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    Default spoonitnow's 5000th Post: Game Theory and Poker

    Yeah so I started this thread the other day knowing that it would take a whole lot of posts before I finished everything I wanted to on the subject. A lot of people start into Mathematics of Poker wanting to learn some game theory to apply to poker, but without a background in mathematics it can be a pretty difficult read. It also doesn't cover some of the more basic games in-depth, opting to leave a lot of that work to the reader. My goal here is to present a series of toy games that are particularly instructive, some that are covered in that book, and some that aren't. I'm doing this for myself just as much as I am anyone else, but I promise that if you work through these games and the discussion of them that you'll learn a lot that you can apply directly to your game right now and see good improvements.

    I think it's worth mentioning that I'm aware that the vast majority of people who open this thread won't work through more than 10-20% of it. That's because less than 10-20% of people who read this thread want to be successful.

    Best of luck, and here's to 5000 more posts.

    The Half-Street Heads-Up 13-Card Fixed-Limit Game

    So here's how this works. I'm going to describe the rules of a poker game to you and ask you some questions. I'm not ready to go to bed yet but am bored as hell which is why I'm doing this. Before I even get started, I already know there's going to be some jackass who wants to chime in and say that this is useless in terms of learning "real" poker or some horse shit like that. You sir, are wrong.

    Okay so here's the deal. We have a 13-card deck that comprises of all of the spades in a normal deck. So there's an Ace, King, Queen, Jack, Ten, Nine, Eight, Seven, Six, Five, Four, Three, and Two in the deck.

    There are two players, Hero and Villain, and Hero is in position with Villain out of position. Each player posts an ante of P/2 before the cards are dealt. Each player receives one card, then Villain is forced to check in the dark. Hero then has the choice to bet 1 unit, or check to showdown. If Hero bets, then Villain has the choice to call and see a showdown or fold. At showdown, whoever has the highest card wins.

    Here are some questions, and I'll probably come up with some more questions later.

    1. How often does Villain have to fold for Hero to have a +EV bluff?
    2. If Villain calls with a 7 or higher, what hands can Hero make a +EV value bet with?
    3. If each ante is 0.25 units, what is the minimal number of hands Villain has to call a bet with so that Hero can't make a +EV bluff?
    4. If Villain calls with an 8 or higher, what is the EV of betting a T for Hero? of checking a T for Hero?
    5. In question 4, is betting ever better than checking for Hero?
    6. If Villain calls with an 8 or higher, what is the EV of betting a J for Hero? of checking a J for Hero?
    7. In question 6, compare betting and checking for Hero. What do you notice? Should Hero bet or check, and why?
    8. If the antes are 1 unit each and Villain calls with J+, how should Hero play his entire range?
    -- 8a. How does the EV of betting hands change as you look at worse and worse hands?
    -- 8b. How does the EV of checking hands change as you look at worse and worse hands?

    That should get you started studying this game.

    A guide to finding the answers is in the next post of this thread, but you're highly advised to work it out on your own to begin with.
    Last edited by spoonitnow; 12-10-2010 at 11:58 AM.
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    1. How often does Villain have to fold for Hero to have a +EV bluff?

    This is just bet/(bet+pot) which here is just 1/(1+P). If you don't understand this, stop reading this thread and go study my mathematics of EV thread.

    2. If Villain calls with a 7 or higher, what hands can Hero make a +EV value bet with?

    Look at his calling range:

    A K Q J T 9 8 7

    We can only bet hands that will beat more hands than it loses to. The worst hand that meets this requirement is a Jack which beats 4 and loses to 3 of the hands in Villain's calling range. Therefore, we can make a +EV value bet with an Ace, King, Queen or Jack.

    3. If each ante is 0.25 units, what is the minimal number of hands Villain has to call a bet with so that Hero can't make a +EV bluff?

    If each ante is 0.25 units then the pot is 0.5 units. We're making a bet of 1 unit, so we need our opponent to fold bet/(bet+pot) percent of the time, which here is 1/(1+0.5) = 1/1.5 = 66.7%. So we need Villain to fold 66.7% of the 12 hands in his range, thus calling with at least 4 hands, so he'll have to call with at least a Jack or better to keep Hero from making +EV bluffs.

    4. If Villain calls with an 8 or higher, what is the EV of betting a T for Hero? of checking a T for Hero?

    First the EV of betting. There are 4 out of 12 cards (A, K, Q, J) that Villain will call with and we lose 1 unit. Then there are 2 out of 12 cards (9, 8) that Villain will call with and we win the pot of P plus the 1 unit he calls. Finally, there are 6 out of 12 cards (7, 6, 5, 4, 3, 2) that Villain will fold and we collect the pot of P. That makes our EV of betting:

    (4/12)(-1) + (2/12)(P+1) + (6/12)(P)

    Now the EV of checking. There are 4 out of 12 cards (A, K, Q, J) that we will lose to at showdown and show a profit of 0 on our check. Then there are 8 out of 12 cards (9, 8, 7, 6, 5, 4, 3, 2) that we will beat at showdown and show a profit of P. Therefore the EV of checking is:

    (4/12)(0) + (8/12)(P)

    5. In question 4, is betting ever better than checking for Hero?

    The simple way to do this is to set the EV of betting to be greater than the EV of checking and see if the equation still makes sense after we simplify things. So if the EV of betting is better than the EV of checking, then:

    (4/12)(-1) + (2/12)(P+1) + (6/12)(P) > (4/12)(0) + (8/12)(P)
    (4/12)(-1) + (2/12)(P+1) + (6/12)(P) > (8/12)(P)
    -4 + 2(P+1) + 6P > 8P
    -4 + 2(P+1) > 2P
    -4 + 2P + 2 > 2P
    -2 > 0

    Since -2 is never greater than 0, we can determine that the EV of betting is never better than the EV of checking in question 4.

    6. If Villain calls with an 8 or higher, what is the EV of betting a J for Hero? of checking a J for Hero?

    This is the same process and everything as question 4. The EV of betting is (3/12)(-1) + (3/12)(1+P) + (6/12)(P) and the EV of checking is (3/12)(0) + (9/12)(P).

    7. In question 6, compare betting and checking for Hero. What do you notice? Should Hero bet or check, and why?

    Let's simplify the EV of betting:

    (3/12)(-1) + (3/12)(1+P) + (6/12)(P)
    (-3/12) + (3/12) + (3/12)(P) + (6/12)(P)
    (9/12)(P)

    Now let's simplify the EV of checking:

    (3/12)(0) + (9/12)(P)
    (9/12)(P)

    What you should notice is that no matter what, they're always the same. That is, the EV of betting here is always the same as the EV of checking. Now would be a good time to think about why you should bet this hand anyway. (If you don't get it, go read this then come back.)

    8. If the antes are 1 unit each and Villain calls with J+, how should Hero play his entire range?

    For this question you need to find the EV of checking and the EV of betting with each individual hand Hero could have to see what the best play is for each hand based on the size of P. Here is an example:

    EV of betting a K =
    (1/12)(-1) + (2/12)(P+1) + (9/12)(P)
    (1/12)(-1) + (2/12) + (2/12)(P) + (9/12)(P)
    (11/12)(P) + (1/12)

    EV of checking a K =
    (1/12)(0) + (11/12)(P)
    (11/12)(P)

    In the above you see that betting a K is always better than checking a K, no matter what the pot size is. Similarly you'll find that you should always bet an A.

    For playing a 7, you should find that betting is better than checking when P > 4/3.
    For playing a 2, you should find that betting is better than checking when P > 1/2.

    That should be a good enough guide for figuring it out. Notice that our range is always polarized. For example, what hands are +EV to bet P = 1.05? The range is {A, K, 6, 5, 4, 3, 2}. It would be worth looking at if Villain can call profitably with a T against this betting range. Note that if so, he can also call profitably with a 7.

    8a. The EV of betting a T, 9, 8, 7, 6, 5, 4, 3 or 2 are all the same.

    8b. The EV of checking decreases as you look at worse and worse hands.
    Last edited by spoonitnow; 12-08-2010 at 04:38 AM.
  3. #3
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    Concepts at Work

    Question 1 asks a basic question that is much more important than a lot of people seem to think. The value 1/(1+P), or bet/(bet+pot) is an extremely important value in poker. In general, it tells us which hands we can make a +EV bluff with in a vacuum. This is pretty basic level stuff, so let's move on.

    Sometimes we'll have an idea of how our opponent plays his range, like here in question 2 we expect him to call with a 7 or better. From that, we can quickly figure out what hands are +EV to value bet in a vacuum. Combined with the concept from question 1, we can form a good estimate of what our betting range should look like and then go from there.

    You can also do the opposite, like in question 3, which is to look at the pot and determine what range your opponent has to be calling with to keep from being super exploitable to bluffs. This is important since you need to be able to gauge this range and know what it means in terms of playing your bluffs as well as your value bets.

    In question 4 we're really starting to figure out specific exploitative strategies for single hands given the tendencies of our opponent. In question 5, we use the information from question 4 to figure out that betting is never better than checking, which we will usually expect with hands that reside in the bottom half of our opponent's calling range.

    Questions 6 and 7 are similar to 4 and 5 except we discover that betting a J (which is the exact center of our opponent's calling range) has the same EV as checking in a vacuum. This means we should bet our Jacks, even though it has the same EV in a vacuum, because of what it will do for our range.

    This last point about betting our Jacks in this scenario (where Villain calls with T or better) is very important. If our opponent sees us betting a Jack, he has 3 things he can do. The first thing he can do is not adjust his play at all, in which case we'll be able to exploit his folding frequency when the pot size is correct and we'll never miss value or value bet a hand that loses money. In short, if he does nothing, life is really easy for us and we're printing money.

    The second thing he can do is call less to make it so that Hero betting a Jack is -EV in a vacuum. The problem with this is that our bluffs become more +EV than they already were at the same time. At this point, you could figure out the EV of our range as currently played if our betting range was {J+, 6-2} and then compare it to the EV of our range if we continue to bet {J+, 6-2} but then he starts folding all of his 8's. If you have any amount of doubt whatsoever, you should do this exercise at least once to see it right in front of you.

    Now we're ready for question 8, which is what we've been building up to the whole time. We quickly find a range that we should always value bet (K+) and then see that we should never bet a Q or J or T (notice again that these hands include the bottom half of his calling range).

    Our bluffing range is interesting also. The first hand we add as a bluff is a 2, which is intuitive and understood by most people, but here we're able to quantify it and prove why bluffing with a 2 is better than a 3. It's not that the EV of betting a 2 is higher than betting a 3, because betting each has the same exact EV. The difference is that checking a 2 is worse than checking a 3. So we gain the most value for our range by betting a 2 and checking a 3.
    Last edited by spoonitnow; 12-08-2010 at 04:21 AM.
  4. #4
    U = bet unit, P = pot size in units

    1. U/(P+U)
    2. A,K,Q,J
    3. 1/1.5 = 2/3, so villain has to call at least 1/3 == 4 cards
    4.
    EV betting: he calls 2/12 we beat, he calls 4/12 we lose, he folds 6/12
    EV(bet) = 2/12(P+U) - 4/12(U) + 6/12(P) = 8P/12 - 2U/12

    EV checking: we beat 8/12, lose to 4/12
    EV(check) = P*8/12 - 0*4/12 = 8P/12

    5. Will 8P/12 - 2U/12 ever be greater than 4P/12? No, unless U is negative (which doesn't make sense).

    6.
    EV betting: he calls 3/12 we beat, he calls 3/12 we lose, he folds 6/12
    EV(bet) = 3/12(P+U) - 3/12(U) + 6/12(P) = 9P/12

    EV checking: we beat 9/12, lose to 3/12
    EV(check) = P*9/12 - 0*3/12 = 9P/12

    7. The EV is the same, so it doesn't matter which he chooses from an EV perspective.

    8a. Once your holding is behind the villain's calling range, you can't profitably bet anymore.
    8b. As your hand gets worse against the opponent's entire range, you can't bet for value once you are behind their calling range. But you can bluff with the worst of your hands, depending on their calling frequency and your bluff size (because fold equity alone will make it more EV than checking).

    This implies that we should bet a polarized range -- bet the top of our range for value, and the bottom for fold equity. The exact values depend on the villain's calling range and the bet size. After some Excel magic for a pot size of 2 units:

    Bet: { A,K,8,7,6,5,4,3,2 }
    Check: { Q,J,T,9 }
  5. #5
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    For example, what hands are +EV to bet P = 1.05? The range is {A, K, 6, 5, 4, 3, 2}. It would be worth looking at if Villain can call profitably with a T against this betting range. Note that if so, he can also call profitably with a 7.
    I asked the above near the end of post #2 of this thread, and I'd like to answer it now and discuss why it's important.

    We determined from question 8 that the list of hands that are +EV for Hero to bet in a vacuum when P = 1.05 are {A, K, 6, 5, 4, 3, 2}. If Villain holds a medium strength hand like a T, then his equity against our betting range is 5/7 = 71.4% since he beats 5 hands and loses. However, he only needs 1/3.05 = 32.8% equity to call.

    This is a perfect example of how when we play in an exploitative manner, our opponent is able to counter by adjusting and exploiting us in return. But how much can he exploit us?

    What if he holds a 4. Then our betting range becomes {A, K, 6, 5, 3, 2}, against which he has 33.3% equity. That means that he can call with a 4 (or higher), which is the vast majority of his range, and it's only correct for him to fold a 3 or 2 (though folding a 2 is obvious since it can never be the best hand).

    What's important to note is that the thought process that got us the range {A, K, 6, 5, 3, 2} was that we just picked all of the hands we thought were +EV to bet in a vacuum and bet them. This is so important because this is the thought process most people reading this will have, and we're going to look at how it could be exploited so you know what to look out for in your opponents as signs that you should change your ranges up.

    Note that if we wanted to play unexploitably against a hand like a T, we'd bluff an amount that would mean he would have exactly 1/3.05 = 32.8% equity, which is betting an Ace and King 100% of the time while betting a 2 about 97% of the time. It's usually good to have a decent idea of what the unexploitable strategy looks like in terms of the ranges involved in spots like this because that serves as a guide if we decide we need to make future adjustments.

    So suppose that he starts calling us a bit wider than he was before. Say he goes from calling with a Jack or better to calling with an Eight or better. What sorts of adjustments should we make? The first adjustment that we saw from above is to bluff less (which is also obvious because now he's calling more). The second adjustment is to value bet more, which in this case means we can value bet a Jack or better.

    Now let's take a moment to go back to the point where he's only calling with a Jack or better and we're betting a range of {A, K, 6, 5, 3, 2}. Let's say that we're afraid our strategy is too obvious and that our opponent will adjust easier. How can we change our strategy so that it's harder for him to adjust without changing the EV of our range very much?

    Again, we should be thinking about value betting more or bluffing less. The value we lose by betting a Q instead of checking it in a vacuum with P = 1.05 is

    (EV of checking - EV of betting)
    ((2/12)(0) + (10/12)(1.05)) - ((2/12)(-1) + (1/12)(2.05) + (9/12)(1.05))
    (0.875) - (-2/12 + (2.05/12) + (9.45/12))
    0.08333

    Then you could compare that to the value we lose by checking a 6 instead of betting it in a vacuum with P = 1.05:

    (EV of betting - EV of checking)
    ((4/12)(-1) + (8/12)(1.05)) - ((8/12)(0) + (4/12)(1.05))
    (-1/3 + (8.4/12)) - (4.2/12)
    0.01667

    Here we see that betting a Q costs us five times as much than checking the 6, so that's something you'd need to take into consideration if you decided to play in a slightly more balanced way.
  6. #6
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    These are my answers after looking only at OP, except for 8) where I first misunderstood that Hero had to choose between blindly checking or betting his entire range. I corrected after I read post #2.

    1) Villain's folding frequency has to be more than U/(U+P).
    2) There are 8 hands in villain's calling range (789TJQKA), and we have to be ahead of > 50% of these for EV to be >0, which is the case when we hold A,K,Q orJ
    3) P=0.5U, so villain has to fold > 2/3rd of the time for a bluff to be +EV. So villain has to fold < 2/3rd of the time to prevent a +EV bluff. So he has to call with at least 4 hands: A, K, Q and J.
    4) EVb=6/12*P-4/12*U+2/12*(P+U)=1/12(8P-2U)=1/6*(4P-U)=2/3*P-1/6*U
    EVc=8/12*P=2/3*P
    5) no
    6) EVb=6/12*P-3/12*U+3/12*(P+U)=1/12*9P=3/4P
    EVc=9/12P=3/4P
    7) Betting and checking are the same as EV is the same. However if we bet, he'll sometimes fold and does not get information about the hand we bet. Since we bet more, he should tend to adjust towards calling more which allows us to value bet our good hands more successfully. If we check more, we can bluff more in the future.
    8) EVba2c=9/13*2U+1/13*(9/12*-U+3/12*3U)+1/13*(10/12*-U+2/12*3U)+1/13*(11/12*-U+1/12*3U)+1/13*-12/12U
    =1/13*[18U+1/12*(-6U-8U-10U-12U)]=1/13*[18U-3U]=1/13*15U=15/13U
    EVca2c=U
    Should bet everything.

    edit: correction after understanding what was really asked:

    EVb(A)=3/12*3U+9/12*2U=27/12*U
    EVc(A)=2U=24/12*U

    EVb(K)=2/12*3U-1/12*U+9/12*2U=23/12*U
    EVc(K)=11/12*2U=22/12*U

    EVb(Q)=1/12*3U-2/12*U+9/12*2U=19/12*U
    EVc(Q)=10/12*2U=20/12*U

    EVb(J)=-3/12*U+9/12*2U=15/12*U
    EVc(J)=9/12*2U=18/12*U

    EVb(2-T)=-4/12*U+8/12*2U=12/12U
    EVc(T)=8/12*2U=16/12*U
    EVc(9)=7/12*2U=14/12*U
    EVc(8)= ... =12/12*U
    EVc(7)= ... =10/12*U
    EVc(6)= ... =8/12*U
    EVc(5)= ... =6/12*U
    EVc(4)= ... =4/12*U
    EVc(3)= ... =2/12*U
    EVc(2)= ... =0/12*U

    So:
    bet: A, K, 2-7
    check: Q, J, T, 9
    bet: 8 for m2m-style metagame purposes until villain adjusts.

    8a) EV of betting decreases from A to T, doesn't matter below that.
    8b) EV of checking decreases linearly from A to 2.
    Last edited by daviddem; 12-08-2010 at 06:49 AM.
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  7. #7
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    Why are you guys using a U anywhere to begin with? The bet size is just 1 and the pot is P.

    Also for clarification, Hero has the option of playing mixed strategies. For example, he could decide to bet 100% of A's and K's, 75% of Q's, 4% of J's, and 71% of 2's if he so pleased.
    Last edited by spoonitnow; 12-08-2010 at 01:03 PM.
  8. #8
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    Quote Originally Posted by spoonitnow View Post
    Why are you guys using a U anywhere to begin with? The bet size is just 1 and the pot is P.
    Because you said he bets 1 unit, so we called U the unit (I didn't look at Gizmo's post before I did that, it's just a coincidence that we both decided to call it U). I just thought it'd be easier to spot a U in the formulas and substitute later rather than have to decide later what is a "1" and what is "1 unit" in the formulas.
    Last edited by daviddem; 12-08-2010 at 02:08 PM.
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  9. #9
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    The antes are also P/2 units, so you would have to write them as PU/2, and then the pot would be PU.

    Hopefully now you see why that's not quite right.

    In general when studying fixed-limit games, we set the pot to be a variable (in this case P) and the bet size to be 1.

    When we're studying big-bet games like NLHE or PLO, we set the pot to be 1 and the bet size to be a variable (usually B or s).

    I hope that's helpful.
    Last edited by spoonitnow; 12-08-2010 at 03:06 PM.
  10. #10
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    Quote Originally Posted by spoonitnow View Post
    The antes are also P/2 units, so you would have to write them as PU/2, and then the pot would be PU.

    Hopefully now you see why that's not quite right.
    OK, maybe I got silly-confused because you said "P/2" and not "P/2 units" in the OP.

    Whichever way, I like to use variables in my formulas and not numbers so that I can change things around later without having to rework everything from scratch.

    Could have called P the pot and B the bet, would have looked prettier I guess.

    Who wants to bet "units" anyway? Next time use $P/2 and $1 in your questions, this way even Chinese and Belgians (and Gizmo) will understand what you mean.
    Last edited by daviddem; 12-08-2010 at 03:25 PM.
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  11. #11
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    Quote Originally Posted by daviddem View Post
    OK, maybe I got silly-confused because you said "P/2" and not "P/2 units" in the OP.

    Whichever way, I like to use variables in my formulas and not numbers so that I can change things around later without having to rework everything from scratch.

    Could have called P the pot and B the bet, would have looked prettier I guess.

    Who wants to bet "units" anyway? Next time use $P/2 and $1 in your questions, this way even Chinese and Belgians (and Gizmo) will understand what you mean.
    You don't need to use both P for pot and B for bet, which is the point. Any bet to pot ratio can be achieved by the conventions normally used that I reference above. Using only one variable for this instead of two makes life so much easier when you start to look at different bet sizes and pot sizes.
  12. #12
    Interesting. It requires a slight shift in mental perspective (for some reason the two-variable way makes more sense to me), but I can definitely see how this would make the math much easier. Just ignore absolute bet and pot sizes, and relate everything to the bet size (limit) or pot size (no limit).

    I was too hung up on being able to plug in absolute sizes for the pot and bet. Which doesn't make sense, since we don't care about absolute sizes, we only ever care about relative sizes anyway. So it makes the math easier, and fits with the concepts of the game. OK, I'm sold.
  13. #13
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    Quote Originally Posted by NightGizmo View Post
    Interesting. It requires a slight shift in mental perspective (for some reason the two-variable way makes more sense to me), but I can definitely see how this would make the math much easier. Just ignore absolute bet and pot sizes, and relate everything to the bet size (limit) or pot size (no limit).

    I was too hung up on being able to plug in absolute sizes for the pot and bet. Which doesn't make sense, since we don't care about absolute sizes, we only ever care about relative sizes anyway. So it makes the math easier, and fits with the concepts of the game. OK, I'm sold.
    Also note that in a big bet game when you set the pot to be 1 and the bet size to be a variable, usually B or s, then the bet size itself is the percent of the pot you're betting.

    In fixed-limit games we can't change the size of the bet, so it makes sense to keep it static. Additionally, in fixed-limit games you describe the pot as the number of bets that are in it, etc.
  14. #14
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    Optimal Exploitative Value Betting

    Optimal exploitative ________ is a game theory term. It essentially means that we are making every play, in this case a value bet, that is not -EV. We're going to use a slightly different game that uses T cards instead of 13 so we can generalize the solution. This is nice because it doesn't change the proof at all.

    Let Villain always call a value bet with n hands such that the worst hand he calls with is h_n, and that he never folds a hand that beats h_n. Then the EV of betting h_k such that 1 <= k <= n is:

    (-1)(k-1)/(T-1) + (1+P)(n-k)/(T-1) + (P)(T-n)/(T-1)

    And the EV of checking h_k is:

    (P)(T-k)/(T-1)

    The the EV of betting h_k is greater than or equal to the EV of checking h_k when:

    (-1)(k-1)/(T-1) + (1+P)(n-k)/(T-1) + (P)(T-n)/(T-1) >= (P)(T-k)/(T-1)

    Now we multiply through by (T-1) to get:

    (-1)(k-1) + (1+P)(n-k) + (P)(T-n) >= (P)(T-k)
    -k + 1 + n - k + nP - kP + PT - nP >= PT - kP
    -2k + 1 + n >= 0
    1 + n >= 2k
    (1+n)/2 >= k

    So we should bet all hands h_k such that k <= (1+n)/2.

    This is a fancy-looking way to say that we should bet when we have 50% equity or more against our opponent's calling range. For example, what if our opponent calls with a T or better:

    A K Q T J
    1 2 3 4 5

    Then n = 5, and we should bet all hands such that k <= (1+5)/2 = 3. So we should bet all hands h_k such that k <= 3, which means we're betting an Ace, King, or Queen.

    Conceptually

    When we're value betting the optimal exploitative frequency, then by definition we open ourselves up to adjustments. The basic adjustment our opponent could make to lower the EV of our value bets is to call less. Our two basic adjustments to protect ourselves from adjustments are either to value bet fewer hands, or to value bet more hands. Each is -EV for our range in a vacuum, and are -EV long term if our opponent never adjusts correctly.

    By value betting so much, we have to bluff some non-zero amount to make life difficult on our opponent. You could do the same type of analysis above to determine the optimal exploitative bluffing frequencies, but I'm going to leave that to the reader to work out.
    Last edited by spoonitnow; 12-09-2010 at 02:05 PM.
  15. #15
    Quote Originally Posted by spoonitnow View Post
    By value betting so much, we have to bluff some non-zero amount to make life difficult on our opponent. You could do the same type of analysis above to determine the optimal exploitative bluffing frequencies, but I'm going to leave that to the reader to work out.
    Thanks for all of the info in this thread, this is fantastic. Here's my shot at figuring out the optimal expoitative bluffing frequency.

    I'll use "j" to represent the opposite numbering of hands that Spoon used for "k": j(1) is the 2 card, J(2) is the 3 card, etc. We're assuming that this is a pure bluff, so if we are called then we lose, therefore 1 <= j <= (T-n+1).

    The EV of checking h_j is:

    (P)(j-1) / (T-1)

    The EV of betting h_j is:

    (P)(T-n-1)/(T-1) - (n)/(T-1)

    We want the EV of betting to be greater than or equal to checking, so:

    (P)(j-1)/(T-1) <= (P)(T-n-1)/(T-1) - (n)/(T-1)
    j - 1 <= T - n - 1 - n/P
    j <= T - n - n/P

    In Spoon's example of the villain that calls with any T or higher, assuming the pot size is 2:

    j <= 13 - 5 - 5/2
    j <= 5.5

    So we can bluff with any 2,3,4,5,6.

    What's interesting (and hopefully was already intuitively understood before reading this far) is that Spoon's equation for value betting is only a function of the villain's calling range. But our bluffing equation is both a function of the villain's folding frequency and the size of the pot.

    Or, in no-limit terms, your optimal exploitative bluffing range should be determined from the size of your bluff bet (relative to the pot) and how often the villain will fold to that bet.
  16. #16
    daviddem's Avatar
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    Quote Originally Posted by spoonitnow View Post
    Optimal Exploitative Value Betting
    Let Villain always call a value bet with n hands such that the worst hand he calls with is h_n, and that he never folds a hand that beats h_n. Then the EV of betting h_k such that 1 <= k <= n is:

    ...

    (1+n)/2 >= k

    So we should bet all hands h_k such that k >= (1+n)/2.
    There is a problem here where you invert your inequality, it should be k <= (1+n)/2

    Villain has n hands in his range and h_n is the worst, so h_1 is the best, and in your example, we should bet h_k with k >= 3 so QKA (not k <= 3)

    So on the same basis, EV of bluffing with h_k and n < k <= T:
    P(T-n-1)/(T-1) - n/(T-1)

    EV of checking is the same as before:
    P(T-k)/(T-1)

    So bluffing is better or equivalent to checking when:
    P(T-n-1)/(T-1) - n/(T-1) >= P(T-k)/(T-1)
    PT-Pn-P-n >= PT - Pk
    k >= n+1+n/P

    So in your example, with a deck of 13 cards and villain calls with T+ (n=5), if the pot size is:
    P=1: k >= 11. We can bluff profitably with h_11, h_12 and h_13 or 4, 3 and 2 (although betting 4 is equivalent to checking it EV-wise)
    P=2: k >= 8.5. We can bluff profitably with {h_9-h_13} or 6-2.
    Last edited by daviddem; 12-09-2010 at 09:26 AM.
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  17. #17
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    Quote Originally Posted by daviddem View Post
    There is a problem here where you invert your inequality, it should be k <= (1+n)/2
    Yeah I caught it later in the text file I'd wrote all that out in but never changed it in my post, thanks.
  18. #18
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    Optimal Exploitative Bluffing

    Through a similar process as above, we determine that for a deck of T cards, Hero bluffing a hand n < k <= T is better than checking when:

    k >= (n/P) + (n+1)

    Interpreting this result is very important and gives a lot of insight into when we should be bluffing and when we should be checking down for showdown value in scenarios like this.

    As Regards the Fundamental Theorem of Poker

    Note that h_n+1 is the best hand that Villain does not call with. Also not that because P > 0 that n/P > 0, so we have k >= n+1, and note that we can never have k = n+1. What this means is that we will never be betting with h_n+1 because it's never better than checking. Note that Villain can never call with worse or fold better if we bet h_n+1.

    When the Pot is Very Large

    When the pot is large, intuitively we should be bluffing more often. When P is large, n/P is very small. As P approaches infinity, n/P approaches zero, and we have k >= n+1, meaning we'll be bluffing every hand worse than h_n+1.

    When the Pot is Very Small

    When the pot is small, intuitively we should be bluffing less often. When P is small, n/P is very small. As P approaches zero, n/P approaches infinity, and we have k >= infinity, meaning that we'll never be bluffing.

    The Total Bluffing Range (Villain calls T+)

    Let's take a look at the scenario in the 13-card game where our opponent is calling a bet with T+. In this game, T = 13 and n = 5:

    Code:
    	A		1		
    	K		2		
    	Q		3		
    	J		4		
    	T		5		
    	9		6		
    	8		7		P >= 5/1 = 5.00
    	7		8		P >= 5/2 = 2.50
    	6		9		P >= 5/3 = 1.67
    	5		10		P >= 5/4 = 1.25
    	4		11		P >= 5/5 = 1.00
    	3		12		P >= 5/6 = 0.83
    	2		13		P >= 5/7 = 0.71
    The first column is the card, the second column is the k-value for each card h_k, and the third column is for what values of P we can bluff. So for betting an 8, which makes k = 7, we need P >= 5/2.

    So for example, if P = 1.1, then the optimal exploitative bluffing range is betting a 4, a 3, and a 2.

    As the pot size increases, it makes sense that we'll be bluffing with more hands since the reward is larger for the same risk. Similarly, as the pot size decreases, it makes sense that we'll be bluffing with fewer hands since the reward is smaller for the same risk.
  19. #19
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    The [0, 1] Half-Street Fixed-Limit Game

    Each player, Hero and Villain, are given a random real number between 0 and 1, inclusive, and this serves as their hand. The player with the lowest number at showdown wins.

    Hero is in position with Villain out of position. Each player posts an ante of P/2 before the hands are dealt. Each player receives their hand, Villain is forced to check in the dark. Hero then has the choice to bet 1 unit, or check to showdown. If Hero bets, then Villain has the choice to call and see a showdown or fold. At showdown, whoever has the highest card wins.

    This is similar to the n-card game, but is much easier to analyze, and is much easier to adapt to "real" poker river situations. We'll be using intervals and interval notation to represent ranges. For example, [0, 0.3) would mean that a range consists of all numbers from 0 to 0.3, including 0 but not including 0.3. We'll also be using U to denote a union of two intervals, so [0, 0.2] U [0.9, 1] would mean all numbers from 0 to 0.2 and 0.9 to 1, including 0, 0.2, 0.9, and 1. For more information, see Interval (mathematics) - Wikipedia, the free encyclopedia.

    Here are some questions to get us started:

    1. If Villain calls with a 0.4 or lower, what hands can Hero make a +EV value bet with?
    2. If each ante is 0.25 units, what is the minimal range Villain has to call a bet with so that Hero can't make a +EV bluff?
    3. If Villain calls with 0.3 or lower and P = 1, what is Hero's optimal exploitative betting range?
    4. What is Villain's optimal exploitative calling range against Hero's range from question 3?
    5. What is Villain's unexploitable calling range (in terms of P)?
    6. What is Hero's unexploitable betting range (in terms of P)?
  20. #20
    1. If Villain calls with a 0.4 or lower, what hands can Hero make a +EV value bet with?

    The top 50% of villain's calling hands are +EV to value bet (not including the exact middle, since that is 0 EV):
    [0, 0.2)

    2. If each ante is 0.25 units, what is the minimal range Villain has to call a bet with so that Hero can't make a +EV bluff?

    villain must fold: 1/(1+P) = 1/(1+0.5) = 2/3

    So, if villain calls with [0, 1/3] than the bluff is not +EV

    3. If Villain calls with 0.3 or lower and P = 1, what is Hero's optimal exploitative betting range?

    For value (top 50% of calling range, inclusive): [0, 0.15]
    As a bluff: (k = our number, P = pot, c = the worst hand he calls with)

    EV(check) = (1 - k)P
    EV(bluff) = (1-c)*P - c = P - Pc - c

    Solve where EV(bluff) > EV(check):

    P - Pc - c >= (1 - k)P
    P - Pc - c >= P - kP
    -Pc - c >= -kP
    kP >= Pc + c
    k >= (P + 1)*c/P

    (1+1)*.3/1 = .6

    So, out total optimal exploitative betting range: [0, 0.15] U [.6, 1]

    4. What is Villain's optimal exploitative calling range against Hero's range from question 3?

    Any hand that has greater than 33.33% equity (getting 2:1 on our call) against the total range is worth calling. Our betting range includes an interval that spans a sum of .55, so the villain's calling range should span 2/3 of our betting range (non-inclusive at the worst end). 2/3*.55 = 0.366...

    So, it is profitable to call [0, 0.15] of hero's betting range. It is also profitable to call with [0.6, .8166...) of hero's betting range. Because it is profitable to call with 0.6, that means it is also profitable to call with [0.15, 0.6].

    So, the optimal calling range is [0, .8166...).

    5. What is Villain's unexploitable calling range (in terms of P)?

    Since the villain is getting odds of 1 : (P+1), then he can unexploitably call [ 0, 1/(P+2) )

    6. What is Hero's unexploitable betting range (in terms of P)?

    I'm stumped here for now. I'll have to think on it.
  21. #21
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    The [0, 1] Half-Street Fixed-Limit Game - Alpha, MoP, OE Value Betting

    Here I'm going to solve a handful of important equations for this game. First I want to talk about an important value in poker math, one so important that you've seen it in tons of places, and one so important that it has its own name. This value is bet/(bet+pot). In our fixed limit games where the pot is P and the bet size is 1, it's 1/(1+P). Typically we refer to this value with the symbol alpha, but here for the sake of ease I'm going to just refer to it as a. So a = 1/(1+P), and that gives us 1-a = P/(1+P), both of which we'll be making use of.

    This game is covered in less detail under example 11.3 on page 116 of Mathematics of Poker. In keeping with their naming conventions, we'll refer to x_1 as the worst hand Villain calls with, y_1 as the bottom of Hero's value betting range, and y_0 as the top of Hero's bluffing range.

    When we're value betting, we always have a hand lower than x_1 by definition. Therefore, the EV of a value bet with hand y has three parts. When Villain holds a hand from 0 to y, we get called and lose our bet and profit -1. When Villain holds a hand from x_1 to y, we get called and win 1 bet for a profit of 1+P. Finally, when Villain holds a hand from 1 to x_1, Villain folds and we profit P. This gives us the following equation for the EV of a value bet:

    EV of Value Betting: (y - 0)(-1) + (x_1 - y)(1+P) + (1 - x_1)(P)

    If we check in the same situation, the EV equation has 2 parts. When Villain holds a hand from 0 to y, we lose and profit 0. When Villain holds a hand from y to 1, we win and profit P. This gives us the following:

    EV of Checking: (y-0)(0) + (1-y)(P)

    The worst hand Hero will value bet with is y_1. At that point, the EV of checking will be the same as the EV of value betting. Therefore, we can set the EV of checking equal to the EV of value betting and simply solve for y, and that will give us y_1.

    (y - 0)(-1) + (x_1 - y)(1+P) + (1 - x_1)(P) = (y-0)(0) + (1-y)(P)
    -y + x_1 + Px_1 -y - Py + P - Px_1 = P - Py
    -y + x_1 - y = 0
    -2y = -x_1
    y = x_1 / 2

    So given x_1, the optimal exploitative value betting range is going to be [0, y_1] such that y_1 = x_1 / 2.
    Last edited by spoonitnow; 12-10-2010 at 10:50 PM.
  22. #22
    Cool thread, thanks spoon.

    I wish I could say I didn't have to read to digest to answer some of these questions, but the truth is I did. Also, I haven't looked at any of the responses. This is what I came up with:

    1.
    We're betting 1 unit into a total pot of P+1, so our opponent needs to fold 1/P+1 percent of the time.

    2.
    Below is my original answer in quotes
    "In order for our value bet to be +EV, he has to call with more hands that are worse than our hand than are better. He's calling with a 7, 8, 9, T, J, Q, K, and A. This means that we can only value bet with a J or better.

    When we value bet with a J there are 4 worse hands that call us (T, 9, 8, 7) and 3 better hands that call us (A, K, Q).

    When we value bet with a T there are 3 worse hands that call us (9, 8, 7) and 4 better hands that call us (A, K, Q, J)

    Therefore, we can only value bet J+ on the river vs a calling range of 7+."

    ***This is wrong though. The above is true in order for our value bet to be more +EV than checking. I believe that in order to answer the question we need to know the value of P. My reasoning is that we can still make a +EV value bet and get called by more worse hands than better hands because the money in the pot compensates this fact. But, how much it compensates this depends on the size of the pot (i.e. the bigger the pot when he folds the bigger the ratio of worse hands/better hands that call us can be)

    3.
    Using our formula from #1 where P= ante * 2 = 0.5,

    In order to make a +EV bluff we need villain to fold 1/(0.5 + 1) = 0.66 = 66% of the time. So, he always has a range of 12 hands (since there are a total of 13 cards and we have 1 of them). We need him to fold 8 of those hands to bluff successfully, so if he calls with 5 hands then we can't make a +EV bluff.

    4.
    Not sure if we're supposed to assume that the antes are 0.25 again or not. I'll do it assuming we don't know the ante I guess.

    Betting:
    When he calls with 8+ and we have the T, he's calling with 6 hands (8, 9, J, Q, K, A) and folding with 6 hands (7, 6, 5, 4, 3, 2).

    So, I believe the EV calculation goes like this:

    EV = (6/12)*(P) + (6/12) * [(2/6)*(P+1) - (4/6)*1]
    EV = 0.5P + 0.1667P + 0.1667 - 0.333
    EV = 0.667P - 0.1667 (so if the ante is 0.25 then EV of betting is 0.1667)

    Checking:
    When we check,

    EV = (8/12)*P = 0.667P (so if the ante is 0.25, then EV of checking is 0.333)

    I think I messed up somewhere in this solution (to #4) because I did a check on it and it didn't come out right. I've already spent a bunch of time and thought on this so I'll come back and do the other problems later.
  23. #23
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    The [0, 1] Half-Street Fixed-Limit Game - OE Bluffing

    By definition, the best hand Hero will bluff with (y_0) will be higher than the worst hand Villain calls with (x_1), so we have x_1 < y_0 < 1. We could prove that y_0 = 1 is never true, but that would take away from more important things, like the optimal exploitative bluffing frequency. So let's look at the EV of betting and checking y_0.

    When we bet y_0: When Villain holds 0 to x_1 we get called, lose, and profit -1. When Villain holds 1 to x_1, Villain folds, we profit P.

    EV of y_0 as a bet = (x_1 - 0)(-1) + (1 - x_1)(P)

    When we check y_0: When Villain holds 0 to y_0 we lose and profit 0. When Villain holds y_0 to 1 we win and profit P.

    EV of y_0 as a check = (y_0 - 0)(0) + (1 - y_0)(P)

    To be bluffing all of the hands that will be +EV, we'll need y_0 to have the same EV as a check or as a bet.

    (x_1 - 0)(-1) + (1 - x_1)(P) = (y_0 - 0)(0) + (1 - y_0)(P)

    Now if we solve for y_0, we'll have the y_0 for the optimal exploitative bluffing percentage.

    (x_1 - 0)(-1) + (1 - x_1)(P) = (y_0 - 0)(0) + (1 - y_0)(P)
    -x_1 + P - Px_1 = P - Py_0
    -x_1 - Px_1 = -Py_0
    x_1 + Px_1 = Py_0
    x_1 (1+P) = Py_0
    y_0 = x_1 (1+P)/P
    y_0 = x_1/(1-a)

    So given x_1, Hero's optimal exploitative bluffing range is [y_0, 1] such that y_0 = x_1/(1-a).
  24. #24
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    The [0, 1] Half-Street Fixed-Limit Game - Unexploitable Solution

    The moment I'm sure you've all been waiting for (well at least NightGizmo). Now we're going to find x_1 (the worst hand Villain calls with), y_1 (the worst hand Hero value bets), and y_0 (the best hand Hero bluffs) for the unexploitable strategy (also known as optimal strategy). Consider the following:

    1. When Hero is playing unexploitably, x_1 will have the same EV for calling or folding.
    2. When Villain is playing unexploitably, y_1 will have the same EV for betting or checking, and y_0 will have the same EV for betting or checking.

    We get this from the definition of unexploitable strategy. It's like if we're playing rock-paper-scissors, and I'm playing unexploitably, then for you the EV of playing any of the 3 options should be the same. If that clears it up, cool. If not, then think on it.

    So we're going to assume both Hero and Villain are playing unexploitably. That means x_1 will have the same EV for calling or folding, y_1 will have the same EV for calling or folding, and y_0 will have the same EV for calling or folding. Now we just have to find all 6 EV equations and set them equal and figure out what to do with that information:

    EV of calling x_1 = (y_1 - 0)(-1) + (1 - y_0)(1+P)
    EV of folding x_1 = 0
    Set them equal:
    (y_1 - 0)(-1) + (1 - y_0)(1+P) = 0
    -y_1 + (1 - y_0)(1+P) = 0
    y_1 = (1 - y_0)(1+P)
    y_1 = (1 - y_0)/a <--- Call this equation 1

    EV of betting y_1 = (y_1 - 0)(-1) + (x_1 - y_1)(1+P) + (1 - x_1)(P)
    EV of checking y_1 = (y_1 - 0)(0) + (1 - y_1)(P)
    Set them equal:
    (y_1 - 0)(-1) + (x_1 - y_1)(1+P) + (1 - x_1)(P) = (1 - y_1)(P)
    -y_1 + x_1 + Px_1 - y_1 - Py_1 + P - Px_1 = P - Py_1
    -y_1 + x_1 - y_1 = 0
    2y_1 = x_1 <--- Call this equation 2

    EV of betting y_0 = (x_1 - 0)(-1) + (1 - x_1)(P)
    EV of checking y_0 = (y_0 - 0)(0) + (1 - y_0)(P)
    Set them equal:
    (x_1 - 0)(-1) + (1 - x_1)(P) = (1 - y_0)(P)
    -x_1 + P - Px_1 = P - Py_0
    -x_1 - Px_1 = -Py_0
    x_1 + Px_1 = Py_0
    x_1(1+P) = Py_0
    x_1(1+P)/P = y_0
    x_1/(1-a) = y_0 <--- Call this equation 3

    We finally get a system of three equations with three variables (remember a is a constant). They are:

    y_1 = (1 - y_0)/a
    x_1 = 2y_1
    y_0 = x_1/(1-a)

    A little bit of simple substitution clears things up, but I'll let the reader go through that. At least until I feel like typing it out.
    Last edited by spoonitnow; 12-11-2010 at 01:10 PM.
  25. #25
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    Here are my answers for Q1-Q4:

    1)
    Let x be Hero's hand with x in [0,0.4]

    The probability that villain's hand is < x is x - villain's hand is better than ours
    The probability that villain's hand is <= 0.4 is 0.4 - villain calls a bet
    The probability that villain's hand is < x when he calls a bet is x/0.4 - villain's hand is better when he calls a bet

    EV of checking:
    (1-x)P
    P-xP

    EV of betting:
    0.6P+0.4((1-x/0.4)(P+1)-x/0.4)
    0.6P+0.4(P+1-xP/0.4-x/0.4-x/0.4)
    0.6P+0.4P+0.4-xP-2x
    P+0.4-xP-2x

    So betting is better than checking when:
    P+0.4-xP-2x > P-xP
    0.4 > 2x
    x < 0.2
    So when our hand has more than 50% equity against villain's range we can value bet profitably.

    2)

    Let y be the worst hand that villain calls with, with y in [0,1]
    Let x be Hero's hand with x in (y,1]

    The probability that villain calls a bet is y

    EV of bluffing is <= 0 when
    (1-y)P-y <= 0
    P-yP-y <= 0
    P/(P+1) >= y
    y <= P/(P+1)

    and since P=0.5, y<=0.3333... so y in [0,0.3333].

    Let's also calculate when bluffing is better than checking with a hand x, a pot P and villain calls with hands <=y:
    P-yP-y > P-xP
    -yP-y > -xP
    yP+y < xP
    x > y(P+1)/P

    3)
    From 1) value bet with x < 0.15
    From 2) bluff with x > 0.6

    So the optimal exploitative range for Hero is [0,0.15) U (0.6,1]

    4) Pot odds are 0.3333. Any hand that has at least this much equity vs Hero's range is worth calling a bet with.
    So what is h_v's equity vs Hero's range? Hero's range is 0.55 wide (0.15+0.4)

    If h_v<=0.15, equity = (0.55-h_v)/0.55 = 1-h_v/0.55, so at the minimum 1-0.15/0.55= 72.7%

    If 0.15<h_v<=0.6, equity = (0.55-0.15)/0.55 = 72.7%, so all these hands have the same equity and are worth calling a bet

    with

    If h_v>0.6, equity = (0.55-0.15-(h_v-0.6))/0.55 = (1-h_v)/0.55. This obviously decreases as h_v increases, and will reach

    the 0.3333 equity threshold is reached for h_v=0.8185

    So villain should call with [0,0.8185]

    extra:
    Note that I started doing question 4 with an EV approach before I realized it could be done simpler than that. In the process, I got this:
    Let y be the worst hand that villain calls with. We are looking for the optimal y for villain.
    Let h_h be Hero's hand when he bets with h_h in [0,0.15) U (0.6,1]

    Let's calculate the EV of villain as a function of y.

    Let v be the event that h_h is in [0,0.15] when Hero bets.
    Let b be the event that h_h is in [0.6,1] when Hero bets.
    Let e be the event that h_v < h_h when Hero bets and villain calls with all h_v < y with y in [0,1]. pr(e) is villain's equity.

    pr(v) = 0.15/(0.4+0.15) = 0.2727...
    pr(b) = 0.4/(0.4+0.15) = 0.7272...
    pr(e)=pr(v)*pr(e|v) + pr(b)*pr(e|b) (where pr(i|j) is the probability of i given j)

    if y <= 0.15, pr(e|v)=pr(h_h<y)*pr((e|v)|h_h<y) + pr(h_h>=y)*pr((e|v)|h_h>=y)
    =y/0.15*0.5 + (1-y/0.15)*1
    =1-3.3333*y
    if y > 0.15, pr(e|v)=pr(h_v>0.15)*pr((e|v)|h_v>0.15) + pr(h_v<=0.15)*pr((e|v)|h_v<=0.15)
    =(y-0.15)/y*0 + (1-(y-0.15)/y)*0.5
    =0.5(1-1+0.15/y)
    =0.5*0.15/y
    =0.075/y
    if y <= 0.6, pr(e|b)=1
    if y > 0.6, pr(e|b)=pr(h_v<0.6)*pr((e|b)|h_v<0.6) + pr(h_v>=0.6)*pr((e|b)|h_v>=0.6)
    =0.6/y*1 + (1-0.6/y) * (pr(h_h<y)*0.5 + pr(h_h>=y)*1)
    =0.6/y*1 + (1-0.6/y) * ((y-0.6)/0.4*0.5 + (1-(y-0.6)/0.4)*1)
    =2.5-1.25y-0.45/y

    Finally, we can calculate villain's EV as:
    EV=y*(pr(e)*2-(1-pr(e))
    EV=y*(3pr(e)-1)

    I could go on an show how to to find the point of max EV by finding the root of the derivative of the above expression, but that's when I realized that there was a much simpler solution. Regardless, I plotted villain's EV relative to y and here it is:


    For 5 and 6, I didn't get it because the definition of unexploitative strategy is not 100% clear in my mind. I thought it was the strategy that prevents the opp from making any +EV play? If so, how do you get to your criteria above?

    edit: OK I think I get it, let me know if this is wrong. Villain's initial strategy is to call only with [0,4] or x_1=0.4 and Hero is exploiting this and playing an optimal strategy against villain, for which Hero's EV of betting y_0(=0.6) is the same as his EV of checking it, and his EV of betting y_1(=0.15) is the same as his EV of checking it. So at this time, villain is being exploited and sure enough, his EV of calling with x_1(0.4) is not the same as his EV of folding it. When he adjusts his x_1 to 0.8185, he is now playing optimally against Hero's strategy and the roles are reversed: the exploiter becomes the exploited, until Hero adjusts again, and so on. Eventually, they should converge to a stable state where all three conditions you listed are met and where as soon as one of them tries to deviate a hair from his strategy, the other one counter adjusts a bit himself, and they both find themselves dragged back like magnets to the equilibrium.

    Now questions: are the equilibria in heads up zero sum games always stable, or are there cases where there are multiple possible equilibria and/or the equilibria are unstable and the players keep bouncing from one to another? Do all these games always converge?
    Last edited by daviddem; 12-11-2010 at 11:49 AM.
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  26. #26
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    y_1 = (1 - y_0)/a (1)
    x_1 = 2y_1 (2)
    y_0 = x_1/(1-a) (3)
    (2) in (3):
    y_0=2y_1/(1-a) (4)
    (4) in (1):
    y_1=(1-2y_1/(1-a))/a
    ay_1=1-2y_1/(1-a)
    (a+2/(1-a))y_1=1
    (a(1-a)+2)*y_1=(1-a)

    y_1=(1-a)/(2+a(1-a))


    so by (2):
    x_1=2(1-a)/(2+a(1-a)) (5)

    (5) in (3):
    y_0=2/(2+a(1-a))

    Not sure if there is a way to make them look better...

    That means x_1 will have the same EV for calling or folding, y_1 will have the same EV for calling or folding, and y_0 will have the same EV for calling or folding
    ^^Some typos spoon, too many calling and folding, not enough betting and checking.
    Last edited by daviddem; 12-11-2010 at 01:09 PM.
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  27. #27
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    Quote Originally Posted by daviddem View Post
    For 5 and 6, I didn't get it because the definition of unexploitative strategy is not 100% clear in my mind. I thought it was the strategy that prevents the opp from making any relatively +EV play? If so, how do you get to your criteria above?
    Imagine both players are playing their own optimal exploitative strategy against each other at the same time. A consequence of this would be that neither player can do better by changing their strategy, because by definition, the optimal exploitative strategy is the best-performing strategy in terms of EV that you can have against a given opponent's strategy.

    So neither player can improve their EV by changing their strategy, and this creates a certain equilibrium. If either player deviates from the equilibrium by changing their play, then they lose EV (since they will be performing worse than the optimal exploitative strategy, which was the maximum EV they could obtain).

    In this equilibrium, neither player can be exploited by a change in strategy. Therefore, their strategies are both unexploitable.

    So what is a change in strategy in this game? Villain has one strategic option, and that's where to place x_1. Hero has two strategic options, and those are where to place y_1 and y_0.

    When Villain is playing his optimal exploitative strategy, the EV of folding x_1 will be the same as the EV of calling x_1. Similarly, when Hero is playing his optimal exploitative strategy, the EV of betting y_1 or y_0 will be the same as the EV of checking y_1 or y_0, respectively.

    So we set the EV of folding x_1 equal to the EV of calling x_1, the EV of betting y_1 equal to the EV of checking y_1, and the EV of betting y_0 equal to the EV of checking y_0. When we do all of this, then both players are playing their respective optimal exploitative strategies, and therefore are both playing unexploitably. The solutions for x_1, y_1, and y_0 in this system of equations gives us each player's unexploitable strategy.

    Quote Originally Posted by daviddem View Post
    (2) in (3):
    y_0=2y_1/(1-a) (4)
    (4) in (1):
    y_1=(1-2y_1/(1-a))/a
    ay_1=1-2y_1/(1-a)
    (a+2/(1-a))y_1=1
    (a(1-a)+2)*y_1=(1-a)

    y_1=(1-a)/(2+a(1-a))


    so by (2):
    x_1=2(1-a)/(2+a(1-a)) (5)

    (5) in (3):
    y_0=2/(2+a(1-a))

    Not sure if there is a way to make them look better...

    ^^Some typos spoon, too many calling and folding, not enough betting and checking.
    Thanks for pointing out the typos. And that's as good as you're going to make them look in text.
    Last edited by spoonitnow; 12-11-2010 at 01:13 PM.
  28. #28
    Quote Originally Posted by spoonitnow View Post
    And that's as good as you're going to make them look in text.
    The bottom terms can be slightly simplified, although it doesn't really look that much prettier in the text.

    2 + a(1-a) = 2 + a - a^2 = -1(a^2 - a - 2) = -1(a-2)(a+1) = (2-a)(a+1)

    y_1 = (1-a)/(2+a(1-a)) = (1-a)/( (2-a)(a+1) )
    x_1 = 2(1-a)/(2+a(1-a)) = 2(1-a)/( (2-a)(a+1) )
    y_0 = 2/(2+a(1-a)) = 2/( (2-a)(a+1) )
  29. #29
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    The [0, 1] Half-Street Fixed-Limit Game - Balanced Play vs. Exploitative Play

    This is my official 5000th post, and I feel like what this post illustrates is pretty important, so I hope some people get a lot out of it.

    Given x_1, the optimal exploitative value betting range is going to be [0, y_1] such that y_1 = x_1 / 2. With this value betting range, how often do we have to bluff so that we're unexploitable?

    We'd like the EV of Villain calling with x_1 to be the same as the EV of folding x_1. Earlier we did this and found that the resulting equation was when y_1 = (1 - y_0)/a. We already know what y_1 will be, so let's change this equation around to make it more convenient for finding y_0 given y_1:

    y_1 = (1 - y_0)/a
    ay_1 = (1 - y_0)
    y_0 = 1 - ay_1

    So given x_1, we should be value betting [0, x_1 / 2], and bluffing [1 - (a/2)x_1, 1] if we want Villain's call with x_1 to be break even.

    Earlier we found that the optimal exploitative bluffing frequency was set by y_0 = x_1/(1-a). Now we're looking at the balanced bluffing frequency being set by y_0 = 1 - (a/2)x_1. The relationship between these two values is very important in the discussion of balance vs. exploitative play.

    Let's take a look at an example scenario. Suppose P = 1.5 and Villain calls with [0, 0.3], making x_1 = 0.3. Note that a = 0.4. Then the optimal exploitative value betting frequency is set by y_1 = 0.15, the optimal exploitative bluffing frequency is set by y_0 = 0.3/(1-0.4) = 0.5 and the balanced bluffing frequency is set by y_0 = 1 - (0.4/2)(0.3) = 0.94. If we plot these values on a number line, we can find some interesting results.


    Note: Don't fall into the trap of thinking that all of this is a lot of theoretical stuff that doesn't apply to "real" poker. In no-limit hold'em, this situation would basically be like we had 2/3 pot left behind on the river in position and Villain had us covered, and after Villain checks to us, we have similar ranges. Additionally, this game can be slightly altered so that we don't have the same ranges.

    I've colored the 5 important segments of the [0, 1] distribution in the above graphic. In the bright red, this is the part of Hero's range that he will value bet with, all of which will show a profit. In the dark red, this is the part of Villain's range that he calls with, but that does not beat the worst hand that Hero value bets. This section exists because Hero bluffs a non-zero percentage of the time, and because of the effect of pot odds.

    The blue section is the range of hands that Hero can bluff with if he wants to exploit Villain with an increased bluffing frequency. By bluffing hands from this section, he increases his EV at the risk of being exploited by Villain calling more.

    The green section is the range of hands that Hero should always bluff. If he doesn't bluff these hands, he loses EV. The problem is that there is no risk to bluffing these hands because Villain cannot exploit Hero by calling more. Moreover, if you don't bluff these hands, you become exploitable by Villain calling less. So basically, you have to bluff these hands because 1) it's +EV compared to checking, and 2) if you don't then you become exploitable.
  30. #30
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    The [0, 1] Half-Street Fixed-Limit Game - Balanced Play vs. Exploitative Play (cont.)

    In the scenario in the post before this one, P = 1.5 making a = 0.4, and we said Villain called with 30% of his range (x_1 = 0.3). Villain is folding too much, so we have a large range where bluffing is better than checking. But what happens if he's calling too much (like a lot of bad players, especially at the micros, do)?

    Let P = 1.5 again (a = 0.4), but let x_1 = 0.7 now. If we do our calculations, we'll find that y_1 = 0.35, the balanced bluffing range is defined by y_0 = 0.86. However, if you look at the EV equations for checking or bluffing when Hero is dealt the hand 1, you'll see that checking is better. This means that there are no hands where Hero can make a bluff that has a higher EV than checking.

    Our balanced strategy includes a decent amount of bluffs because we are value betting with such a wide range. However, those bluffs are -EV in a vacuum, and work only to protect us from Villain adjusting by calling less (and folding more).

    This means that against people who call too much and will not adjust to you never bluffing, you should probably never bluff since it costs you money without earning you anything.

    Showdown Value and the Size of the Pot

    Conventional wisdom says that the larger the pot, the more valuable your showdown value is, so the wider of a range you should be checking behind on the river.

    To test this, let's use this same scenario with P = 1.5 and x_1 = 0.7 to look at what happens to our checking range when the pot gets bigger. If you notice, the range from y_1 to y_0 is our checking range, so we could quantify the size of this range just by using (y_0 - y_1). Here we're going to let y_0 be the balanced version with y_1 = 0.35. The following are some values for P and (y_0 - y_1) when x_1 = 0.7 and Hero plays a balanced strategy with y_1 = x_1 / 2:

    1.5 0.5100
    1.6 0.5154
    1.7 0.5204
    1.8 0.5250
    1.9 0.5293
    2.0 0.5333
    2.1 0.5371
    2.2 0.5406
    2.3 0.5439
    2.4 0.5471
    2.5 0.5500

    And as expected, when the pot size increases, the size of our checking range increases, gradually approaching 0.65. Note we can only check 65% of hands because we're value betting the other 35%.
    Last edited by spoonitnow; 12-12-2010 at 12:01 AM.
  31. #31
    Quote Originally Posted by spoonitnow View Post
    2. If Villain calls with a 7 or higher, what hands can Hero make a +EV value bet with?

    Look at his calling range:

    A K Q J T 9 8 7

    We can only bet hands that will beat more hands than it loses to. The worst hand that meets this requirement is a Jack which beats 4 and loses to 3 of the hands in Villain's calling range. Therefore, we can make a +EV value bet with an Ace, King, Queen or Jack.
    I think this is wrong.
  32. #32
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    Quote Originally Posted by kfaess View Post
    I think this is wrong.
    Why? When you make a value bet, you should have more than 50% equity against villain's calling range {AKQJT987} The lowest card that has more than 50% equity against this range is J because if you bet a J, you win 4 times against T987 and loose 3 times against AKQ, so you win more than 50% of the time. If you bet a T, you win 3 times vs 789 and you loose 4 times vs AKQJ, so you win less than 50% of the time, so a T does not fit the requirement. So J is the lowest card you can make a value bet with.
    Last edited by daviddem; 12-13-2010 at 01:07 AM.
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  33. #33
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    Quote Originally Posted by kfaess View Post
    I think this is wrong.
    Surely if you think something this basic is wrong then you could explain why?

    But it's not wrong so you might have a hard time doing that.

    So what about it is confusing you?
  34. #34
    A J is the lowest card you SHOULD make a value bet with, I agree with that. However, that's not what the question asks. The question asks what is the lowest card we can make a +EV value bet with. There is a big difference there.

    What if for some reason P=100 units? We're betting 1 into 100 and he's folding half his hands. Is this not +EV?

    A J is the lowest card we can make a value bet with where a bet is better than checking, imo. That doesn't necessarily mean that a bet with a T is -EV.

    You might say that in the example where P=100 and we bet with a T that's not a value bet. Well, I think its more of a value bet than a bluff and its compensated by the fact that we win a ton of dead money when he folds.
  35. #35
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    edit: the below is completely wrong, so feel free to not read because it just confuses things. I just leave it here to avoid disrupting the continuity of the thread.

    In the case you mention, what happens is that a T becomes part of your bluffing range. For a bluff to be profitable, you need villain to fold more than 1/(P+1) or 1 out of 101 times, so basically ALL the hands that are not in your value betting range will be in your bluffing range. And of course in that case it is +EV to bet a T, but that does not make it a value bet. It is just a profitable bluff. Or to put it yet another way: a value bet does not mean "any bet that is +EV". A value bet is by definition a bet with a hand that has more than 50% equity vs the calling range of villain (and yes of course a value bet is +EV).

    Maybe it will be clearer if you consider the extreme case where villain NEVER folds. In that case you can never bluff profitably. And even if the pot is 100 and the bet is 1, you should not bet a 7 because long term it is -EV.

    note: you can bet an 8, or check an 8, EV-wise it is the same
    Last edited by daviddem; 12-13-2010 at 11:58 AM.
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  36. #36
    I guess the difference in opinions comes down to semantics of definitions. Obviously not all bets that are +EV are value bets...that's not what I'm saying.

    You define a value bet as a bet where we have more than 50% equity vs a calling range. In one book I've read the author defines a value bet as a bet to get worse hands to call and a bluff as a bet to get better hands to fold (and he plays high stakes btw). Which definition is correct? I'm not really sure.

    What exactly are we trying to do when betting a T? With a T he never folds better hands, but he does call with worse hands. By the definition I've read that would make it a value bet and not a bluff. You want to call that a bluff? I'd argue that's more like a thin value bet. You could also argue that its a bet to collect dead money, which would be neither a value bet or a bluff.
  37. #37
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    OK you're right I messed up: T can never be a bluff because he never folds any better hand. Please ignore my previous post. However, it cannot be a value bet either, because he calls with better most of the time. That is why you should always check your T even if the pot is 100, or 1000.

    OK so villain calls with A-7. The pot is 100. You have a T. You can check or bet 1.

    When you check your T:

    EV = (8/12)*100 = 66.666
    (When he has 2-9, you win 100, and there is a 8/12 chance that he holds something in that range. When he has A-J, you loose nothing because you did not bet)

    When you bet your T:

    EV=(5/12)*100 + (3/12)*101 - (4/12)*1 = 66.583
    (He folds 2-6 and you win 100, or he calls with 7-9 and you win 101, or he calls with A-J and you loose 1)

    So checking your T has a better expectation.
    Last edited by daviddem; 12-13-2010 at 12:06 PM.
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  38. #38
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    In the question I was clearly asking about the relative EV, not the absolute EV, and no one ever showed any sign of being confused, so all of this is really unnecessary.

    For people who are following along in Mathematics of Poker, you'll notice that betting T has a negative EV when using ex-showdown EV, and this correlates to a showdown EV that's negative when betting by looking at the difference between the profit of betting and of checking.

    I'm sure you'll get a lot out of studying the games in this thread. Best of luck.
    Last edited by spoonitnow; 12-13-2010 at 01:10 PM.
  39. #39
    Quote Originally Posted by kfaess View Post
    You define a value bet as a bet where we have more than 50% equity vs a calling range. In one book I've read the author defines a value bet as a bet to get worse hands to call and a bluff as a bet to get better hands to fold (and he plays high stakes btw). Which definition is correct? I'm not really sure.
    Value bet -- the second definition is an oversimplification of the real goal, which is to bet when you have more than 50% equity against his calling range. His range might include better hands than you that will call, but as long as you are ahead of the calling range then your value bet is successful (because it is better than checking behind).

    Think about it -- a value bet, by definition, profits when the villain calls. For every dollar we put into the pot, the villain has to match it. If we have more than 50% equity, we get our dollar back plus some percentage of the villain's dollar. But if we have less than 50% equity, then for every dollar we put into the pot we get less than 1 dollar back at showdown. To make matters worse, it's safe to assume that the villain's calling range is stronger than his total range, so we also get a smaller percentage of the original pot.

    Bluff -- true, you are trying to get better hands to fold, but that's also incomplete. First off, you need to get enough hands to fold to make it worth your bet -- folding out 10% of his range, all of which beats you, is a losing proposition if you bet the size of the pot as a bluff. To bluff profitably, you should only bluff if it has a higher EV than checking behind.
  40. #40
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    in question three, it seems that our particular holding influences what the bottom point of villain's calling range has to be in order to prevent us making +EV bluffs. ie if we hold an ace, his (forced) checking range is 2-K, and 2/3 of that range is 2,3,4,5,6,7,8,9 (8 combos), meaning a Ten would be the worst card he would be calling to prevent +EV bluffs. do we just assume, because the question specified that we were bluffing, that we have a 2? in which case his checking range is 3-K, and 2/3 of that range is 3-T.
  41. #41
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    bump

    can anyone please explain to me what is going on in the "Optimal Exploitative Value-Betting" section? more specifically, what on earth do things like "h_n" and "h_k" refer to?
  42. #42
    I think h_k and h_n are just representations of the actual cards represented by the numbers n and k. So think of "h_n" as "hand n". In the example at the end of that post, spoon assigns Ace to 1, King to 2, etc., so if "n" was "2", then "h_n" is "king".
  43. #43
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    ah ok thanks. still doesn't compute. guess i should have paid more attention in high school
  44. #44
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    Bump 'cause it's hugely relevant.

    What's up with the rule that Hero is IP, but Villain is forced to check in the dark? How is that different than Hero being OOP? Is it because Villain can’t raise?

    Also, enough people have posted their work on this that I'm not going to fill the thread with the same math and reasoning.

    Can we take this a step further?

    Just adding onto Villain's options... What happens in the [0, 1] game when Villain is allowed to raise?

    So the action goes: 0) Villain checks in the dark, 1) Hero bets or checks, 2)Villain raises, folds or calls, 3)Hero folds or calls.

    Hero's ranges include: b/c, b/f, c/c, c/f

    Villain's ranges:
    When Hero bets, Villain will
    raise [0, 0.1)U[0.95, 1] ; call [0.1, 0.2] ; fold (0.2, 0.95)

    raise call fold raise
    0 ------- 0.1 ------ 0.2 ------ 0.95 ------- 1


    When Hero checks, Villain will
    bet [0, 0.3)U[0.85, 1] ; check behind [0.3, 0.85)

    bet check bet
    0 ----- 0.3 ------- 0.85 ----- 1


    1) Given Villain’s ranges, with what hands can Hero make a +EV value bet?
    The worst Villain calls with is 0.2, so Hero can value bet
    [0, 0.1] with 0.1 being 0 EV.

    2) Given Villain’s ranges, with what hands can Hero make a +EV b/c?
    When facing a bet Villain will raise 15% of the time, so Hero can b/c 7.5% of the time for value
    [0, 0.075] with 0.075 being 0 EV.

    3) When Hero bets, how often does Hero have to b/f for Villain to have a +EV bluff?
    a = 1/(1+Q) where Q = P+1 (Hero added a bet to P)
    a = 1/(2+P) where P is the initial pot with 2 antes of P/2
    If P/2 = 0.25, P = 0.5
    a = 1/(2.5) = 2/5 = 40%

    4) Given the answers to 2 & 3, what % of Hero’s starting range should b/f?
    Let x be the % of hands that hero b/f
    x/(0.075+x) = a
    x = 0.4*(0.075+x) = 0.03 + 0.4x
    x – 0.4x = 0.03
    0.6x = 0.03
    x = 0.03/0.6 = 0.05 = 5%
    So 5% of hands can be added to Hero’s betting range, in addition to the 7.5% of value hands.

    5) So what is Hero’s betting range?
    7.5% + 5% = 12.5%, of which 10% can be bet for value
    [0, 0.1]U[0.975, 1]

    6) So what’s Hero’s checking range?
    Everything that is not in the betting range
    (0.1, 0.975)

    7) Given Villain’s ranges, with what hands can Hero make a +EV c/c?
    When checked to, Villain will raise 45% of the time, so Hero can c/c (0.1, 0.225] with 0.225 being 0 EV.

    8) When Hero checks, how often does Hero have to c/f for Villain to have a +EV bluff?
    a = 1/(1+P) where P is the initial pot with 2 antes of P/2
    If P/2 = 0.25, P = 0.5
    a = 1/(1.5) = 2/3 = 66.7%

    9) Given the answers to 6 & 8, what % of Hero’s starting range should c/f?
    Hero checks 0.975 – 0.1 = 0.875 = 7/8 = 87.5% of the time
    7/8 * 2/3 = 14/24 = 0.583 = 58.3%

    10) What is Hero’s range to c/c?
    87.5% - 58.3% = 29.2%
    (0.1, 0.392]


    OK, so why the discrepancy between the answer to 7 and the answer to 10?

    Is my thinking completely bunk, here?

    Is it that 7) is really asking about Hero's value range to c/c, while 10) is asking about a balanced range?
    Last edited by MadMojoMonkey; 12-16-2012 at 01:06 AM.
  45. #45
    spoonitnow's Avatar
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    I'm not going to read all of that because I don't have time, but if you're interested in a more complicated game, you should look into the full-street fixed-limit AKQ game. I'm pretty sure that I've broken it down at some point.
  46. #46
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    After 20+ hours of going through the full street game, I can assure you that the above assumptions are total bunk. I still have a lot of confusion. The core of which is that it's hard to establish a range when there are multiple streets of betting. There is not clear cut division of starting range that I have proved is correct. I can only assume different split points and show that one is better than another.

    This is something that I can't get off my mind... It's so simple, but so complex.

    FWIW, with the full street [0, 1] game, Villain only b/c's with the top 2 best hands, and b/f's only the 1 hand dictated by the call value in the half-street game. This value is the line between c/c and c/f. This minimizes Hero's EV. I found this by trial and error, not proof. I don't know how many divisions in Villain's range are appropriate. The above is based on assuming villains range is something like:
    |--b/c--|--c/c--|--b/f--|--c/f--|
    right now
  47. #47
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    Quote Originally Posted by MadMojoMonkey View Post
    FWIW, with the full street [0, 1] game, Villain only b/c's with the top 2 best hands, and b/f's only the 1 hand dictated by the call value in the half-street game.
    What in the hell are you talking about? There are no top 2 best hands with [0,1] interval ranges.
  48. #48
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    There are when I'm really drunk and just start plugging numbers into Excel, and I set the granularity to 0.001... lol Wasted post.

    The major take-away is that I'm still solving the full street game and my initial crude applications of half-street solutions to the full street system was giving bad results (at least on some of the values I've checked).

    The only 100% consistent result is that the lowest hand which can be bet for value is always the average of the limits of the caller's calling range. I.e. if the opponent will make the call with the range [0.2, 0.4], then the lowest number to V-bet into that range is (0.2 + 0.4)/2 = 0.3. Very intuitive.

    I'm still dealing with the issue of getting really odd results, but I have solved Hero's range for 2 distributions of Villain's range.

    |--b/c--|--c/c--|--b/f--|--c/f--| polarized bet/merged check
    AND
    |--b/c--|--c/c--|--c/f--|--b/f--| polarized bet/polarized check

    The really odd thing I've noticed is that once Villain checks, Hero just owns the crap out of him. Villain has a limited range, but Hero's is still undefined. Hero can never make a -EV check behind, but can make V-bets and bluffs which are more +EV than checking (quite similar to the half-street game, really)...The power of showdown value, at work.
  49. #49
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    Quote Originally Posted by MadMojoMonkey View Post
    The only 100% consistent result is that the lowest hand which can be bet for value is always the average of the limits of the caller's calling range. I.e. if the opponent will make the call with the range [0.2, 0.4], then the lowest number to V-bet into that range is (0.2 + 0.4)/2 = 0.3. Very intuitive.
    This kinda translates into "I need at least 50% equity against villain calling range to make a v-bet", doesn't it?
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  50. #50
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    Quote Originally Posted by daviddem View Post
    This kinda translates into "I need at least 50% equity against villain calling range to make a v-bet", doesn't it?
    Exactly.

    Some interesting things about EV calc's:
    1) All EV calc's start positive (nut hands are winners), and
    2) always have a slope less than or equal to 0 (good hands win more than, or at least as much as, bad hands),
    3) whether Hero bets, calls, checks or folds (no line will make a worse hand have more EV than a better hand when both are played correctly).

    For the mathematically literate:
    Given that EV is a function of hand x and betting line L (where L includes all prior actions and the current action), then:
    All EV(x|L) are monotonically decreasing functions.

    conclusions, taking into account that a fold is always 0EV:
    If the EV of a call falls below 0 for any value, then a fold is less -EV than a call for all greater values. (If it's -EV to call with hand x, then it is -EV to call with any hand that can't beat x.)

    If the EV of any bet is below 0 for any value, then Hero can not bluff for exploitation. (When Villain calls too much, Hero exploits by not bluffing.)
  51. #51
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    Fun fact: in the full street AKQ game, Hero has to bet his K sometimes when playing unexploitably.

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