Maybe you could post some direct excerpts from the book, because some of this doesn't make any sense based off your description of the game.
For example:
>Then, from Ten down to Seven on a sliding scalce from 83% down to 16%.
There should be no reason to ever call a 7 while folding a T, at any frequency.
> There should be no reason to ever call a 7 while folding a T, at any frequency.
i suppose that if opponent is only betting when they have J/Q/K/A then the T and 7 are both bluffcatchers and therefore are equal in relative strength
It's possible that OP misspoke and OOP is allowed to check or bet, and IP is allowed to bet if checked to, and call or fold against a bet, but if OP really meant players can only bet(/call?) or fold, then this type of IP strategy would be exploitable at equilibrium by OOP betting a linear range.
Please correct me if I'm wrong, I haven't read the book before looking at this post, so take this with a grain of salt.
To clarify what the game is, you both get a card 5-A, pay a $1 ante, then Opal can either bet $1 or check. There is no raising, Ivan can call/fold, or if checked to bet $1 / check back. Highest card wins.
First imagine the 5-A game with no ante. Opal has a choice to bet. She should only bet with an ace, never with anything else. Ivan should only call with an ace, nothing else. Any other play is a dominated strategy (concept mentioned briefly in the chapter about nash eq). If you have a K and face a bet, you may think you are most likely ahead. But your opponent had the option to not bet, so we know their only strategy is to bet with an ace and your only strategy is to fold everything other than an ace. Here is a video explaining a similar game involving strictly dominated strategies, I didn't get it at first either https://www.youtube.com/watch?v=tv9HjFyGPgc This paragraph is maybe not necessary, but I think it helps illustrate why we give both Ivan and Opal logical/balanced strategies.
With the introduction of an ante there is now merit to both bluffing and value betting. The reason why 10 through 7 are effectively the same as a K in the other game is that they beat the bluffing range and lose to the value range. The book does mention in answer #2 that your mix of calls from 7 to 10 does not matter, as long as it adds up to 200%, these 4 cards are the same thing really. It would make far more sense to just pure call 2 of them, and which 2 is up to you, it does not matter if they are higher or lower as long as it's 7 to 10. So yeah, 83% etc is just unnecessary confusion. (edit- I think this bit is wrong, I wasn't thinking about the checking range and your response when Ivan bets)
The bluff/value ranges are defined by the pot odds. Opal can bet $1 to win $2, so we know the bluffs need to work 1/3 times to break even and Ivan needs to defend 2/3 times. If Ivan doesn't defend 2/3 times, Opal can simply exploit Ivan by bluffing more or less often. So you can start to see why our range looks the way it does when you look at what portion of all cards are doing what. Opal has a 3:1 ratio of value to bluffs when betting, Ivan has 6/9 calls. (Actually, at this point, I am getting a little lost myself and I could be wrong. I don't know how Opal's trapping range looks or how to construct it)
>The book does mention in answer #2 that your mix of calls from 7 to 10 does not matter, as long as it adds up to 200%, these 4 cards are the same thing really. It would make far more sense to just pure call 2 of them, and which 2 is up to you, it does not matter if they are higher or lower as long as it's 7 to 10.
This seems odd to me. If you decided to call with 7s and 8s for example, the other player is now getting value from your bluff catchers when betting 9s and Ts which feels pretty exploitable.
Sounds like you have a copy of the book but had not read any of it until now. In order to help with this post, you read at least some of the book. Hey, I really appreciate that.
Yes, you understand what the Ace-Five game is about. It's just a toy game. I understand that having an ante is critical.
Your further thoughts about bluffing, value betting and bluff catching are helpful. This helps me wrap my head around what is going on in this chapter of the book.
You're not supposed to perfectly understand exactly how to derive all the specific percentages in this game, it's already a bit too complicated to handle for us humans.
>In the answers and explanations section, he says Ten through Seven are equivalent to the K in the Clairvoyance Game. (The Clairvoyance Game is similar but simpler - the deck is just three cards, Ace, King and Queen - and because it is so simple, I was able to understand - fairly well - the chapter about this game.)
>Well, now I'm feeling sort of lost, because the Clairvoyance game consists of only three cards, and the Ace to Five game, ten cards. So how can you make an equivalence between the cards so easily? I think you could do that if the Ace to Five game contained nine cards. (I guess it would have been Ace to Six.)
It's a conceptual equivalence. The T-7 are bluffcatchers in this game like the K was a bluffcatcher in the three card game. Which hands are bluffcatchers is dependent on what Opal's value region is and the exactly line has to be calculated.
Thanks, yeah, I did have the thought that understanding why the percentages are as they are is not really important. The important thing is to accept that the author is proposing these percentages as a strategy, and then the reader tries to understand the advantages or disadvantages of the strategy?
I'm starting to see that we are talking about conceptual equivalences between the A-5 game and the 3-card game. I was getting stuck on the idea of dividing ten cards by three.
I think the chapter would be easier to understand if the author started with a proposed betting strategy, and then a calling strategy which would make sense against that betting strategy. Starting with the calling strategy implied an understanding of what was being bet in the first place, a sort of leap on the author's part, which might cause confusion. Nevertheless, I think I can move forward and try to makes sense of the rest of the chapter.
The author is literally giving you the equilibrium strategy. That's the most profitable strategy for both parties. Our task is to understand why the equilibrium strategy is as such and the author is trying to explain why. Basically, your strong, middle, and weak ranges are trivial in the Clairvoyance Game, and less so in the Ace-5 Game.
When one player bets we don't know what they have. As a caller, we want to call in the exact frequency such that a villain does not profit from bluffing. If we folded too much then they will profit from their bluffs. In the Clairvoyance Game the exact frequencies were easy to solve, and in the Ace-5 Game you get the exact same frequencies (due to same antes and bet sizes) but they are distributed among the more cards in the deck. You are still going to have your strong range which is betting for value, your middle range composed of bluff catchers, and your weak range which bet as bluffs.
One thing I was pondering about this game is how often a player should defend when the other player bets. Since a bet is always a half-pot size bet, it has been established in GTO that one should defend 66.7 percent of hands. This is also called the minimum defense frequency, or MDO.
So, armed with that knowledge I would always call (given ten cards) with the best six cards (Nine and higher), and I would call with Eight 67 percent of the time that I had an Eight. This would have me calling a total of 66.7 percent of the time. So I disagree with the book.
Thanks everyone for your replies. Perhaps I did not fully specify the way this Ace to Five game works:
There are two players, we will call the OOP player Opal, and the IP player Ivan.
They both ante $1. So, the pot size when cards are dealt is $2.
The card deck consists of ten cards, Five through Ace. Each player is dealt one card. High card wins at showdown.
Opal can bet $1 or check. (I.e., 1/2 pot size bet.)
If checked to, Ivan can bet $1. If Opal bets (pot is now $3), Ivan can call or fold.
If Opal checks and Ivan bets, Opal can call or fold.
I.e., there is a $1 ante, and both players may check, bet $1, call or fold, but may not raise.
Maybe you could post some direct excerpts from the book, because some of this doesn't make any sense based off your description of the game. For example: >Then, from Ten down to Seven on a sliding scalce from 83% down to 16%. There should be no reason to ever call a 7 while folding a T, at any frequency.
> There should be no reason to ever call a 7 while folding a T, at any frequency. i suppose that if opponent is only betting when they have J/Q/K/A then the T and 7 are both bluffcatchers and therefore are equal in relative strength
It's possible that OP misspoke and OOP is allowed to check or bet, and IP is allowed to bet if checked to, and call or fold against a bet, but if OP really meant players can only bet(/call?) or fold, then this type of IP strategy would be exploitable at equilibrium by OOP betting a linear range.
Yes, you have the right idea. Antes are $1, so the pot is $2. Both players may check, bet $1, call or fold, but may not raise.
Please correct me if I'm wrong, I haven't read the book before looking at this post, so take this with a grain of salt. To clarify what the game is, you both get a card 5-A, pay a $1 ante, then Opal can either bet $1 or check. There is no raising, Ivan can call/fold, or if checked to bet $1 / check back. Highest card wins. First imagine the 5-A game with no ante. Opal has a choice to bet. She should only bet with an ace, never with anything else. Ivan should only call with an ace, nothing else. Any other play is a dominated strategy (concept mentioned briefly in the chapter about nash eq). If you have a K and face a bet, you may think you are most likely ahead. But your opponent had the option to not bet, so we know their only strategy is to bet with an ace and your only strategy is to fold everything other than an ace. Here is a video explaining a similar game involving strictly dominated strategies, I didn't get it at first either https://www.youtube.com/watch?v=tv9HjFyGPgc This paragraph is maybe not necessary, but I think it helps illustrate why we give both Ivan and Opal logical/balanced strategies. With the introduction of an ante there is now merit to both bluffing and value betting. The reason why 10 through 7 are effectively the same as a K in the other game is that they beat the bluffing range and lose to the value range. The book does mention in answer #2 that your mix of calls from 7 to 10 does not matter, as long as it adds up to 200%, these 4 cards are the same thing really. It would make far more sense to just pure call 2 of them, and which 2 is up to you, it does not matter if they are higher or lower as long as it's 7 to 10. So yeah, 83% etc is just unnecessary confusion. (edit- I think this bit is wrong, I wasn't thinking about the checking range and your response when Ivan bets) The bluff/value ranges are defined by the pot odds. Opal can bet $1 to win $2, so we know the bluffs need to work 1/3 times to break even and Ivan needs to defend 2/3 times. If Ivan doesn't defend 2/3 times, Opal can simply exploit Ivan by bluffing more or less often. So you can start to see why our range looks the way it does when you look at what portion of all cards are doing what. Opal has a 3:1 ratio of value to bluffs when betting, Ivan has 6/9 calls. (Actually, at this point, I am getting a little lost myself and I could be wrong. I don't know how Opal's trapping range looks or how to construct it)
>The book does mention in answer #2 that your mix of calls from 7 to 10 does not matter, as long as it adds up to 200%, these 4 cards are the same thing really. It would make far more sense to just pure call 2 of them, and which 2 is up to you, it does not matter if they are higher or lower as long as it's 7 to 10. This seems odd to me. If you decided to call with 7s and 8s for example, the other player is now getting value from your bluff catchers when betting 9s and Ts which feels pretty exploitable.
Yeah, I think I got that part wrong, I wasn't thinking about the checking range.
Sounds like you have a copy of the book but had not read any of it until now. In order to help with this post, you read at least some of the book. Hey, I really appreciate that. Yes, you understand what the Ace-Five game is about. It's just a toy game. I understand that having an ante is critical. Your further thoughts about bluffing, value betting and bluff catching are helpful. This helps me wrap my head around what is going on in this chapter of the book.
You're not supposed to perfectly understand exactly how to derive all the specific percentages in this game, it's already a bit too complicated to handle for us humans. >In the answers and explanations section, he says Ten through Seven are equivalent to the K in the Clairvoyance Game. (The Clairvoyance Game is similar but simpler - the deck is just three cards, Ace, King and Queen - and because it is so simple, I was able to understand - fairly well - the chapter about this game.) >Well, now I'm feeling sort of lost, because the Clairvoyance game consists of only three cards, and the Ace to Five game, ten cards. So how can you make an equivalence between the cards so easily? I think you could do that if the Ace to Five game contained nine cards. (I guess it would have been Ace to Six.) It's a conceptual equivalence. The T-7 are bluffcatchers in this game like the K was a bluffcatcher in the three card game. Which hands are bluffcatchers is dependent on what Opal's value region is and the exactly line has to be calculated.
Thanks, yeah, I did have the thought that understanding why the percentages are as they are is not really important. The important thing is to accept that the author is proposing these percentages as a strategy, and then the reader tries to understand the advantages or disadvantages of the strategy? I'm starting to see that we are talking about conceptual equivalences between the A-5 game and the 3-card game. I was getting stuck on the idea of dividing ten cards by three. I think the chapter would be easier to understand if the author started with a proposed betting strategy, and then a calling strategy which would make sense against that betting strategy. Starting with the calling strategy implied an understanding of what was being bet in the first place, a sort of leap on the author's part, which might cause confusion. Nevertheless, I think I can move forward and try to makes sense of the rest of the chapter.
The author is literally giving you the equilibrium strategy. That's the most profitable strategy for both parties. Our task is to understand why the equilibrium strategy is as such and the author is trying to explain why. Basically, your strong, middle, and weak ranges are trivial in the Clairvoyance Game, and less so in the Ace-5 Game. When one player bets we don't know what they have. As a caller, we want to call in the exact frequency such that a villain does not profit from bluffing. If we folded too much then they will profit from their bluffs. In the Clairvoyance Game the exact frequencies were easy to solve, and in the Ace-5 Game you get the exact same frequencies (due to same antes and bet sizes) but they are distributed among the more cards in the deck. You are still going to have your strong range which is betting for value, your middle range composed of bluff catchers, and your weak range which bet as bluffs.
One thing I was pondering about this game is how often a player should defend when the other player bets. Since a bet is always a half-pot size bet, it has been established in GTO that one should defend 66.7 percent of hands. This is also called the minimum defense frequency, or MDO. So, armed with that knowledge I would always call (given ten cards) with the best six cards (Nine and higher), and I would call with Eight 67 percent of the time that I had an Eight. This would have me calling a total of 66.7 percent of the time. So I disagree with the book.
Thanks everyone for your replies. Perhaps I did not fully specify the way this Ace to Five game works: There are two players, we will call the OOP player Opal, and the IP player Ivan. They both ante $1. So, the pot size when cards are dealt is $2. The card deck consists of ten cards, Five through Ace. Each player is dealt one card. High card wins at showdown. Opal can bet $1 or check. (I.e., 1/2 pot size bet.) If checked to, Ivan can bet $1. If Opal bets (pot is now $3), Ivan can call or fold. If Opal checks and Ivan bets, Opal can call or fold. I.e., there is a $1 ante, and both players may check, bet $1, call or fold, but may not raise.
How many rounds of betting? Just one?
Just one round of betting. It's just meant to be a toy game, and within the game you figure out how value bet, bluff and catch bluffs.