"Game theory optimal" (as poker players like to call it) is not really the optimal strategy. It's the nash equilibrium strategy assuming the other players are also playing a nash equilibrium strategy. As soon as one player deviates from the nash equilibrium it's not optimal anymore :).
It's interesting how this affects play. If a player is bluffing slightly less than they should be, the adjustment is drastic, you should actually never call with hands that do not beat their value hands. If they are bluffing optimally, you are supposed to call using what is known as "minimum defense frequency". What's interesting about this is the minimum defense frequency is based on what the strongest hands you can possibly have in that situation are and the opponents possible hands do not even factor into it. It's required to prevent your opponent from bluffing with any two cards profitably.
To do the math, if you are on the river and the opponent bets 100 into 100, for this to be profitable for them they need to win 50% of the time or more. If your opponent is bluffing optimally, you need to call with 50% of the strongest hands you have in that specific situation (if you don't know what hands you have in a specific situation that's a problem) and sometimes they can be dogshit like King high.
But, very important to note, very few players actually bluff enough and if they bluff less than they should you should only ever call when your hand actually beats their range of possible value hands. (value vs bluff is kind of a difficult thing to communicate, generally it's value if you want your opponent to call)
Most players don't bluff enough as a result of most players calling too much! When they call too much you should obviously not bluff! This leads to very boring games of poker.
>”Game theory optimal" (as poker players like to call it) is not really the optimal strategy. It's the nash equilibrium strategy assuming the other players are also playing a nash equilibrium strategy. As soon as one player deviates from the nash equilibrium it's not optimal anymore :).
This is not really what most people mean when they say “optimal strategy”. It’s true that exploitative play will make more money if you know exactly what your opponent is doing and if they keep doing it despite it not working. Neither of these will generally hold in an actual game.
The reason it’s called “optimal strategy” is because it works no matter what your opponent is doing. It will not make as much as a strategy tailored perfectly to your opponent but it will never lose to anyone under any circumstances, assuming infinite games (assuming infinite games just so we can ignore variance). The worse case the strategy has is break even to anyone else using it.
I'm kind of curious how accidental collusion could work out. Like imagine multiple players playing in such a way that they help each other - but purely out of ignorance on how best to play the game!
It sounds like you are maybe describing a dominant strategy? Oh, wait, no, you are saying that if you play the Nash equilibrium, then no other strategy among opponents will do better against it than the Nash equilibrium would, and therefore (as the game is zero sum) the worst their choice of strategy can make you get on average, is breaking even?
Ok, now that I (I think) understand your comment: I don't think you have to know exactly what non-Nash strategy someone is playing in order to exploit it. I don't think trying to estimate how someone is likely deviating from the Nash equilibrium, in order to try to exploit it, is necessarily a mistake. I think it could be feasible for someone to get better returns on average by noticing off-Nash patterns of play in other players, than playing Nash regardless would? (Not that I could win this way. I couldn't.)
A Nash equilibrium is a pair of strategies (or, one strategy for each player) such that no player can get a better result on average from deviating from it.
If one player isn’t playing a strategy that is part of any Nash equilibrium, then the best response might also not be part of any Nash equilibrium.
If all other players are playing a strategy from a given Nash equilibrium, then you can’t do better (in expectation) than you would if you were to play the strategy for you in that Nash equilibrium.
(A game may have multiple Nash equilibria. Possibly one such equilibrium could be better for you (or for everyone) than another.)
> What's interesting about this is the minimum defense frequency is based on what the strongest hands you can possibly have in that situation are and the opponents possible hands do not even factor into it.
This is actually not quite true. MDF is purely a formula based on pot size and the bet size (pot size / pot size + bet size). The fact that it doesn't consider various ranges is why it's not really useful - it was a simplified formula used to try to understand the game before solvers existed.
There are situations where your opponent can bet any two cards profitably and you do have to fold - imagine they bet the size of the pot, but have the better hand 99% of the time, you're simply forced to let them bluff the 1% of the time they're bluffing. MDF is a pre-solver concept and not an especially useful concept in the modern game.
I'm pretty sure mdf applies to rivers when you are last to act. I'd be interested in being proven wrong however if you have solver output that shows it. I remember studying solver output and seeing it in action.
I know that before the river there are range advantages that make defending mdf a losing play.
What's true is that equilibrium strategies typically converge to solutions where the better makes the caller indifferent between calling and folding. In the toy example I've given where the betters range is so strong, the caller should always fold, the better now has an incentive to add more bluffs to the range to take advantage of the folds. Then the caller will want to call more. This might converge to the MDF which might be what you're suggesting, assuming we started with ranges that could have enough bluffs given the runouts.
If you open up the solver, and give one player only Ace-Ace as their starting range, and the other player a pair of twos, and the board Ace-Ace-Three-Three-Three, then the pair of twos will fold 100% on river and will not call at MDF.
I think another way to say this is that MDF works only if you're in a spot where you have hands that are strong enough to call. If you play every hand, and you see every river in that 100into100 situation, you shouldn't call with 50% of your hands because your hand range is too wide for that to be profitable.
So you can't make a ton of mistakes say "MDF" and call off, you have to have done the right things in previous streets to end up with a range that can call at MDF. That range (and those street actions) require an understanding of GTO (and the adjustments needed when someone isn't playing GTO).
You're writing something that is at best accidentally misleading and at worst confidentially incorrect.
What is your formal definition of "optimal strategy"? A Nash equilibrium is considered optimal in the sense that it's a state where no one can gain an advantage by deviating from the equilibrium.
Sure, if your opponents don't play the Nash equilibrium, there is room to exploit that deviation and potentially gain more than what you would get from playing the Nash equilibrium. However, you also make yourself exploitable in return, so I don't think you're presenting the whole picture here.
Poker players typically optimize for making the most money in expectation per hand. Either way, I'm certain that the exploits I've described for players that bluff too much or call too much are correct so I'm not too worried about being slightly off. Poker strategy is about heuristics.
For practical poker, I'd formally define "optimal strategy" as the strategy that maximizes profit per time (or per game) for a set of opponents, including also the actions needed to "explore" and discover any bias before exploiting it.
Assuming at least one of opponents is not playing Nash equilibrium (which is a very solid assumption), playing the Nash equilibrium becomes suboptimal as it doesn't exploit the exploitable as much.
In the narrow range of poker variants (all heads up, ie only two players, not full ring like all but the last few hands of the Main Event) where it's meaningful to talk about a truly optimal game, any theoretical optimal play will still take money from all humans it's just slower (but with zero risk) compared to exploitative play.
In live cash games, speed matters, you want to take all the available chips before the fish realise they're out-matched, but to protect yourself the optimal play, if you could memorise it, would be safer because it can't be exploited yet it does take the opponent's chips.
Poker players are gamblers. So "safer" isn't really what they were going for anyway.
Trying to understand what you are getting at made me realize why I do not gamble I am just not smart enough or lucky enough. I had a friend try and convince me he knew a sure fire way to beat roulette but in the end the house always wins. He eventually had to admit he had a gambling addiction and quit doing it all together.
Like anything else, Poker is a skill that you can learn with time and practice.
It is not really related to your smartness or luck (doesn't apply to _everyone_ of course but I'd wager that the average HN reader is already smart enough for poker)
> When they call too much you should obviously not bluff! This leads to very boring games of poker.
I don't know, poker theory is all about optimal ranges and Nash equilibrium, but there's something satisfying (and very practically important, since if all your opponents even understand the phrase Nash equilibrium you should find a different game) about trying to make the most money against an opponent who calls or bluffs way too much.
Modern Poker Theory by Acevedo was the premier book on this, but I've been out of the game a few years. Idk if I'd fully trust his charts given modern sizing theory, but it's going to improve your game if you understand the concepts. If you're really serious, you want to get a solver: GTO+, PIOSolver, or GTOWizard (online version).
Chess manages in practice to be a game of imperfect information, like poker. Obviously it is explicitly a game of perfect information on the board, but the hidden information is all psychological. For instance: "do they actually know this opening/endgame? do they see a tactic? did they take a long time because they calculated that it's a good move, or are they bluffing by making it seem like they calculated something? are they actually better than me or just acting like they are?". etc.
It's true that "bluffing has minimal value against best play", but no human is in that situation. Even super-GMs will play "bluffs" if they are behind (or playing a lower-rated player and sure they can recover later. or just for fun if the stakes are low).
And that's not even mentioning optimal strategy under time pressure. For instance some of the Lichess tournaments are structured such that winning fast is more valuable than winning slowly because the resulting score comes from how many wins you can get in (or in other cases, how big of a streak you can get). So people will play in a way that optimizes for winning quickly by taking big bets / bluffing / creating chaos with un-calculated gambits, especially if they have a good reason to believe they're better than their opponents.
But every serious chess game is a matter of time allocation. The best players in the world are unable to calculate as much as they want in every position: Ultimately there is bluffing and risk taking. See the last game of the last candidates, where both Fabi and Ian have to win to get into a playoff, and they get themselves into extremely complicated positions, where accurate play just takes too long for a human. At an 8 hour time limit, the game is very different than at the time the players actually had, as ultimately Fabi just couldn't calculate to the end on every position he knew was key.
It wasn't the most accurate game of the tournament, but the most instructive as far as the psychology of chess goes
That being said, when Super GMs play a bad move against a lower rated GM, they quite often gets a pass. They don't capitalize, simply because they assume the move is excellent.
It's interesting how this affects play. If a player is bluffing slightly less than they should be, the adjustment is drastic, you should actually never call with hands that do not beat their value hands. If they are bluffing optimally, you are supposed to call using what is known as "minimum defense frequency". What's interesting about this is the minimum defense frequency is based on what the strongest hands you can possibly have in that situation are and the opponents possible hands do not even factor into it. It's required to prevent your opponent from bluffing with any two cards profitably.
To do the math, if you are on the river and the opponent bets 100 into 100, for this to be profitable for them they need to win 50% of the time or more. If your opponent is bluffing optimally, you need to call with 50% of the strongest hands you have in that specific situation (if you don't know what hands you have in a specific situation that's a problem) and sometimes they can be dogshit like King high.
But, very important to note, very few players actually bluff enough and if they bluff less than they should you should only ever call when your hand actually beats their range of possible value hands. (value vs bluff is kind of a difficult thing to communicate, generally it's value if you want your opponent to call)
Most players don't bluff enough as a result of most players calling too much! When they call too much you should obviously not bluff! This leads to very boring games of poker.