WARNING: SPOILERS ARE COMING
In Game of Thrones Season 6 Episode 10, the Sparrows are holding a trial for Cersei Lannister, which they expect her to attend. It is clear that the Sparrows have more military strength than Cersei. They have the Faith Militant, as well as the King’s army. Cersei, at this point, only has her zombie bodyguard, The Mountain.
When Cersei does not show up for trial, the High Sparrow is unconcerned and thinks that he has won; namely, that he may now take Cersei prisoner by force. However, Margaery Tyrell reaches a different conclusion. Margaery says to the High Sparrow, “Cersei understands the consequences of her absence, and she is absent anyway, which means she does not intend to suffer those consequences.” The High Sparrow responds with a shrug.
In David Sklansky’s book, The Theory of Poker, he discusses the concept of multiple-level thinking. Below is a rough idea of what it means to think on each level, together with an example from Texas Hold’em.
Level 0: What do I have?
Ex: I have a straight.
Level 1: What do I think my opponent has?
Ex: I think my opponent has 2 pair or something similar.
Level 2: What do I think my opponent thinks I have?
Ex: I think my opponent thinks that I was going for a draw. It could have been a flush or a straight. It is the river, and the 3rd flush card did not come, so s/he might think I have a busted flush draw. It is also possible that s/he thinks I hit my straight.
Level 3: What do I think my opponent thinks I think s/he has?
Ex: My opponent thinks that I think s/he has 2 pair. On the river, when my opponent bets, I raise, and s/he re-raises. I must realize that my opponent thinks that when I raised, I thought s/he was betting and would call a raise with two pair. If I am re-raised s/he likely has a hand that could beat the completed draws that would raise against two pair.
Level 4: What do I think my opponent thinks I think s/he thinks I have?
As you can see, the concept gets complicated quickly. Fortunately, by the time we reach Level 3, we are thinking deeper than most other players. It is important to both go through this analysis ourselves as well as having an idea of what level our opponent is thinking. If your opponent only thinks up to Level 1, s/he will miss opportunities where s/he may either bluff credibly or value bet thinly.
In our Game of Thrones scenario above, we can get an idea of what level Margaery is thinking on.
Level 0: We have the Faith Militant and the military forces of the realm.
Level 1: I think that Cersei has The Mountain. The Mountain is strong, but not as strong as our forces.
Level 2: I think that Cersei knows that we have a stronger fighting force. But she challenged us anyway. Margaery says, “Cersei understands the consequences of her absence.” Here, Margaery realizes that the only way Cersei’s actions make sense is if Cersei’s hand is stronger than Margaery initially thought.
The High Sparrow stopped at Level 1. When Cersei raised, he stuck to his initial conclusion. He failed to consider why she was raising when it looked obvious that she had been beaten. This turned out to be a huge mistake. If Margaery Tyrell sat down at your poker table, you might want to find a better game. In the game of thrones, you have to know what level you are playing on.
So in practice how would you decide what level of thinking to operate on? Do you simply try and think ahead as many “moves” as possible like in chess, or would you try to have a more general “reaction function” to the information set you have, i.e. the set of info of all past reactions of the opponent, and the current cards at play?
In practice I want to have an idea of what level my opponent is thinking on and then think one level above them. It is not useful to think about what my opponent thinks my hand is if they are a player who only considers the strength of their own hand. To determine what level my opponent is thinking on I need to observe their actions across multiple hands. This is why it is so important to be observant even when not playing in a hand.
You can use this thought process to plan for future betting rounds. For example, if I think my opponent is the type of player who will pay off large bets with just a pair of aces in no-limit holdem, he is probably thinking on level zero. He is only considering that he has aces. On the flop when calculating whether I have the correct odds to call a bet with my draws I should include the potential bets I will get in on later streets if I hit my draw. Against an opponent who is thinking on level one I am not going to get paid off on my draws as often because they will potentially read my hand as a completed straight/flush. Note that in this case because I know they are thinking on level one, I can now consider bluffing some percent of the time when a potential draw is completed. I could bluff with a missed straight draw when a 3rd flush card hits. Against a player thinking on level zero I would rarely consider bluffing in this spot.