2M All-In into $5 Pot! WWYD? Daniel Negreanu's No-Limit Hold'em Challenge! (Poker Hand Analysis)
deep learning machine learning arxiv explained neural networks ai artificial intelligence paper poker negreanu daniel negreanu realkidpoker flop turn river holdem libratus pluribus rebel facebook poker bot nash equilibrium overbet hand range level raise hole cards aces quads twitter analysis bluff nuts
#ai #technology #poker

Daniel Negreanu posted a set of very interesting No-Limit Hold'em situations on Twitter. I try to analyze them from the perspective of a poker bot. See how such bots think about the game and approximate Nash equilibria.

https://twitter.com/RealKidPoker/status/1337887509397741568
https://twitter.com/RealKidPoker/status/1337899147337244673
https://twitter.com/RealKidPoker/status/1337904860721606656

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 Hi there. Today I want to bring to you a little bit of a different video. The video right now is supposed to be sort of a motivational lead up to the next video I want to release. And the next video is going to be about Facebook's new rebel algorithm, which is an algorithm that solves two player zero sum imperfect information games. So it is very similar in to the alpha zero algorithm or the alpha go algorithm, just that line of algorithms that combine search and learning. But whereas the alpha line is in perfect information games, so games where you can see everything like chess or go, the rebel algorithm is an imperfect information games. And one example of this is poker, so heads up, heads up poker, like heads up, Texas hold them no limit, let's say in this case, is a two player zero sum, let's assume the house doesn't take rake two players zero sum, imperfect information game, which this algorithm rebel can solve better than apparently anything before it. And Daniel Negrano, who is a, you know, a longtime poker pro has released these polls on Twitter, which I found just to be very interesting. So the timing was very fitting. And I thought I sort of make a lead up video to the next paper video, just to sort of get you into the, the thinking if you've never if you've never played poker at sort of beyond an amateur level, I sort of want to motivate you what makes this game so interesting, because it seems pretty simple at the start. Okay, so here we go. The Daniel Negrano poses the following question, poker question for you all. And maybe I should briefly explain how the game works for anyone who doesn't know there. And if you have one minute, if you know, just jump ahead one minute or so. So at the beginning, you get two cards, your opponent gets two cards, you don't know the opponent's cards, the opponent doesn't know your cards, then success, successively, on the board, they're going to be revealed first three cards at a time, which is called the flop. Then there's one other card, which is called the turn. And then there's another card, which is called the river. And there are four betting rounds. So there's one betting round pre flop, which is when no cards are on the table, there's one betting round at the flop, one at the turn and one at the river. And then if the players are still in and haven't folded, the cards are revealed and scored according to the normal rules of poker. So your two cards and the table five cards, you get to choose any five of those seven to make up the poker hand, whoever has the better poker hand wins. Okay. So in this situation here, you have aces. So your whole cards are two aces, which is, you know, the best pre flop hand, but the board is ace, aka eight, four, four, so ace King eight, four, and four. So that's the board at which gives you a full house aces with fours, okay, which is the second best hand that's possible on this board. So you have the second best hand that usually you would be happy to put all your money in into this board. Because the only hand that's better than you is if your opponent has two fours. So that's is a possibility, right? But it's a very, very, very slim possibility. So you might think I want to put all my money into here. But now, you know, now comes the tricky part is you put all your money in here, because you say, well, there's only really one hand that beats me. Okay, but you have to think ahead and say, how often does my opponent have that hand? And crucially, crucially, how often are they going to give me their money while not having this hand? So let's say your opponent has an eight and a nine, okay. And, and so they have a pair of eights, which, you know, they might think, you know, I have a pair pair is okay. But you put in a lot of money, they're probably going to fold that hand, right? So if you put in a lot of money here, they're not giving you any money. So if now, let's say they have like two kings, which is a very strong hand on this board. But if you put in like exorbitant amounts of money, still, they're going to conclude, well, it's, it's not worth it. Like, there are still better hands I'm going to fold. So all of this, it's not just a question of which cards do you have, it's not even a question which cards your opponent has. It's, it's a it's a question also of how much money do you put in because that regulates very much how the strategies are. I hope I hope you can sort of see that. So you always have to think about what possible cards could my opponents hold? And which of these cards are they willing to put in how much money into the pot? And then from that, you can determine is that profitable for me or not? In this particular situation, there are $5 already in the pot. So all the previous betting rounds, they get collected into what's called the pot. So the pot here, in this case is $5. And your opponent, your opponent bets $2 million. Okay, so $2 million on the pot into a pot of five, it's obviously a constructed scenario. But your opponent now puts up 2 million, okay, so you have to put in 2 million into a pot that's now $2 million and $5. So if you let's say if you fold, you lose whatever you put in of these $5. So you shouldn't think that sunk cost anyway, you should simply think I put in 2 million in order to win five plus the 2 million the opponent puts in. Okay. So obviously, this is exactly the reverse of what we looked at. Now your opponent is putting in a ginormous amount of money, okay, and you, you have the second best hand. So this this get now gets interesting. Now there is an additional complication here, would you call or fold against the guy who always goes in on the river every hand, okay, this is an additional information, somehow, you know, that this person always goes in on the river. So on the river, they always shove all their money all in. That's what you know. Now, a lot of people would lean to an easy call here, a lot of people would say, of course, they're going to all in with any like any, any time they're on the river. So of course, I'm going to call it the second best hand, there are many, many hands, and if they're going to do this with all hands, but that's not the case. They're just because they always go all in on the river, every hand, I think this is slightly under specified. It's every hand where they get to the river, right. So here, a smart opponent, let's say this is a smart opponent. But for some reason, someone kidnapped their dog and threatens to kill the dog if they don't always go all in on the river. But other than that, they're very smart player. So they, they now also know that they always go all in on the river because you know, they always go in all in on the river. So what they will do is, once they're on the flop and the turn, they will only ever continue with hands where they would go all in all in on the river, right. And they are they not only they not don't always have 2 million in the end on the table, they might have smaller values. So when they are on the flop, and when they are on the turn, they are very much aware that they have this giant amount of money, and that they must go all in if they reach the river. So conceivably, they would fold every hand that they weren't willing to go all in on the river. So they they won't have just any cards, they that that seriously skews their distribution of cards that they could hold because they make that inference, right. So now you can sit here and say, Okay, it's conceivable that they actually hold off on, you know, most of their cards, they would fold most of their cards on the on the flop or turn, given that they must always go all in all in on the river. So let's actually look at the turn. So let's imagine we do not know that this is a four, right. So we the last decisions are made here, right here, when it's the when it's the turn. Here your opponent will only go to the river with cards where they feel that they can then fully go all in all the way, right? That's because they also know they go all in every time they reach the river. So the question is, what possible range could they do this with? And one possibility is like they they do it. If they know they have 2 million. It's a very risky move to go all in on the river, right. So conceivably, I'd say they would not do it with two fours because they can't possibly know that another four is coming, the chances so incredibly slim. However, of course, that strategy now also changes the range of hands that you continue to the river with. So you can be you knowing that the opponent will only go to the river with cards where they could go all in on the river also will change your distribution. But just in this particular situation, I would say the following. If this is the case, the opponent can't possibly know that there's another four coming. Therefore, their range here, if it includes two fours, if it includes those, it will also include something like two kings, it will also include something like ace four, or king four, like conceivably because those maybe not but two eights maybe. But at least two kings, so their range is conceivably. Yeah, if it includes two fours, it must include two eights and two kings, right? Because these are strictly better at the turn. It could even be any ace because that blocks you from having an ace. So if they can have fours at the end, they can also have kings and eights. And just because they can have those hands, it probably makes for a for a good call here on the river because you are beating kings and eights on on the river. Specifically, the fours are much more unlikely because the four is actually in the deck since we we already know it's coming right here. So in this case, I would call because of those whole reasoning, not because I have the second best hand, right? I hope you can sort of see how this back and forth goes. So you assume that your opponent is smart, your opponent assumes that you are smart. And then you sort of reason 123 levels in depth. And of course, if you reason to infinity, that becomes a Nash equilibrium. And that's exactly what this rebel algorithm approximates. I would have guessed that this situation is much more interesting if you reverse the board. So if the board was something like 4484444 ace, King eight or something like this, where your opponent clearly already has the best possible hand before they enter the river, that would make would would make it quite a bit more interesting, I believe. And I, I don't know what the analysis would be. But let's go on to the next 10. So that would be my guess would be called. I haven't, as you can see, I haven't answered yet. I will after the video. But it's irrelevant because the most comments I read are just like inferring very simple things, which are, as I say, irrelevant. So the follow up question here is their same situation $5 in the pot, two million opponent bets 2 million all in on the river board is the same, you have aces, would you call or fold against a player you know nothing about? Okay, so here's a player you know nothing about. Now, the you know nothing about is so now you like, now you have to estimate probabilities that the person is brain dead and things like this, right. But what you can do, what you can do is always just estimate sort of the Nash equilibrium strategy of the situation, and maybe go with that, because at least then you cannot lose an expectation. So if you fact if you like, factor in the fact that the person might be dumb or brain dead or something like this, then if you mess up these probabilities, you are in fact exploitable. Though, you know, the exploitability only matters if that situation happens over and over and over and over again, whereas I think this is going to happen to you at maximum once. However, same situation, but your opponent does not go all in on the river every hand, you know nothing about them, right, the board happens as it is. And all of a sudden, this person pushes 2 million. Now let's analyze this. So you might think, hey, this person pushes 2 million in a pot of $5. They must hold the nuts very, very, very often for this to be profitable, right. So they probably hold the two fours right here. But then again, if you infer that you might want to go ahead and fold those aces, okay, you fold the aces. So your opponent thinks about this, and they realize, wait a minute, if I can get them to fold aces, which is the second best hand on this board, right, I should probably push this much money a lot more often, because I can, you know, like I can get them off aces, I can probably get them off most hands that they are in this situation with right on this board, a ace, King eight, we don't know the colors, but there are a lot of hands that get to the river in this situation. So I can bluff them off a lot of them by simply pushing 2 million in the pot, right. But then it's this old game, you push 2 million to win $5. This has to work very often. In fact, this has to work now it has to work like for for 399,000 out of 400,000 times to break even right, if it if it doesn't work even one time. Yeah, so if Europe, if you fold anything but the but the absolute nuts, your opponent might actually just hold a single four, because then they know you don't have two fours. And then they know you can't possibly have the best hand, then it can push you off of it. But then, right, they if they bluff a certain amount of time, if they don't need to bluff often for you to actually make it profitable. And if they do, in fact bluff, so let's let's assume they just bluff if they have a four, because they know you can't have both fours because they have one. So you can never have the best hand. And they think if they bet 2 million, they can push you off any hand. Now you go ahead and you say, wait a minute, if they bluff whenever they have a single four, they're much more often going to have a single four, like maybe they have a four, for nine or something like this, they're much more often going to have a hand like this, and two fours just combinatorically, right. So maybe they're actually on a bluff pretty often here if they do this every single time they have a four. So I can actually call, it doesn't even matter that I have aces, right, I can call with any any hand that hits anything on the sport is probably going to to beat though, if they have a four, they have trips. So let's say if they bluff with any hand, I can call with any hand. And they will think about this and say, maybe I shouldn't bluff with any hand, right, I should probably moderate that because the other person will adjust. If they bluff with a four, they have trip fours. So I even if they bluff with a four, I might only and it is a bluff like if you have a foreign you bet 2 million here, that's a bluff, like you're clearly trying to get someone off of like aces. Because it's not like you don't bet for value 2 million into $5 with this. So I will only call with aces kings eights, ace for king four eight four stuff like this, because they all beat a single four, right. And now the question becomes, again, how so there is there is the number of hands I will call with like aces, kings, and so on. Ace for how these are a subset of hands, or maybe not like this as subset of hands, probably a large subset of all the hands that I would hold on the river like that I would get to the river with right here. And they are going to push me off of those hands with with any large bet. But this this bet is really meant to get me off of those strong hands. So the question is, how often do they do this with a four in order to still be profitable. So we get back to this sort of inference of how often can this be a bluff for me to legitimately call here. And that factors in how often I am on the river and how often on the river I hold one of these hands that I could conceivably catch a bluff with. So you can see that a lot of a lot of stuff is going in here. Me personally, I would say that I know nothing about this person, I would probably fold in this in this case, because if I assume they're smart, they must know that they can only pull this 2 million into $5 thing very, very few times if they don't have the absolute nuts in this case. And if they don't have the nuts, it almost it almost doesn't matter what they have, they probably have a single for and then yeah, the number of hands that I can have on the river that are going to catch a bluff with a single for is just too large for them to often bluff right here. Of course, if we both play, if if the person plays Nash optimal, then I have like some assignment to call or fold, right, probability of call, probability of fold that I would do in this particular situation. And and it's going to be break even. Okay, last question, though that might not be true. I might have actually a fixed binary decision here. No, because that influences their strategy to Yeah. Um, last question, same thing. But now, which hand would be better to have if you choose to call? So you, you choose to call. But now, which hand would you rather have in that situation? Would you have King for or aces? So some people might say, well, aces, clearly, because aces here is the better hand than King for right? aces is full house aces full of force and King for is forceful of Kings. So let's say you imagine you have King for why would you want to have King for you would want to have King for because now your opponent can't have two fours anymore. Okay, so the possibility of your opponent holding two fours is off the table because there are only four fours in the deck. And the so you're blocking that possibility that your opponent has two fours. So they cannot have the nuts possibly. They it's much more probable now that in fact, they have a single four, right? And they are trying to push you off of something like aces, you see. So it's a bit the same situation as before. And we can we can remark that King for is also in this hands that we would call with. But so are the aces. Now, it all again boils down to what's the frequency of them folding here and that boils down to what's the proportion of hands that you have here plus what's the frequency of them that you call with. So the question is, would you rather have aces or King for and why would you why would you rather have aces? What would be reasons that you would rather have aces? Well, if your opponent is smart, they might think that and I haven't thought this through before, but let's just try to figure this out together. Your opponent. So if you'd rather have aces than King for that must mean that your opponent would do this conceivably with hands that you beat with aces, but not with King for like you you decide to call that's a given you decide to call so now everyone reveals their cards. And so if you say you'd rather have aces, that means you think that your opponent would do this kind of stuff with something like that Kings or or eights or something like this, something that would beat King for but not beat aces. So your opponent your opponent might be smart and think, wait a minute. If this person has an a four, right, then they will think that I cannot possibly have two fours. And therefore they will call with a single four, even if I bet 2 million they will think who I have the four and therefore they can't have the four. So this must be one of those rare times where they bluff, right. And then they might say, well, but I have two eights, right, I have two eights, I beat a single four. And therefore, I can actually get money out of anyone that's trying to catch my bluff because they have a single four. So now the question is, how often does anyone on the river here have a single four. And again, this is where I go and say that board would be probably more interesting if it was this way around, because it's much more conceivable that anyone has a single for laying around. If the flop was this already, though, King for conceivably as you hit the king on the flop, and then you somehow get through to the river while scoring two fours, but it's just not as likely that you still have the four around. But still, you can sort of see the thinking, right. So the opponent might think, wait, they're going to call me with any old four, especially with like, also with like King for I have eights, I beat things like ace for King for I beat a single for my opponent's gonna think I only do the 2 million things with two fours. My opponent's gonna have a four, they will infer that I can't have a four, they will call me because they think I'm bluffing and ta da da da. Okay, so you can see that it goes pretty, pretty deep. And then in that case, they will push with the eights. And in that case, you much rather have the aces right here, because they don't know whether you have the four or not, right. But if you have the aces, again, you do not have the four. And it is very possible that your opponent has two fours. And after all, it's 2 million into a pot of $5, they would only they have to have a very good hand very often for this to be profitable. Okay, so this, this kind of thinking is is what computation of an ash equilibrium in effect boils down to. So we're going to see, I don't know what the correct answers to this is, by the way, I the even the rebel source code isn't open source for poker, the code is open source, but the implementation for poker isn't and I think the checkpoints for poker aren't. So maybe we won't we won't find out, I would love to hear your opinions on this, maybe I am completely wrong right here. But this is about what an algorithm like that has has to do. And I hope I've sort of given you an overview of why this sort of games are interesting, what these algorithms need to think about, and why it is so much harder than something like chess or go, not that the game itself is harder, but you have to constantly reason about things that you do not know. And you constantly have to assign probabilities and combinatorial fractioning, how often does this happen? How often does this happen? And then you have to adjust each time when you adjust your strategy, you have to think that your opponent can make the same conclusions, given the observed state, and they can also adjust their strategy. So that's the difficulty. Those are the questions I would say you go vote, see what other people have to say. And maybe Daniel will let us know once the polls are over. Alright, so that was it for me. Thanks a lot for watching. And I hope to have the next video out very, very soon about rebel. Bye bye.
 Hi there. Today I want to bring to you a little bit of a different video. The video right
 now is supposed to be sort of a motivational lead up to the next video I want to release.
 And the next video is going to be about Facebook's new rebel algorithm, which is an algorithm
 that solves two player zero sum imperfect information games. So it is very similar in
 to the alpha zero algorithm or the alpha go algorithm, just that line of algorithms that
 combine search and learning. But whereas the alpha line is in perfect information games,
 so games where you can see everything like chess or go, the rebel algorithm is an imperfect
 information games. And one example of this is poker, so heads up, heads up poker, like
 heads up, Texas hold them no limit, let's say in this case, is a two player zero sum,
 let's assume the house doesn't take rake two players zero sum, imperfect information game,
 which this algorithm rebel can solve better than apparently anything before it. And Daniel
 Negrano, who is a, you know, a longtime poker pro has released these polls on Twitter, which
 I found just to be very interesting. So the timing was very fitting. And I thought I sort
 of make a lead up video to the next paper video, just to sort of get you into the, the
 thinking if you've never if you've never played poker at sort of beyond an amateur level,
 I sort of want to motivate you what makes this game so interesting, because it seems
 pretty simple at the start. Okay, so here we go. The Daniel Negrano poses the following
 question, poker question for you all. And maybe I should briefly explain how the game
 works for anyone who doesn't know there. And if you have one minute, if you know, just
 jump ahead one minute or so. So at the beginning, you get two cards, your opponent gets two
 cards, you don't know the opponent's cards, the opponent doesn't know your cards, then
 success, successively, on the board, they're going to be revealed first three cards at
 a time, which is called the flop. Then there's one other card, which is called the turn.
 And then there's another card, which is called the river. And there are four betting rounds.
 So there's one betting round pre flop, which is when no cards are on the table, there's
 one betting round at the flop, one at the turn and one at the river. And then if the
 players are still in and haven't folded, the cards are revealed and scored according to
 the normal rules of poker. So your two cards and the table five cards, you get to choose
 any five of those seven to make up the poker hand, whoever has the better poker hand wins.
 Okay. So in this situation here, you have aces. So your whole cards are two aces, which
 is, you know, the best pre flop hand, but the board is ace, aka eight, four, four, so
 ace King eight, four, and four. So that's the board at which gives you a full house
 aces with fours, okay, which is the second best hand that's possible on this board. So
 you have the second best hand that usually you would be happy to put all your money in
 into this board. Because the only hand that's better than you is if your opponent has two
 fours. So that's is a possibility, right? But it's a very, very, very slim possibility.
 So you might think I want to put all my money into here. But now, you know, now comes the
 tricky part is you put all your money in here, because you say, well, there's only really
 one hand that beats me. Okay, but you have to think ahead and say, how often does my
 opponent have that hand? And crucially, crucially, how often are they going to give me their
 money while not having this hand? So let's say your opponent has an eight and a nine,
 okay. And, and so they have a pair of eights, which, you know, they might think, you know,
 I have a pair pair is okay. But you put in a lot of money, they're probably going to
 fold that hand, right? So if you put in a lot of money here, they're not giving you
 any money. So if now, let's say they have like two kings, which is a very strong hand
 on this board. But if you put in like exorbitant amounts of money, still, they're going to
 conclude, well, it's, it's not worth it. Like, there are still better hands I'm going to
 fold. So all of this, it's not just a question of which cards do you have, it's not even
 a question which cards your opponent has. It's, it's a it's a question also of how much
 money do you put in because that regulates very much how the strategies are. I hope I
 hope you can sort of see that. So you always have to think about what possible cards could
 my opponents hold? And which of these cards are they willing to put in how much money
 into the pot? And then from that, you can determine is that profitable for me or not?
 In this particular situation, there are $5 already in the pot. So all the previous betting
 rounds, they get collected into what's called the pot. So the pot here, in this case is
 $5. And your opponent, your opponent bets $2 million. Okay, so $2 million on the pot
 into a pot of five, it's obviously a constructed scenario. But your opponent now puts up 2
 million, okay, so you have to put in 2 million into a pot that's now $2 million and $5. So
 if you let's say if you fold, you lose whatever you put in of these $5. So you shouldn't think
 that sunk cost anyway, you should simply think I put in 2 million in order to win five plus
 the 2 million the opponent puts in. Okay. So obviously, this is exactly the reverse
 of what we looked at. Now your opponent is putting in a ginormous amount of money, okay,
 and you, you have the second best hand. So this this get now gets interesting. Now there
 is an additional complication here, would you call or fold against the guy who always
 goes in on the river every hand, okay, this is an additional information, somehow, you
 know, that this person always goes in on the river. So on the river, they always shove
 all their money all in. That's what you know. Now, a lot of people would lean to an easy
 call here, a lot of people would say, of course, they're going to all in with any like any,
 any time they're on the river. So of course, I'm going to call it the second best hand,
 there are many, many hands, and if they're going to do this with all hands, but that's
 not the case. They're just because they always go all in on the river, every hand, I think
 this is slightly under specified. It's every hand where they get to the river, right. So
 here, a smart opponent, let's say this is a smart opponent. But for some reason, someone
 kidnapped their dog and threatens to kill the dog if they don't always go all in on
 the river. But other than that, they're very smart player. So they, they now also know
 that they always go all in on the river because you know, they always go in all in on the
 river. So what they will do is, once they're on the flop and the turn, they will only ever
 continue with hands where they would go all in all in on the river, right. And they are
 they not only they not don't always have 2 million in the end on the table, they might
 have smaller values. So when they are on the flop, and when they are on the turn, they
 are very much aware that they have this giant amount of money, and that they must go all
 in if they reach the river. So conceivably, they would fold every hand that they weren't
 willing to go all in on the river. So they they won't have just any cards, they that
 that seriously skews their distribution of cards that they could hold because they make
 that inference, right. So now you can sit here and say, Okay, it's conceivable that
 they actually hold off on, you know, most of their cards, they would fold most of their
 cards on the on the flop or turn, given that they must always go all in all in on the river.
 So let's actually look at the turn. So let's imagine we do not know that this is a four,
 right. So we the last decisions are made here, right here, when it's the when it's the turn.
 Here your opponent will only go to the river with cards where they feel that they can then
 fully go all in all the way, right? That's because they also know they go all in every
 time they reach the river. So the question is, what possible range could they do this
 with? And one possibility is like they they do it. If they know they have 2 million. It's
 a very risky move to go all in on the river, right. So conceivably, I'd say they would
 not do it with two fours because they can't possibly know that another four is coming,
 the chances so incredibly slim. However, of course, that strategy now also changes the
 range of hands that you continue to the river with. So you can be you knowing that the opponent
 will only go to the river with cards where they could go all in on the river also will
 change your distribution. But just in this particular situation, I would say the following.
 If this is the case, the opponent can't possibly know that there's another four coming. Therefore,
 their range here, if it includes two fours, if it includes those, it will also include
 something like two kings, it will also include something like ace four, or king four, like
 conceivably because those maybe not but two eights maybe. But at least two kings, so their
 range is conceivably. Yeah, if it includes two fours, it must include two eights and
 two kings, right? Because these are strictly better at the turn. It could even be any ace
 because that blocks you from having an ace. So if they can have fours at the end, they
 can also have kings and eights. And just because they can have those hands, it probably makes
 for a for a good call here on the river because you are beating kings and eights on on the
 river. Specifically, the fours are much more unlikely because the four is actually in the
 deck since we we already know it's coming right here. So in this case, I would call
 because of those whole reasoning, not because I have the second best hand, right? I hope
 you can sort of see how this back and forth goes. So you assume that your opponent is
 smart, your opponent assumes that you are smart. And then you sort of reason 123 levels
 in depth. And of course, if you reason to infinity, that becomes a Nash equilibrium.
 And that's exactly what this rebel algorithm approximates. I would have guessed that this
 situation is much more interesting if you reverse the board. So if the board was something
 like 4484444 ace, King eight or something like this, where your opponent clearly already
 has the best possible hand before they enter the river, that would make would would make
 it quite a bit more interesting, I believe. And I, I don't know what the analysis would
 be. But let's go on to the next 10. So that would be my guess would be called. I haven't,
 as you can see, I haven't answered yet. I will after the video. But it's irrelevant
 because the most comments I read are just like inferring very simple things, which are,
 as I say, irrelevant. So the follow up question here is their same situation $5 in the pot,
 two million opponent bets 2 million all in on the river board is the same, you have aces,
 would you call or fold against a player you know nothing about? Okay, so here's a player
 you know nothing about. Now, the you know nothing about is so now you like, now you
 have to estimate probabilities that the person is brain dead and things like this, right.
 But what you can do, what you can do is always just estimate sort of the Nash equilibrium
 strategy of the situation, and maybe go with that, because at least then you cannot lose
 an expectation. So if you fact if you like, factor in the fact that the person might be
 dumb or brain dead or something like this, then if you mess up these probabilities, you
 are in fact exploitable. Though, you know, the exploitability only matters if that situation
 happens over and over and over and over again, whereas I think this is going to happen to
 you at maximum once. However, same situation, but your opponent does not go all in on the
 river every hand, you know nothing about them, right, the board happens as it is. And all
 of a sudden, this person pushes 2 million. Now let's analyze this. So you might think,
 hey, this person pushes 2 million in a pot of $5. They must hold the nuts very, very,
 very often for this to be profitable, right. So they probably hold the two fours right
 here. But then again, if you infer that you might want to go ahead and fold those aces,
 okay, you fold the aces. So your opponent thinks about this, and they realize, wait
 a minute, if I can get them to fold aces, which is the second best hand on this board,
 right, I should probably push this much money a lot more often, because I can, you know,
 like I can get them off aces, I can probably get them off most hands that they are in this
 situation with right on this board, a ace, King eight, we don't know the colors, but
 there are a lot of hands that get to the river in this situation. So I can bluff them off
 a lot of them by simply pushing 2 million in the pot, right. But then it's this old
 game, you push 2 million to win $5. This has to work very often. In fact, this has to work
 now it has to work like for for 399,000 out of 400,000 times to break even right, if it
 if it doesn't work even one time. Yeah, so if Europe, if you fold anything but the but
 the absolute nuts, your opponent might actually just hold a single four, because then they
 know you don't have two fours. And then they know you can't possibly have the best hand,
 then it can push you off of it. But then, right, they if they bluff a certain amount
 of time, if they don't need to bluff often for you to actually make it profitable. And
 if they do, in fact bluff, so let's let's assume they just bluff if they have a four,
 because they know you can't have both fours because they have one. So you can never have
 the best hand. And they think if they bet 2 million, they can push you off any hand.
 Now you go ahead and you say, wait a minute, if they bluff whenever they have a single
 four, they're much more often going to have a single four, like maybe they have a four,
 for nine or something like this, they're much more often going to have a hand like this,
 and two fours just combinatorically, right. So maybe they're actually on a bluff pretty
 often here if they do this every single time they have a four. So I can actually call,
 it doesn't even matter that I have aces, right, I can call with any any hand that hits anything
 on the sport is probably going to to beat though, if they have a four, they have trips.
 So let's say if they bluff with any hand, I can call with any hand. And they will think
 about this and say, maybe I shouldn't bluff with any hand, right, I should probably moderate
 that because the other person will adjust. If they bluff with a four, they have trip
 fours. So I even if they bluff with a four, I might only and it is a bluff like if you
 have a foreign you bet 2 million here, that's a bluff, like you're clearly trying to get
 someone off of like aces. Because it's not like you don't bet for value 2 million into
 $5 with this. So I will only call with aces kings eights, ace for king four eight four
 stuff like this, because they all beat a single four, right. And now the question becomes,
 again, how so there is there is the number of hands I will call with like aces, kings,
 and so on. Ace for how these are a subset of hands, or maybe not like this as subset
 of hands, probably a large subset of all the hands that I would hold on the river like
 that I would get to the river with right here. And they are going to push me off of those
 hands with with any large bet. But this this bet is really meant to get me off of those
 strong hands. So the question is, how often do they do this with a four in order to still
 be profitable. So we get back to this sort of inference of how often can this be a bluff
 for me to legitimately call here. And that factors in how often I am on the river and
 how often on the river I hold one of these hands that I could conceivably catch a bluff
 with. So you can see that a lot of a lot of stuff is going in here. Me personally, I would
 say that I know nothing about this person, I would probably fold in this in this case,
 because if I assume they're smart, they must know that they can only pull this 2 million
 into $5 thing very, very few times if they don't have the absolute nuts in this case.
 And if they don't have the nuts, it almost it almost doesn't matter what they have, they
 probably have a single for and then yeah, the number of hands that I can have on the
 river that are going to catch a bluff with a single for is just too large for them to
 often bluff right here. Of course, if we both play, if if the person plays Nash optimal,
 then I have like some assignment to call or fold, right, probability of call, probability
 of fold that I would do in this particular situation. And and it's going to be break
 even. Okay, last question, though that might not be true. I might have actually a fixed
 binary decision here. No, because that influences their strategy to Yeah. Um, last question,
 same thing. But now, which hand would be better to have if you choose to call? So you, you
 choose to call. But now, which hand would you rather have in that situation? Would you
 have King for or aces? So some people might say, well, aces, clearly, because aces here
 is the better hand than King for right? aces is full house aces full of force and King
 for is forceful of Kings. So let's say you imagine you have King for why would you want
 to have King for you would want to have King for because now your opponent can't have two
 fours anymore. Okay, so the possibility of your opponent holding two fours is off the
 table because there are only four fours in the deck. And the so you're blocking that
 possibility that your opponent has two fours. So they cannot have the nuts possibly. They
 it's much more probable now that in fact, they have a single four, right? And they are
 trying to push you off of something like aces, you see. So it's a bit the same situation
 as before. And we can we can remark that King for is also in this hands that we would call
 with. But so are the aces. Now, it all again boils down to what's the frequency of them
 folding here and that boils down to what's the proportion of hands that you have here
 plus what's the frequency of them that you call with. So the question is, would you rather
 have aces or King for and why would you why would you rather have aces? What would be
 reasons that you would rather have aces? Well, if your opponent is smart, they might think
 that and I haven't thought this through before, but let's just try to figure this out together.
 Your opponent. So if you'd rather have aces than King for that must mean that your opponent
 would do this conceivably with hands that you beat with aces, but not with King for
 like you you decide to call that's a given you decide to call so now everyone reveals
 their cards. And so if you say you'd rather have aces, that means you think that your
 opponent would do this kind of stuff with something like that Kings or or eights or
 something like this, something that would beat King for but not beat aces. So your opponent
 your opponent might be smart and think, wait a minute. If this person has an a four, right,
 then they will think that I cannot possibly have two fours. And therefore they will call
 with a single four, even if I bet 2 million they will think who I have the four and therefore
 they can't have the four. So this must be one of those rare times where they bluff, right.
 And then they might say, well, but I have two eights, right, I have two eights, I beat
 a single four. And therefore, I can actually get money out of anyone that's trying to catch
 my bluff because they have a single four. So now the question is, how often does anyone
 on the river here have a single four. And again, this is where I go and say that board
 would be probably more interesting if it was this way around, because it's much more conceivable
 that anyone has a single for laying around. If the flop was this already, though, King
 for conceivably as you hit the king on the flop, and then you somehow get through to
 the river while scoring two fours, but it's just not as likely that you still have the
 four around. But still, you can sort of see the thinking, right. So the opponent might
 think, wait, they're going to call me with any old four, especially with like, also with
 like King for I have eights, I beat things like ace for King for I beat a single for
 my opponent's gonna think I only do the 2 million things with two fours. My opponent's
 gonna have a four, they will infer that I can't have a four, they will call me because
 they think I'm bluffing and ta da da da. Okay, so you can see that it goes pretty, pretty
 deep. And then in that case, they will push with the eights. And in that case, you much
 rather have the aces right here, because they don't know whether you have the four or not,
 right. But if you have the aces, again, you do not have the four. And it is very possible
 that your opponent has two fours. And after all, it's 2 million into a pot of $5, they
 would only they have to have a very good hand very often for this to be profitable. Okay,
 so this, this kind of thinking is is what computation of an ash equilibrium in effect
 boils down to. So we're going to see, I don't know what the correct answers to this is,
 by the way, I the even the rebel source code isn't open source for poker, the code is open
 source, but the implementation for poker isn't and I think the checkpoints for poker aren't.
 So maybe we won't we won't find out, I would love to hear your opinions on this, maybe
 I am completely wrong right here. But this is about what an algorithm like that has has
 to do. And I hope I've sort of given you an overview of why this sort of games are interesting,
 what these algorithms need to think about, and why it is so much harder than something
 like chess or go, not that the game itself is harder, but you have to constantly reason
 about things that you do not know. And you constantly have to assign probabilities and
 combinatorial fractioning, how often does this happen? How often does this happen? And
 then you have to adjust each time when you adjust your strategy, you have to think that
 your opponent can make the same conclusions, given the observed state, and they can also
 adjust their strategy. So that's the difficulty. Those are the questions I would say you go
 vote, see what other people have to say. And maybe Daniel will let us know once the polls
 are over. Alright, so that was it for me. Thanks a lot for watching. And I hope to have
 the next video out very, very soon about rebel. Bye bye.