****"I went through the calculation for a game with 3 possible hands: AA, AK, KK. "


I'm sorry but I do not have any view on a game of this type. I do not do game theory exercises mainly


because I do not know how to and do not wish , right now, to learn. For all I know your calc's are correct. My


question is "What relevance is there to Casino poker games such as Holdem or stud?" ****


I was trying to show you how an optimal strategy can be generated for a simple example, because it is very complicated to work out for heads-up hold'em or stud. However, what I was trying to show you was the method for devising such a strategy. The example at the bottom of a simple game was just to show that the formula I gave at the beginning worked.


So the formula, which determines the expectation for each of your opponent's possible hands based on the frequency with which you bet each possible hand is applicable to hold'em or stud. You have to adjust it to take into account raises, and you have to work backward, in all likelyhood, from the last round, but the idea is the same.


***I do not know a lot about a lot of things so maybe i do not know why the optimal strategy "is one for which all


the Yj's are equal to -1."


I believe the optimal strategy is one for which player one realizes the greatest expectation. ***


This may explain why you refuse to accept the arguments being presented. You are taking "optimal" to be an equivalent term to "greatest expectation", but that's not what it is. You should instead think of it as "unexploitable". That is why the idea is to get all of the Yj's (or at least as many as possible) to be equal to -1. You want your opponent's actions to make as little difference as possible. That is why when deciding how often you should bluff on the end, you want to bluff at a frequency which makes the odds against you bluffing equal to the pot odds your opponent is getting to call. Game theory optimal strategy would be something akin to that, but applied to all possible hands across all rounds of betting. This means that devising a heads-up hold'em GTO strategy is incredibly difficult, but possible in theory. It may never be done, but nonetheless, it could be done. Just like mapping out all the possible games of chess that could ever be played may never be done, but would still be possible given sufficient time. It may be impractical to come up with a GTO poker bot, but that doesn't make it impossible.


I'm still unsure whether a GTO strategy is possible in a multi-way game, but a rules-based adaptive program would probably work much better in a ring game anyway, since it is easier to make a program that plays better than an average field of opponents than it is to make one that can beat a world champ heads-up. Short of devising a GTO bot, I don't believe we are anywhere close to making a computer which could beat a top pro poker player in a heads-up match. It may eventually be done, but only once we have ai which is able to accurately mimic human thought patterns (perhaps a very elaborate neural net would be capable of learning to beat a human pro). This is not the same thing as chess, because in chess you can evaluate a particular position and it is always the same every time you see that position. In poker, you have to adapt to your opponent, and without having a GTO strategy, the computer would have almost no chance of keeping up with a pro. Again, this argument is based on current ai technology.