we would expect, the OOP strategy seems very, very strange. We observe 2 main things that are initially unexpected:

• Kings is betting sometimes;

• The bigger sizing is never used, not even by Aces.
  

What is going on here? How can this be the case? Betting with Kings contradicts everything we are taught about how poker works. You’re only supposed to bet with a hand if you can 1) get called by worse (value bet) or 2) get a better hand to fold (bluff). This bet with kings accomplishes neither. In fact, recall that because of the blockers in this game, when OOP holds Kings, it can only run into Aces (better hand – runs into it half of the time) or Queens (worse hand, runs into it half of the time). Aces is the nuts so it always raises; and Queens has 0% equity so it will never call a bet.

The reason why KK bets in this toy game is exactly the 3rd for betting in poker. Let’s understand what it is about. 

If you look above at the IP player’s response against the small bet from OOP, you can see that QQ raises sometimes facing this bet. Why do you think that happens? 

The most likely answer you’ll give is that QQ raises to balance the Aces. By doing this, IP gives incentive for the OOP to call the raise sometimes with Kings, which gives value to the IP’s Aces. 

That’s wrong though. OOP never calls a raise with Kings:

Hm. Tough one. We need to continue digging.

Now I’m going to teach you a study technique that is very simple but yet extremely powerful in understanding why things happen at equilibrium. If you want to understand why something is happening in a solver solution, force it not to happen and then see how strategies adjust. You can achieve that with the Node Locking function.

Node Locking is a PioSolver feature that allows the user to alter the strategies in any given node, thus forcing the solver to adapt to that new strategy. Then you can lock the node, which means that you don’t allow the solver to change the strategy for that node – only unlocked nodes will be updated if you run the solver again. 

So let’s do that. If we want to understand why IP raises with QQ in a situation where OOP never calls with a bluffcatcher, all we gotta do is force IP to not raise with QQ and then we see what happens.

When QQ is forced to always fold against a small bet, then the strategy for the OOP player completely changes:

Now Aces always bets big, and Kings pushes the betting from a mixed strategy to a pure bet.  

Hm. When QQ was raising sometimes, KK was indifferent between betting small and checking – both options made the same amount of money. Now that QQ is not raising, KK bets all the time. Without the QQ forcing Kings to bet/fold sometimes, the EV of betting with Kings becomes superior to checking.

We can conclude then that QQ raises to prevent KK from betting all the time. Without the bluff raises, OOP is able to exploit the IP player by always betting with the middling strength hand. If IP doesn’t do anything about it, KK can make more money by betting small than it does by checking. 

Okay. But how is that the case? How can KK make more money by betting than checking, when betting simply either runs into a better hand that raises, or a worse hand that folds? Perhaps the math can help us.

The EV of betting with KK for 1/4 size is very easy to calculate. Due to blocker effects, only 2 things can happen when betting – either KK gets raised by AA and has to fold; or it gets a fold from QQ and wins the full pot.

EV(Betting with KK) = 0.5 4 – 0.5 1 = 2 – 0.5 = 1.5 bb.

The EV of betting with KK is 1.5bb per hand – we win the full pot half of the time, and half of the time we lose our bet (1bb).

What’s the EV of checking? That is also relatively straightforward to calculate. When we check with KK, only 2 things can happen – either we face a bet or we face a check. Since villain is supposed to make us indifferent to calling a bet, facing a bet doesn’t net us anything – EV when facing a bet is zero. When we face a check on the other hand we win the full pot at showdown, since our opponent’s checking range is 100% made of hands we beat (Queens).

EV(Checking with KK) = EV(Facing Bet) + EV(Facing Check) = EV(Facing Check)

From the solution, QQ should check back 66.66% of the time. Since we run into QQ half of the time, the EV of facing a check is calculated as follows:

EV = 0.5 0.66666 1 4 = 33.33% 4 = 1.3333

When we hold KK, we run into QQ half of the time. QQ checks 66.66% of the time, so we get to showdown 33.33% of the time. When we do get to showdown, we have 100% equity, therefore we win the full 4bb pot. 33.33% of 4 bbs = 1.3333 bb.

Have you figured it out yet? If not, I can explain.

By getting QQ to fold all the time, KK never gets bluffed. By getting QQ to fold, KK guarantees always winning the pot against worse hands. Always winning the pot against the worse hand increases the EV of KK because the alternative is checking, and when KK checks, it faces a bet sometimes from QQ. Since IP contains value hands in it’s range, QQ is able to bluff KK 33% of the time. Because of that, KK only gets to win the pot against QQ 33.33% of the time – the 50% frequency it runs into it times the 66.66% checking frequency. 

Have you ever heard of the term “equity realization”? I bet you have. Equity Realization is a variable that calculates how much of a hand’s equity actually gets turned into money. 

In this toy game, the pot is 4bb, and KK has 50% equity – beats QQ and loses to AA, each being half of IP’s range. If KK had 100% equity realization, then it would turn it’s 50% equity into 2bb of EV – 50% of the 4bb pot. However, the EV of KK in the original solution is 1.333, which constitues a 66.66% Equity Realization.

Equity realization is almost never 100% for bluffcatchers because they get bluffed by worse hands all the time. Let that sink in for a moment. You can currently be beating many different hands, but you can’t turn that equity into real money because those very hands you are beating will force you out of the pot sometimes.

If you have absorbed everything we discussed so far, you’re ready to understand the 3rd reason for betting:

We should bet to prevent worse hands from bluffing us effectively.

That’s how KK draws EV from betting in this toy game – if QQ never raises, then betting becomes superior to checking because by betting it never gets bluffed and thus captures the full pot against the worse hands. Checking is inferior because it’s an alternative where KK gets bluffed sometimes, and thus has a decreased equity realization. 

There is one more detail we haven’t explored yet. KK pays a price to get access to this strategical option: half of the time it puts money into the pot and runs into the nuts. That’s why KK’s equity realization isn’t 100% even when it always wins against the worse hands – it is now sacrificing 1bb every time IP has AA, which is something checking doesn’t require. 

That’s the reason why if we only allow OOP to bet 50% pot, then KK always checks – the sacrifice becomes too big, and therefore not worth it. The sacrifice only becomes worth it at 33% pot size.

Now you can understand why QQ raises in the original solution. QQ must prevent KK from doing this trick. Not raising can’t be the optimal strategy because IP can bluff raise and decrease the EV of KK. By raising a little bit, QQ is able to reduce the EV of betting with KK to the exact EV of checking it, making KK indifferent. 

That however increases the EV for AA – when AA bets the bigger sizing, it never gets any value from QQ, as it should always fold. But now that IP is raising sometimes with QQ against small bet, betting small with AA becomes the highest EV play, because AA can capture some value from the IP’s bluffs. Increasing the EV of AA however is a price IP is willing to pay, because the alternative – not reducing the EV of KK – is even worse.

There it is. You understood everything there is to know about this toy game. Let’s recap:

• If QQ doesn’t raise, KK makes more money betting because it never gets bluffed, which is a better alternative than checking, where it does get bluffed sometimes. 

• IP realizes this and plays a strategy that makes KK indifferent between betting and checking. It can accomplish this by bluff raising QQ a little bit.

• By bluff raising with QQ, IP generates incentive for OOP to bet small with the nuts. Even though OOP is now using a smaller betsize and thus getting less value from bluffcatchers, the added value gained from the air region compensates and makes betting small superior to betting bigger.

• IP makes a trade off – it increases the EV of AA by bluff raising and decreases the EV of KK. That is okay because It’s higher EV for IP to make KK indifferent than it is to make AA indifferent. 
  

Now I’m going to blow your mind a little bit further. You’ve just learned the reason why blocking bets exist in poker. You know those small bets solver throws out seemingly everywhere in the game tree? 10% pot flop cbets, 25% pot turn delayed cbets? Yeah, that’s the reason why. 

There is only 1 unique circumstance where blocking bets are not a part of equilibrium strategy. Blocking bets exist on the flop OOP, on the flop IP, on the turn OOP, on the turn IP and on the river OOP. The only spot where it doesn’t exist at equilibrium is on the river IP. This is because being on the river IP is the only circumstance in the NLHE game tree where you have the option to win the full pot against all the hands you’re currently beating. Checking back guarantees 100% equity realization. In every single other circumstance, you are at risk of getting bluffed by worse hands after you check. 

Now this is where all of this theory talk turns into something (extremely) valuable and practical for you to use at the tables. You’ve just understood that betting to make worse hands fold is actually a thing in poker, because the alternative (everywhere except on the river IP) is checking and potentially getting bluffed. This can only work if you bet small, because you’re putting money into the pot (which checking does not) and sometimes you lose that money by running into better hands. You’ve also understood that an optimal opponent understands you can try this trick, and the way they have to prevent you from doing this all the time is by playing exactly the amount of bluff raises that makes your middling hands indifferent between betting and checking. In our toy game, that means bluff raising with QQ exactly 6.666% of the time.  

That’s impossible to do in real games.

In real games, one of the following 2 things are going to happen:

• Your opponent doesn’t find enough bluff raises;

• Your opponent finds too many bluff raises;
  

In scenario 1, you should murder such opponent by always block betting your middling strength hands, and perhaps taking hands you would otherwise check and block bet instead, as your EV will increase a lot. At the same time, you should size up with your value bets (just like we saw AA doing above after the node lock), because the opponent is not giving you enough value with their bluffs to justify betting small with your nuts.

In scenario 2, you should now always bet small with all your nutted hands, as your opponent is adding too much EV to that line for you. With regards to your bluffcatchers, it depends how much the opponent is actually raising. It could be that its correct for you to check your bluffcatchers, as the extra bluffs are only decreasing your EV. Or it could also be that there is so many bluffs that they actually increase your EV (because you can bet/call very profitably) and therefore you should bet.

Luckily for you data shows that people fall much more often in the easier, first scenario. Most players underraise and overfold relative to solver when facing small bets. And I’m not just talking about the river – people underraise flops and turns facing small bets, in almost all lines. 

Take a look at this turn spot OOP – we open SB, BB calls. Flop comes 877r, and the action goes check check. Turn is the Qh. This is the GTO solution:

Against the small size, IP is supposed to bluff raise with many non intuitive hands, like K9o, KTo, KJo:

If we node lock IP to never raise with these hands, while maintaining everything else the same, this is how the solver adjusts the delayed cbet strategy:

If you want to make more money in poker, leverage the 3rd reason for betting to exploit a population that is failing miserably to make your middling hands indifferent between betting and checking. Make those worse hands fold and enjoy a substantial boost to your winrate.

The AKQ toy game is one of the topics I teach in my GTO Blueprint course. If you enjoyed this article, then my course is for you. With over 12 different modules and dozens of videos, I teach you not only the fundamentals of poker theory (like this toy game) but also give you practical advice (heuristics) on how to construct your ranges at the tables. Check it out here.

| For a more detailed and mathematical analysis of the AKQ toy game, refer to the book Mathematics of Poker, chapter 15, example 15.2.

Thanks for reading. See you next week.
Until then – don’t play the game. Play the Metagame.

Saulo