*The painter ChatGPT “A dramatic scene of a poker game in progress, featuring a player engaged in a Texas Hold’em bluff.”
Texas Hold’em and Utility Functions
Recently, I’ve started tapping away at the free Texas Hold’em game on my smartphone (although I’m writing this article even though I’ve never played it, so you might wonder). In a previous post, I explained how to determine a betting strategy for Texas Hold’em using expected utility theory, but I would like to add some supplementary notes based on what I noticed while actually playing the game.
In fact, the idea of determining a betting strategy to maximize an expected utility function is rarely used in Texas Hold’em. It seems common to consider a betting strategy as maximizing the expected value (EV) rather than the expected utility function. This may be because the nature of the game makes it common to bet all your chips (all-in), and the Kelly criterion and CRRA utility functions limit the range of strategies available by trying to avoid situations in which all chips are lost.
Also, unlike cash games, the value of chips gradually inflate in tournaments, so you have to come up with a strategy to grab as many chips as possible in as few games as possible. In addition, in tournaments, only the top 10% of players will receive any prize money, so if you are below that rank, whether you are in the top 11% or last place, there is no difference in the sense that you will not receive any prize money. Therefore, there is little need to think about risk management, and it may be more reasonable to judge only based on the expected value of how many chips you can take from your opponent.
However, from an economic perspective, devising a betting strategy in gambling that maximizes expected value is considered close to insanity. For example, suppose there is a bold play in which you double your bet if you win with a probability of \(\small p=0.51\). In this case, the optimal betting ratio under the Kelly Criterion was
\[ \small f^{\ast} = 2p-1 = 0.02. \]
However, in a strategy that maximizes expected value (risk-neutral utility function), the optimal betting strategy is \(\small f^\ast = 1\), making a full bet the optimal solution. In Texas Hold’em, a betting strategy that maximizes expected value (EV) is determined by this same criteria.
The reason why this discrepancy occurs is that in Texas Hold’em, the chips on the table do not represent the entire bankroll. If you have $1,000 in your bankroll, you would put about $100 in chips on the table (or maybe less), and if you go all-in and lose all your chips, you would have to buy new chips to continue the game. Thinking in this way, the utility function when losing all-in might be \(\small u(\$0)\) rather than \(\small u(\$900)\). Thinking in this way, it may be possible to apply a utility function that includes risk aversion to Texas Hold’em. An expectation maximization strategy corresponds to an approximation in which the chips on the table are negligibly small compared to the bankroll.
When calculating the Nash Equilibrium (GTO strategy) in Texas Hold’em, it seems that a strategy is constructed to maximize the expected value, but you will notice that there is no theoretical basis for saying that this “strategy that maximizes the expected value” is the optimal solution for this problem. It is the same as an axiom or principle in academic science; it has no theoretical basis, but can be thought of as a premise that is established with the feeling that it would be better if you think about it normally. However, from an economic perspective, this assumption seems very unnatural. If you are dissatisfied with the strategy that is derived from the Nash equilibrium that maximizes the expected value, you may want to question this assumption.
That’s all I wanted to add. For the rest of this article, I’ll share some of my thoughts on playing the game.I should note that the game I play is probably attended by beginner-level players (I am a beginner myself), so this may not generally apply.
High Limp-in Rate
Poker books often argue that beginners are far more likely to limp in (call a bet equivalent to the big blind), making them easy targets for chip thieves. Even if they are beginners, many people would read at least one of these books before starting the game, so the author thought that in reality, people who limp in are probably in the minority.
My impression after actually playing the game is that there are a lot of players who limp in, and I myself find myself limping in (despite having read several books on the subject). It’s a mysterious phenomenon where I can’t apply the lessons from the books I read…It’s also a very strange phenomenon, as for some reason everyone seems to be doing things that many authors tell people not to do. This may be due to the fact that, as mentioned in Section 1, people assume that the chips on the table are their entire bankroll and are afraid to bet aggressively.
Of course, there are also many players who can be assumed to practice what is written in the book. While he folds in most games, he is a player who tries to control the bet amount by raising aggressively when he gets a strong hand. Nash equilibrium strategies based on game theory recommend a strategy close to a tight-aggressive playing style, so many books argue that beginners tend to be loose-passive and would be better off consciously adopting a tight-aggressive playing style. Therefore, you might think that a player who plays with a tight-aggressive style of play is a step ahead of beginners. However, my impressions from playing the game is the following.
Tight Aggressive Players Are Not Strong As I Think
Of course, that doesn’t mean that a properly played tight aggressive player is weak, but I got the impression that a tight aggressive player who is just trying to imitate it is just as weak as a beginner limper. After playing a few games, it seems like players who play a textbook tight-aggressive style are subtly noticed. Tight aggressive players almost always have strong hands when they raise and come out strong, so if you don’t have a hand strong enough to counter them, you should just fold. As a result, tight-aggressive players often end up not winning many chips even with strong hands.
On the other hand, if you have a hand that can counter or is better than the opponent, you can slow play and they will automatically raise their bet, so you can just follow through with a call. It seemed to me that in many cases, if you were noticed as a tight-aggressive player, you would lose a lot of chips in this way. Also, it doesn’t work well with beginner limpers, and I think that players who mostly just call can get stung by them without realizing that they happen to have a strong hand (straight or flush). Tight aggressive players need to be patient as their chips slowly decrease, waiting for a strong hand to come their way, but just when they think they’re getting it, they may be stung by slow play and end up losing a lot of chips, leaving them at a loss.
In order to properly utilize tight aggression, it is necessary to consciously increase one’s participation rate in the game. In this case, there is a certain probability that you will participate even if you have a questionable hand, but if you do not appear to lose your aggressive style at this time, your questionable hand will be noticed. As a result, in order to be successful in tight aggression, you need to be able to effectively bluff questionable hands to make them look like strong ones. However, this bluff can be very difficult to manage, and when a fast-track tight-aggressive player tries it, it looks like this.
Bluffing Battle
You might think that bluffing is an advanced technique and that beginners would rarely use it, but in reality, I get the impression that there are many players who bluff frequently, even at tables that seem to be full of beginners. However, the way they use bluffs is often puzzling. This means making incredibly aggressive bets with weak cards from their hand or community cards that are not even close to being a winning combination. Of course, it would be great if they could fold all of their opponents, but in reality, I frequently saw the following phenomenon:
Suppose that two players folded preflop and four players limped in (one of them was me), and after the flop and turn, everyone just checked until the river. Probably none of the players had a good hand, and I was fully prepared to fold if someone raised with a pair of threes. I thought that everyone would check and the game would go to a showdown, but the last player to bet on the river suddenly made a 20bb raise. Given the situation with the common cards, it was clearly a bluff, but I wasn’t keen on it and folded it easily.
Oddly enough, the next player reraised to 40bb and then the other player raised the bet by going all-in. All of them were clearly bluffing, and the last one may have thought that if he went all in he could make the other two fold. However, that did not happen, and the player who made the original raise went all-in, and then the player who reraised to 40bb also went all-in. After the showdown, everyone had high cards, and the winning player’s kicker was also a questionable 9. The other two players had hands like 64o and 52o that looked like they had no chance of winning.
In the first place, such a situation would never arise when betting real money, but I can’t help but feel that the reaction to this bluff being countered with another bluff lacked rationality. Bluffing is done with the aim of forcing your opponent to fold when you do not have a strong hand, but if you fail to make your opponent fold you should retreat quickly and cut your losses, and whenever possible avoid revealing any cards to avoid giving away any evidence of your bluff. Once a player has gone all-in, the game is bound to go to showdown, so there is no point in raising the bet, and you will end up showing your hand to the other players. If you think about it this way, as soon as a player goes all-in, your bluff has failed and you should probably back out.
Perhaps in real life, bluffing is not often done with obviously weak hands, but is more often done in situations where it is unclear whether you are winning or your opponent is winning. Even if you bluff, it seems like it’s not worth it if you have too weak a hand. Also, if your bluff is called or countered with a raise, it means that your bluff has been discovered or your opponent is confident in their hand. If you are bluffing, you have to pretend to be an assassin, and if you can’t kill your enemy in one hit, you have to run away quickly, otherwise you’ll end up saying, “He’s tricky guy!”.
Finally, if a player who enjoys bluffing is the only bluffing player at the table, they can quite often force their opponents to fold regardless of their hand if there are a lot of passive players. However, when there are multiple players at a table who like to bluff, the tables often get caught in a bluffing battle, and it is not uncommon to see players lose all of their chips as the game goes to showdown with weak hands. I haven’t tried this myself, so I don’t know if this is a method that has a chance of success.
All-in Preflop
I wrote that in games with a lot of beginners, the proportion of limping-ins is quite high, and I noticed that there are a certain number of players who go all-in from the SB or BB position as a way of attacking players who limp-in frequently. In this case, the player who limped in will usually be folded and lose 1bb worth of chips. Because of this type of strategy, people may be told to fold rather than limp in. Also, from the perspective of a tight aggressive player, an opponent that is unpleasant is a player who hides a strong hand and intentionally limps in (apparently this is called a trappy). Trappy is a player who lets tight aggressive players get away with it and tries to make a comeback in the second half, so this may be a strategy to eliminate such players as much as possible.
However, what’s interesting is that when a player is suddenly all-in pre-flop, there are a considerable number of players who don’t fold but instead go all-in. Moreover, when I look at the cards at the time of showdown, they often aren’t that strong. I wonder what causes this phenomenon, but perhaps limpers are timid people who rarely raise or bet, and at the same time, they hate to lose. In short, in a deep stack game where a player has around 50-100bb of chips, going all-in preflop is clearly a bluff, but these people may not want to tolerate an obvious bluff, or may be willing to take on a challenge.
As a result, there is a good chance that if you go all-in suddenly, you will lose all your chips and be forced out (of course, the opposite can also happen). If you think about it this way, a strategy of going all-in and trying to get some chips from the limper is like picking up coins in front of a bulldozer, and may not be a very good strategy. Also, there are poker books that contain specific strategies for fighting all-ins pre-flop, so you may be targeted and stung by more experienced players at the table. In the end, when there are a lot of limpers, many of the players are entering the game with weak hands, so rather than forcing your way through with tight-aggressive, it may be safer to change your strategy to loose-aggressive and allow the probability to converge steadily.
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