*The painter ChatGPT “A dramatic poker scene featuring a close-up of a player’s hand holding two Ace cards in a Texas Hold’em game.”
Although the title is “Bet Sizing in Texas Hold’em,” it is actually a compilation of fragmentary writings that could not be included in the previous trilogy:
I don’t think I’ll have any opportunities to feature them in the future, so even though they’re not very cohesive, I’ll post them here anyway.
How to Consider Bet and Raise Sizes
When considering the specifications of the game of Texas Hold’em, one thing that seems very difficult is how to represent the player’s action of raising (betting). According to the rules, a player can increase his bet by any number of chips as long as it exceeds the minimum raise. There is often no limit to the number of raises. In theory, it might be possible to let users enter any number into a text box and perform a validation check. In reality, it is likely that there are quite a few players who decide the number of chips roughly, and it may not be uncommon for them to include chips that are left over from their hand and that do not round up into a bet or raise.
However, it is suspected that there are many players who play in a manner (guideline) that is somewhat established by convention. For example, when making an opening raise pre-flop, you should always specify 3 to 5 times 1bb (big blind), and when re-raising, you should specify 3 times that amount (7bb to 12bb).You should never limp in (call the big blind) or make a minimum raise (2x pre-flop). Also, the minimum raise is reset each time the round changes, such as from pre-flop to flop, or from flop to turn, so the opening bet for each round can again be specified from 1bb. However, in reality, the opening bet is not usually 1bb, and it seems that numbers such as 33%, 50%, or 75% of the pot size (the total bets of all players) are often chosen.
If you read poker books, you will find almost no mathematical explanation of how this size should be determined. It is simply a matter of conventional wisdom or a choice made using game theory; there appears to be no theory in place for determining the size in the first place. This is because the concept of expected value (EV), explained in a previous post, cannot be used alone to determine the size of your stake. If you just want to maximize your expected value, then going all-in would be the only option if the expected value is positive. To do this, we must use the concept of risk aversion. Let us develop a tentative theory on this.
Suppose there is a certain amount of chips \(\small \pi\) in the pot. A player has a stack of \(\small s\) and the question is how to calculate the optimal bet amount \(\small b = fs = \varphi\pi\) from it. \(\small f\) is the percentage of the current stack, and \(\small \varphi\) is the percentage of the pot (the 33%, 50%, 75%, etc. ratios mentioned at the beginning). Let’s assume that this player is maximizing a mean-variance utility function for the current stack. That is, determine \(\small b\) to maximize
\[ \small E[u(b)] = \mu(b) -\frac{\gamma}{2}\sigma^2(b). \]
If you use the Kelly criterion, you set \(\small \gamma=1\), and if you are maximizing expectation (assuming a risk-neutral utility function), you set \(\small \gamma=0\).
For simplicity, assume there are \(\small n\) players and that they all call regardless of the amount you bet. If you estimates your probability of winning to be \(\small p\), then you can calculate your expected return \(\small \mu\) on your stack when you bet \(\small b\) with
\[ \small \mu \times s =p \times (\pi+b\times (n-1))+(1-p) \times (-b). \]
Divide both sides by \(\small s\) to get
\[ \small \begin{align*} \mu &= p \times \left(\frac{\pi}{s}+f\times (n-1)\right)+(1-p) \times (-f) \\ & = p \times \left(\frac{\pi}{s}+f\times n\right)-f. \end{align*} \]
The variance of returns can be calculated similarly and we get
\[ \small \sigma^2 = p(1-p) \left(\frac{\pi}{s}+f \times n\right)^2. \]
Hence, the optimization problem to be solved is:
\[ \small \begin{align*} &\max_{f} \; E[u(f)] = \mu(f) -\frac{\gamma}{2}\sigma^2(f) \\ &\text{s.t.}\quad 0 \leq f \leq 1. \end{align*} \]
Since the constraints are simple, in order to maximize expected utility as an unconstrained problem, we equate the Lagrangian to the expected utility and differentiate with respect to \(\small f\), which gives us the condition:
\[ \small \frac{\partial L}{\partial f} = \frac{\partial E[u(f)]}{\partial f}=np-1-n\gamma p(1-p)\left(\frac{\pi}{s}+f \times n\right) = 0 \]
that the optimal solution satisfies. By solving it, we get
\[ \small f^{\ast} = \frac{1}{n}\left[\frac{1}{\gamma}\frac{p-\frac{1}{n}}{p(1-p)}-\frac{\pi}{s} \right], \quad 0 \leq f^\ast \leq 1. \]
If \(\small \gamma=0\), then the optimal solution is to go all-in if \(\small p\times n>1\), otherwise to check. To calculate the ratio to the pot, we simply multiply it by \(\small s/\pi\) (this ratio is called SPR (Stack-Pot Ratio)), and get
\[ \small \varphi^\ast = f\frac{s}{\pi} = \frac{1}{n}\left[\frac{s}{\pi}\frac{1}{\gamma}\frac{p-\frac{1}{n}}{p(1-p)} – 1 \right]. \]
It may be hard to visualize just from the formula, so let’s consider a concrete numerical example. Assume that a player has a stack of \(\small s=90\) and there are already \(\small \pi=30\) chips in the pot. There are three players left and you estimate your chances of winning at 50%. What is the optimal bet amount? Let’s calculate the case for the Kelly criterion \(\small \gamma=1\). Since
\[ \small f^\ast = \frac{1}{3}\left[\frac{0.5-1/3}{0.25} -\frac{30}{90} \right] = \frac{1}{9}, \]
the optimal bet amount is \(\small b^\ast = f^\ast s = 10\). If we calculate the ratio to the pot, we get \(\small \varphi=1/3\), which gives us the answer that we should bet 33%. By performing these calculations and determining a value for \(\small \gamma\) that fits your imagination, you will be able to determine a fairly consistent bet size.
In reality, the probability that your opponent will call or fold depends on \(\small b\), and if you can get your opponent to fold, your probability of winning \(\small p\) will also change. The expected return should probably be calculated as:
\[ \small \mu \times s =p(b) \times \left(\pi+b\times \sum_{i=1}^{n-1}c_i(b)\right)+(1-p(b)) \times (-b) \]
by expressing the probability of calling for a bet \(\small b\) by a player other than yourself as \(\small c_i(b)\) and the probability of winning as \(\small p(b)\). This type of analysis would be impossible without building a model of your opponent’s decision-making, and because it is more of an art than a theory, it is probably impossible to come up with an easy-to-understand formula for determining bet sizing.
How To Use Pocket Pairs
When playing the game, it seems like there is a certain number of players who like pocket pairs. These are players who go all-in pre-flop with aggressive moves even with low-ranking pocket pairs such as 33 and 55. Even if they don’t go all-in, some players bet aggressively even when the flop shows a card of higher ranking than a pocket pair. Statistically speaking, the winning probability of a low-ranking pocket pair is not high, so it is not appropriate to use it as an all-in from pre-flop (the exception is when there is only one person going all-in and the stack is small, but it seems that it is okay to go all-in because the winning probability is 50/50 against high-ranking hands such as AKo), and if a common card of a higher rank than a pocket pair is played, the winning probability is not high either, so it does not seem like a rational action (unless it is a bluff). In this article, I would like to write about how I think it would probably be best to make use of it in this way.
Among pocket pairs, the higher rankings (AA, KK, QQ, JJ) have a higher chance of winning, so you should bet and raise aggressively. It could be said that these hands have a chance of winning even if you go all-in preflop (although if your stack is close to 100bb, I think it would be better to just play AA and KK). If you are going all-in pre-flop, it is better to wait until you have received a re-raise in response to a raise, or after waiting until chips have accumulated in the pot from later positions such as the SB or BB, rather than doing so immediately from an earlier position. This is because it will increase the number of chips you can win. It feels like a waste to have such a strong hand and have everyone fold and only get 1.5bb. Even if you can’t go all-in preflop, you can go all-in on the flop if the community card situation is not bad.
To be honest, I don’t really know how to handle a mid-ranking pocket pair (TT, 99, 88, 77). Going all-in pre-flop doesn’t give you a high chance of winning, and the chances of getting a higher-ranking common card on the flop don’t seem low either. This may be a hand that allows you to be aggressive when it seems like all the remaining players have not hit their cards on the flop. Of course, if you get three of a kind on the flop, you have a good chance of winning, but it’s not easy to get a strong bet from your opponent, and if you make a strong bet yourself, your opponent may be more likely to suspect that you might get three of a kind. In this case, a better strategy may be to bet as if you had one pair and wait for your opponent to raise or make a strong bet.
With low-ranking pocket pairs (66, 55, 44, 33, 22), it is not appropriate to go all-in preflop (unless you are in a tournament and your stack is under 10bb and you have no choice), and I don’t think there is any need to bet aggressively for any reason other than as a bluff. The uses of these cards might be as follows: For example, suppose you have AKo in your hand and the flop shows A, K, and 5 as community cards. With a high ranking two pair on the flop, you’re probably just trying to win the game and wondering whether to go all-in on the flop or how to make a good CB and get more chips out of your opponent. Most people probably don’t worry about what to do if a player has three of a kind for 5s.
It may be that low-ranking pocket pairs have an advantage in situations like this (which are not very likely to occur statistically) because they are more likely to draw big bets from aggressive players. In this case, rather than betting aggressively as if you had three of a kind, you can encourage the aggressive player to take the risk by pretending to have a pair of Aces, for example. Thinking about it this way, the author’s opinion is that low-ranking pocket pairs are more of a hand to hide from the opponent rather than a hand to show off the fact that it is a strong hand.
Preflop All-In
To be honest, when I first started playing the game, I didn’t have a good impression of players who went all-in pre-flop. If there is an inconvenient truth about Texas Hold’em, it is that by abusing all-ins pre-flop, the game becomes a game of chance with no skill involved, and every player has the right to do so. In fact, there are cases where players who have lost a lot of chips in a game will go all-in pre-flop without any consideration, and you might think, “Hey, stop disturbing the game and just get out of the table.”
However, recently I’ve come to think that how well you can play pre-flop all-ins may also be an important factor in Texas Hold’em. I wrote that when loose-passive players play against each other, the element of luck is strong, and because they are timid, they tend to bet small and have few opportunities to win a lot of chips. However, the chips will move significantly between these players when the game goes all-in pre-flop. Playing all-ins well pre-flop may be the most efficient strategy for a loose-passive player.
In addition, if you are running low on chips during a tournament, you may find yourself in a situation where your only strategy is to either fold or go all-in. Generally, when you have less than 10bb of chips (known as a short stack), your strategy options are almost gone and you have to endure, praying that you’ll get a hand that allows you to go all-in. When you think about it this way, the skill of being able to play pre-flop all-ins well may be surprisingly important in Texas Hold’em. I plan to look into this theoretically in the future (using game theory and Monte Carlo simulations), but I won’t go into the details here (or rather, I don’t have the ability to explain it at this point).
How to Play against Tight Passive Players
Although the title is “tight passive,” I will describe the characteristics of tight passive and tight aggressive players and consider countermeasures. Tight aggressive is known as the standard playing style used by advanced players. A balanced tight-aggressive is, in a sense (in a game-theoretic sense), a player with no weaknesses, and one that may be difficult to beat in the first place.
Unlike loose players, tight players will generally fold preflop except with limited hands. It seems that a preflop non-fold rate (known as VPIP) of around 5-20% is standard. The range of participating hands depends on the player, but generally looks something like this:
Pocket Pairs: AA,KK,QQ,JJ
Suited: AKs,AQs,AJs,KQs,KJs,QJs
Off-suited: AKo,AQo,AJo,KQo
Of course, there are times when players will enter the game with weak hands in order to include bluffing (even in this case, they will often have Axes, Kxes, Suited-Connectors from JTs to 76s, or low-ranking pocket pairs), but this is probably not a very common occurrence. When two tight players remain on the flop, the hand range is narrow, so it is likely that the situation will turn into a psychological battle of trying to read each other’s hands. There doesn’t seem to be much difference between tight passive and tight aggressive preflop play.
The difference between tight passive and tight aggressive is in their strategy from the flop onwards. Tight aggressive players bet aggressively, including bluffing, and use a similar strategy to loose aggressive players, except for their entry hand range. In contrast, a tight passive rarely bluffs and tends to choose to check or fold if they do not have a hand on the flop. Tight-passive players tend to be cautious when betting on potential flushes or straights, even when exposed to favorable community cards, and tend to fold to oversized bets on the river.
Tight passive players are generally risk averse and tend not to make large stakes bets unless they are certain they will win. This style of play is rarely seen among advanced players, and is known as NIT (apparently meaning rock). NIT players enter preflop with hands like AA, KK, QQ, and AKs, and fold in most games. Off the flop their strategy is similar to that of the tight passives, but with these hands they will often be able to play in an advantageous position even if the community cards are missed. However, this playing style requires extraordinary patience, as it can sometimes involve folding continuously for 2-3 hours (100-200 games), so it may not be a play style that most people can pull off. Also, although the NIT is certainly strong, they tend to be looked down upon as poker players in the sense that they are players who do not compete.
Let us look at counter strategies for tight passive and tight aggressive players. If you’re a loose player and you enter most games the easiest thing to do is just ignore them. This means that they will not play in 80% of the games, so you will earn chips in the games these players are not playing and fold when these players bet high preflop. When playing against a tight player, it is best to only do so if you have the same hand range as the tight player. If you have a premium hand or a high ranking pocket pair, you might want to go all-in preflop and try your luck. Tight players will get fed up with this happening over and over and will likely walk away.
If you are also a tight player and need to play it fair, or if you are loose but willing to take it seriously if you have a hand, the countermeasures for them is like the following:
- In games where these players are participating, try to narrow your hand. A tight passive player will not be aggressive in raising preflop, but would be better off choosing hands that are okay for a 3-bet range. If a tight passive open-raises, it is better to reraise and become the aggressor.
- On the flop, you should bet the CB. Unlike loose passive, you may use bluff bets to make your opponent fold. If it is a community card that gives you the potential for a flush or straight, you may want to consider using a bluff even if you don’t have those cards in your hand. If you are faced with a strong bet or your raise is returned, it may be time to back out.
- Unlike loose passives, they tend to make high-ranking one pair, two pair, and three-of-a-kind using pocket pairs, so if you have a hand that can beat these hands, you should make aggressive value bets.
- These players may not bet or raise aggressively (fake) even if they have a strong hand, so if they don’t fold on the flop, it may be best to refrain from betting or raising too aggressively on the river or turn.
- If your opponent makes a strong bet, it is unlikely to be a bluff, so if you don’t have any cards to counter it, you should generally fold.
Since a well-balanced tight-aggressive strategy is close to a Nash equilibrium strategy, you have no choice but to play a well-balanced tight-aggressive strategy yourself. The countermeasures against an overly aggressive tight-aggressive strategy are as follows:
- In games where these players are participating, try to narrow your hand. If you try to force the aggressor, it will lead to a raising battle, so it is better not to force a raise above the 3-bet range unless you are okay with going all-in (AA, KK, etc.).
- As with loose aggressive, your opponent will bet on their own even if you don’t bet yourself, so the basic strategy is to follow along by calling. Unlike loose aggressive, if you already have a strong hand, there is a high chance that a large raise in response to a CB will be folded, so if you want to make the pot bigger, it is best to call as long as your opponent has made a large enough CB.
- It is rare for a player to have a weak hand at the preflop stage, so if they use a bluff, it will be if they miss the flop (for example, if they had AKo but neither A nor K came out). It is a good idea to observe the flop’s CB and see if there is a difference between a value bet and a bluff bet. If it is a bluff-like CB, you can try to raise and bluff catch it.
- Tight-aggressive players do not easily fold even in close-fought situations, such as when comparing high cards or kickers, so even if you think you have a good chance of winning (and it’s not a close-fought situation to begin with), it may be a good idea to appear to be thinking long and hard before raising or calling right away.
- Even if you think a tight aggressive player is bluffing too much (being too aggressive), the hand itself is not weak, so unless you have a decent hand (such as in a high card or one pair kicker match), try not to force a hero call.
I understand that it is better to play tight preflop, but being a tight player may require a certain amount of patience. Furthermore, because I lack patience, I enter even with a random hand, and my VPIP exceeds 60%… I don’t think it has to do with my knowledge, but rather with my personality, where I just can’t be patient…
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