Challenging a Fundamental Principle of Game Theory
Posted by Nick Howard
Posted by
Nick Howard
posted in
High Stakes
Challenging a Fundamental Principle of Game Theory
In honor of Easter Sunday, I thought why not make the theory gods roll over in their empty graves!
Here's are two excerpts from posts made by a member who was trying to devise closer-to-optimal 3-bet ranges:
MOP defines a GTO strategy pair as two strategies who are maximally exploiting each other.
One of the main tenets of game theory is that you always take the line which is highest EV against your opponent. You never really take a lower EV line for the sake of balance.
Rules are usually made to be broken, but this is one i've always just assumed was made of steel. I think we're all probably in agreement here more or less, but i'm challenging us to prove it using two recent examples that have me a bit thrown. By sheer coincidence, i came across both of them while cruising Tyler Forrester's videos. (you're off the hook tyler!)
Ex 1: Tyler 4bets and calls off AKo CO vs BB for about 115bb effective. In video he says the EV of calling the 5bet is about neutral. When a member asks why he 4bet if it's 0EV to call 5bets, his main reason was for range expansion -- adding AK to the call 5-bet range incentivizes BB to 5-bet more aggressively, which in turn raises the EV of CO's KK-AA as 4-bets. I'm assuming tyler took this line b/c he also thinks its the most +EV play for AK (anticipating villain will play a call 4-bet range), but it got me thinking: if we increased stack depth to the point where raising 3-bets with AK became slightly -EV relative to calling, could it be possible that the existance of AK in the 4-bet range still raised the EV of AA-KK enough to justify taking the lower EV line?
Ex 2: Tyler does a CREV analysis MP v BTN coldcall, 872 flop. He concludes that it's sound for MP to check his entire range OTF. Sauce comes into the thread and agrees, saying that he thinks MP should devise his range on this texture in a way that allows him to realize as much equity with overcards as possible. This seems like a very solid argument. At the same time, with ranges being relatively symmetrical on this flop, I think i could make strong arguments that if MP starts by checking his whole range, BTN would be correct to respond passively enough that we would see MP regaining incentive to c-bet some premium hands, since they don't appreciate BTN taking a street out of the game at a high frequency. I suppose that's another thread in itself, and i could be wrong, but i've played with the ranges enough that i think i can give the counter-argument at least a good run for its money.
Cliffs:
If we assume that a particular hand is -EV in line 1 relative to line 2, does that automatically mean that line 1 is a sub-optimal strategy?
Is it possible that hands can perform ritual sacrifice to serve the EV of the range on the whole? Sorry for being dramatic but i think it's fun to treat hands as people.
happy egg hunting guys!
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I think that it's tough define to decide what the EV of a hand actually is in the following sense:
Let's say you have KJ on 9TQ3r in position. Is the EV of raising a turn bet higher than the EV of calling one? How do you know?
One the one hand it's hard to argue against raising a bet when you hold the best possible hand, and if this were the river then calling would be dominated by raising, but on the other hand if you always raise the turn bet with KJ then you never have the nuts on the river unless the board pairs. With reasonable stack depth, this would seem to suggest splitting your range between turn raises and turn calls because you're actually indifferent to doing either versus some kind of optimal opponent. It could be that the highest EV play is to call some and raise some.
I think the paradox in your thinking is coming by treating AKo, for example, as AKo, and not as 12 AKo's.
Your thoughts seem to actually be arguing a lot of the equilibrium in poker being heavily mixed.
Btw:
"Sorry for being dramatic but i think it's fun to treat hands as people"
Just sayin'...Good point, but i'm functioning under the assumption that neglecting to mix hands that should be mixed will only show up as -EV vs a nemesis opponent, not an optimal opponent.
So under that assumption , if a hand is -EV in a certain line vs an optimal villain, it already indicates that it does not belong to a mixed strategy
I don't really understand how what you're saying is all that much different than saying "We must bluff so that our opponent is incentivized to call our river bet and this is made up for by the times that we have value"?
I mean I see some differences but it just seems like you're re-iterating the concept of range and if you find that you are taking -EV lines with certain to "make up" for your other hands since the -EV line incentivizes Villain to change his strategy then it would seem that by definition we are pretty far from the equilibrium. The equilibrium is the point where nobody is incentivized to do anything, I think what you're talking about here is just how we end up finding it.
I'm not sure i understand you. I'm assuming villain is playing optimally. And i'm assuming that we're aware certain hands in our line our slightly -EV, but wondering if it could ever be possible that the EV that it gives to other parts of our range could still make this the best strategy for us on the whole. Ex 2 is probably the better one to look at b/c the wider we allow villain to bet when checked to OTF, the less equity we realize with overcards, and overcards make up a ton of our range.
I think the mistake in this challenge against the EV Maximization Principle is that you're implicitly assuming the AKQ game, where equities are symmetrical. If you look at the example of 872r and some other ones which similarly fit, I think you're talking about situations where ranges are asymmetrical, and when this is the case I think that you could be taking the highest EV line with all of the hands in your range and your statement would still be true.
It can't be proved with your examples, since poker isn't solved. But it has been proven definitively before in terms of game theory, so I'm not sure what the point would be in trying to do it again here.
Instead, I think the only point would be to be more careful with the way in which we explain things so as not to obfuscate the point.
There's really no point in trying to make the BB 5-bet more aggressively or to raise the EV of some portion of the Villain's range. Let's assume that the Button is supposed to have a 4-bet range andd the BB is supposed to have a 5-bet range. If the BU doesn't 4-bet enough, then the BB will in turn stop 5-betting enough... in which case, the BU then will have an incentive to 4-bet more. This continues until an equilibrium is reached. You can sometimes look at the frequencies of each player and see if something seems off. In other words, maybe the range that we though was correct for the BU, meant he was only 4-bettling like 5% which seems a bit low. Things like that even though they're a bit arbitrary.
When thinking about how to play a single hand like AK here, I think it helps to imagine how different parts of the Villain's range reacts when we 4-bet or call. I break it down into hands that we dominate, hands that dominate us, hands that call which we beat (not dominated), hands that call that beat us (not dominated), and then the same with different portions that fold. This is more or less the different ways that we realize our equity as EV, but it's easier than trying to do an insane EV calculation in our head. In other words, if we get a lot of worse hands that we dominate to 5-bet or call our 4-bet, then this is a big plus. If we get lots of hands that we dominate to fold, than this is a negative. Maybe we'll get hands with decent equity against us to fold even though we beat them, which isn't really a bad thing either. I tend to kinda weigh all these factors and make my decision from there. But the main point is that you're expressing your reasons in terms of why the hands plays best in terms of EV, instead of expressing your reasons as having primarily to do with your opponent being incentivized to play a balanced range.
"It can't be proved with your examples, since poker isn't solved. But it has been proven definitively before in terms of game theory, so I'm not sure what the point would be in trying to do it again here.
Instead, I think the only point would be to be more careful with the way in which we explain things so as not to obfuscate the point."
Totally agree with that!
I need preflop help. I get what you're saying about reaching equilibriums through 4betting/5betting frequencies ... but when i think about it terms of "4betting less incentivizes BTN to 5bet less", initially it just resonates to me as exploitative in the sense that we would exploitatively 5-bet bluff villains who 4-bet too many bluffs, and vice versa for villains who don't 4-bet bluff enough.
I guess i'm wondering if what you're saying also directly applies to finding an equilibrium in the value hands that we put in the 4bet range, and if there's no room to budge on a hand like AK if we found that it is -EV vs an optimal villain, relative to flatting a 3-bet (even if taking the slightly -EV 4-bet line had the effect of increasing the EV of KK-AA as 4bets, relative to them flatting 3-bets). Or is that just an impossible event?
Here was another quote from tyler that seems relevant here:
Also, the term EV is being thrown around quite loosely in a lot of these discussions, but I know that Matthew Janda used to harp on the idea that you absolutely cannot convert your equity into expected value. You can probably decide if it's positive or negative though?
never seen the kid get so mad tbh. miss that guy.
yeah, I'm kinda with you. It seems a bit strange for MP to check his entire range. This would only make sense if his range was much weaker than the Buttons. I think your critique seems good to me.
On the other hand, on these type of boards, the in position player has a lot of incentive to bet wide because it puts the oop player in a touch spot, but I think you would really need to have a sense of what each players ranges were from preflop
One of the most important reasons of why we are C betting is too weaken our range. If you as a preflop raiser, check your hole range on a dry flop your range becomes too strong and the preflop caller should only bet some nuts and air when checked to.
However, my opinion is that it is okay to check 100 percent of your range when being oop on flops that smashes your opponents´s range much harder than yours.
I have not had time to watch his CREV video yet by the way.
If a requirement of an optimal strategy by definition is that it cannot contain any -EV hand values, i want to rephrase the question:
why cant the highest EV strategy for a player include any -EV hand values?
this way we cant just state "bc it defies the definition of optimal!"
It can be correct to take a - EV line with a hand when playing GTO poker depending on what your perception of EV is.
Think about it in this way instead: When playing GTO poker you should never take a line that will make your stack smaller after the hand has ended, than if you have had taken another line instead.
Ugh, you actually got me to rethink ... and do some calculations - with unexpected results. :)
Take the following scenario: we´re on the river, we have 8 combos, 6x nuts, 2x air, Villain has a bluffcatcher. We bet 0.48x pot (say we have $48 left in a pot of $100 and we´re forced to shove or check), which means, Villain needs ~24.5% equity. Now we have two options, we can either bluff too much (when we bet both air-combos, we have a bluff% of 25%) - and offer Villain the right odds to call 100%, or we can bluff too little (1 combo = 14% bluff-ratio), forcing Villain to fold 100%.
I purposely chose numbers that are very close, still our nemesis opponent will either call 100% or fold 100% - because it´s his range of optimal choices.
When we bluff too much and Villain calls 100%, our bluffs obviously are -EV. Still, our overall EV is higher for our entire range:
EV (bluff too much) = (6/8 * 1.48) + (2/8 * -0.48) = 0.99
EV (bluff too little) = (6/8 * 1) + (1/8 * 0) = 0.7
Does that actually mean, we would be right in choosing the -EV-route for our bluffs - in the given scenario - to support our valuerange? Or did I make a mistake in my calculation?!
If I didn´t make a mistake, then it seems as if our mistake from bluffing too much is smaller (for our overall range) than our mistake from bluffing too little, we should actually make the "smaller mistake", even though it means, we´re playing single hands for -EV.
I am sorry but I do not understand. How can we bluff too much when our range consists of 75 percent of value combos in your example? Our range is too value heavy and we should shove with our entire range always.
We should always take the line that makes our stack biggest after the hand has ended. If I am in the BB and button opens, I will defend with all hands that has an EV of less than -1 bb:s post flop. Same with your example, when playing GTO you should take the line that maximizes your chip amount after the hand has ended. Which in this example makes it GTO correct to shove.
BigFiszh, your calculations seem correct, but I don't think the conclusion is very useful. You're basically comparing two unbalanced strategies (bluff too much or too little) while assuming that villain exploits the imbalance maximally, and conclude that overbluffing is less exploitable than not bluffing at all. This is not surprising since the optimal bluffing frequency in this situation is obviously very close to all 2 combos.
Neither of those strategies are optimal, though. When bluffing enough to make villain indifferent (1.946 combos or 97.3% of the times we have a bluffing hand) the EV of our range is 0.99325 > 0.99.
EDIT: Seems like GT wrote the same comment already.
I am not a GTO expert but I believe this:
If we are using the "4 bet-fold AK" example: If I as the villain, play GTO and use a "sometimes call 4 bets oop strategy", then my "calling 4 bet " range will be created so that opponent not will have a plus EV C bet opportunity post flop.
I would for example sometimes call the 4 bet with AA. I would also, with a polarized range lead flops that hits my range really well.
This at the same time as my very polarized "5 bet range" will be indifferent of getting called.
Can you give an example of where "the highest EV strategy for a player includes a -EV hand value?"
There are no real contractions here. Only contradictions in your head since you are knee deep in semantics.
For instance Ex 1, if you want to show a contradiction, assume that 4-bet calling AKo is +EV at 100bb, 0EV at 115bb and -EV at 150bb. Now come up with GTO preflop strategies for both players (feel free to make them up, as long as they are consistent with your proposed example/contradiction) and show me a real contradiction.
Yes!
Because then that player could unilaterally improve his EV by removing the -EV part of that strategy and replace it by a 0EV (folding) or +EV strategy and unlaterally improve his overall EV, contradicting that he was playing GTO.
"Because then that player could unilaterally improve his EV by removing
the -EV part of that strategy and replace it by a 0EV (folding) or +EV
strategy and unlaterally improve his overall EV, contradicting that he
was playing GTO."
What about my example? The "correct" line seems to shove - although the bluffs are -EV. Obviously I took some restrictions, but the principle should still hold - regardless of the model, right?
Ok, but this is still ignoring that that the existence of -EV hands in the range seem to be increasing the EV of other hands in the range (Ex 2 is the clearest demonstration of this). The question is not whether any hand in isolation has incentive to change for it's own benefit, it's whether or not the hand has incentive to change for the benefit of the EV of the strategy on the whole. The challenge is to prove that the last sentence is a mutually exclusive event without simply pointing to a rule. Why does the rule exist?
The way you're expressing it is still using the definition of optimal support that adding a -EV hand value to a range would be detrimental to the EV of the strategy on the whole. This is sort of like saying "the sky can't be green, because it's blue". What's the credibility of blue in a vaccuum? What are the qualities that make it so in relation to the full spectrum?
In the non-GTO example of bigfiszh where he was bluffing with 2 combos instead of 1.95, he caused his opponent to always call, while being -EV on his bluffs and increasing his overall EV. I don't see the distinction between these examples, apart from the fact that bigfiszh was willing to put up exact numbers and you still haven't. The quoted situation can happen, but it is not GTO, and does not improve the overall EV. Hence it does not contradict anything.
This is very ill-defined, and you're likely mixing up GTO and non-GTO in ways that you don't understand. The 'Fundamental Principle' that you are talking about is a property of a GTO strategy pair. It is assumed that your opponent also plays a GTO strategy, when you are talking about "incentive to change for the benefit of the EV of the strategy on the whole" you are assuming that your opponent plays a perfect counterstrategy to your strategy, so you start with the assumption that neither of you is playing GTO, then you change your own strategy to a new strategy (GTO or non-GTO, you tell me) and then you derive a contradiction from assuming that everyone was playing GTO the whole time?
This is very easy, we assume that we play a GTO strategy pair (N, G), where N is your strategy and G is my strategy. My strategy G is fixed, and doesn't change no matter what you try, I will always realize my game value V.
Now you want to change your strategy by way of going from the optimal action X with hand range H to the non-optimal action Y. This deviation will make me a positive amount of W when it occurs. So if P(Y) is the chance that you take this action against my strategy, my game value has just gone up from V to V+P(Y)*W > V.
This is why the rule exists.
It is a mathematical theorem that a blue sky exists. And a blue sky can't be green by definition. The problem is that you are confusing blue and green skies.
I enjoyed this thoroughly. top notch counter.
It seems that there is no good argument for the OP at this point, since the basis of the argument, by definition of containing any -EV hands in the proposed strategy, already goes against the theorem that optimal strategies do exist, and they don't contain any -EV hand values.
Do you agree with applying this conclusion for Ex 2:
"If we did find that some premium hands were -EV in a strategy where we checked our entire range OTF (vs an optimal villain), it automatically means that the net gain from the EV of overcards in this range (relative to an optimal strategy) will not offset the loss of EV from the premiums?"
Brave that you give up so quickly.
I'm not sure what this even means, it sounds too vague and non-GTO. If you were to write it down more clearly it would probably be similar to bigfiszh bet 48 in 100 situation, since checking your entire range is unlikely to be optimal.
Starting with a scenario where both players are playing a balanced game where they maximize the expectation of each individual hand against the counter strategy
Your question is, can one of the players improve the expectation of his total strategy by taking a lower EV line with one of his hands?
The idea is that player 2 would be forced to adjust to the new strategy of player 1, and that the resulting outcome would be more favorable to player 1 than the original game
However, the reason that this can't be true is because player 2 is not forced to make adjustments! If he keeps playing the same strategy as before your total expectation has decreased. (Because you gave in some EV with one hand and the EV of all the other hands are the same). The only reason player 2 would adjust his strategy, is if it increases his overall expectation. In that case your overall expectation must be lower than it was before.
So, no loss leaders in GTO.
However, from an exploitative view your point sometimes makes sense. In practize it might not always be best to aim for the highest EV line with each hand, because giving up some EV with some hands can get your oponnent to make wrong adjustments (Or keeps him from fixing bad strategys)
To give one example, consider an opponent who is folding way too often to 3bets. The max exploit line might be to 3bet any 2. However it quickly becomes obvious that you are exploiting him and he will fix his mistakes. If you 3bet somewhat higher than the theoretical optimum but dont overdo it, thus giving up some direct EV with the hands you do not 3bet, he might not realize that you are exploiting him and therefore he keeps playing bad, thus making your longterm EV of your strategy higher
The last sentence goes against the OP's assumption. The assumption is that we give EV away with some hands to raise the EV of other hands in the range. The goal is to prove whether or not the net effect of this tradeoff can ever yield an EV equal to or higher than what is considered to be the only valid construction for an "optimal" strategy.
Under the definition "all hands must be in their most +EV line for optimal to be reached", we ignore that overcards in Ex 2 are actually realizing a higher EV than they could in any other line, by benefiting from the potentially -EV play of the strong hands that are checking to protect them. If overcards are higher EV in this strategy than they are in the "optimal" strategy, then that already proves that the definition we're using for optimal is inconsistent.
Nick, imagine that villain continues to play exactly the same strategy without any adjustment to the fact that our checking range is stronger. We arrive on the flop with a marginal hand once again. How has our EV in this very hand increased? Surely it's still the same, since we're facing the same actions from the same ranges at the same frequencies.
At the core of it, checking strong hands isn't (directly) about other parts of our range. The reason we want to check them is because it's a good line for getting value since villain is putting in money with too wide of a range after our check (probably correctly, to exploit our weak checking range). The protection comes as kind of a side effect since villain has to throttle his betting to avoid being exploited by our slowplays.
Again, checking has to increase the EV of our strong hands to make it a viable play and to make villain adjust his ranges. Otherwise the situation would play out like Schulti described: villain plays exactly like before and the EV of every hand stays the same, except that we lose EV with some of our strong hands. (Unless villain can now find an even better strategy).
Our opponent calls 48 to make a pot of 196, 24.49% or 12/49. So for a breakeven call for him we need to have 12 bluffs for every 37 value combo. Since we have 6 value combos, we must bluff 12/37*6= 1.9459 combos as a bluff. This implies a bluffing frequency of 12/37*6/2 = 0.9730. Our opponent now calls with a frequency to make our bluffs indifferent, we risk 48 to win 100, so he calls with a frequncy of 100/(100+48) = 25/37 = 0.6757.
Our overall EV becomes (6+ 12/37*6)/8 = 147/148 = 0.9932
Since our opponent is indifferent between calling and folding, for the calculation of our GTO strategy we can assume that he always folds, so we take down the pot whenever we bet.
Which is of course larger than the non-optimal values of 0.99 and 0.7.
Recall that nobody is forced to play a pure strategy. If you want to force everyone to play a pure strategy, there is no longer a guarantee that a nash equilibrium exists. For instance in rock-paper-scissors none of the pure strategies are optimal.
"Recall that nobody is forced to play a pure strategy."
OK, this was the missing point in my thought process, instead of bluffing either 1 or 2 combos (100% each), we could take a strategy where we bluff each combo only a fraction of all times to get to the optimal frequency. Thanks!
It is fascinating that a solution of a problem sometimes can be right there in front of your nose without you understanding it because it is too simple and obvious. Thank´s for helping out GameTheory!
Most of the EV in the AKo hand Tyler was talking about comes from the parts of the game tree where villain folds or calls.
In the 872r flop spot, Tyler and I just agree that checking is a way we won't get slaughtered on the flop, neither of us think checking with our range is optimal.
Thanks Ben. I think there's a lot of merit in not having to bother with mixed strategies OTF using the check 100% strategy. I know that's a cop-out but its real.
Nick: I don't think you know what a "mixed" strategy means. It does NOT mean that you have a checking and a betting range. It means that you play a SINGLE HAND by both betting and checking (or really any two different actions) because both actions have the EXACT SAME EV -- and it's balances correctly by playing the same hand in two different ways.
I understand. I'm saying the EV we sacrifice by not having to play a mixed strategy with certain overcard bluffs if we do install a c-betting range OTF is a small price to play for the practicality of being able to check our whole range. it saves a lot of energy.
you can have a betting range plus a checking range and NEVER have a mixed strategy.
what makes you think that having a betting range here is hard? why can't you just bet most of your strong hands, check-raise your really strong ones, and check-call your medium strength ones.
I'm concerned with balancing overcard x-c's and c-bets at good frequencies. It's a headache that's alleviated somewhat if we eliminate the c-bet range. We'll still likely have to play a mixed strategy with overcard x-c's but it's not as complicated adding another line.
I'd be really surprised if no c-bet range was optimal on the vast majority of flops, but it's easier to control your ranges and i think that's really valuable.
Also, as tyler has pointed out, what he might give up in EV by taking sub-optimal unorthodox lines, he gains back by being able to maneuver better than his opponents ott/otr in those lines.
But why are you concerned about balancing overcard check-calls and c-bets at good frequencies?
I think this logic only makes sense where the EV of both lines is very close which you can make up for by not playing your ranges poorly later. I don't think there's really a reason to think that the EV of the two lines for all hands would be close here.
I don't understand what you're saying. What do you mean by "maneuver" better? This would only make sense if Tyler, independent of his opponent, would make more costly mistakes by trying to play a GTO strategy.
if i play a c-bet range, i'm going to need to bluff some overcards. I'm also going to need to x/c some and x/f some. If i don't balance the c-bet range effectively, it's not worth using one.
basically that Tyler will exploit well in lines where he's more familiar with the distribution of what the ranges actually look like than villain is. Tyler will have a better idea of villain's range more often than villain does of Tyler's
It seems to me that you'll substitute one problem for another because now you have to balance a check-raising range.
By the way, for all of the talk about whether or not to c-bet out of position, nobody is talking about the maximal exploitive strategy against it. Does anybody have any ideas?
What should our betting range be in position? The first obvious iteration would be "bet all of the hands we would call or raise with and balanced with our bluffs" but this doesn't seem all that great for multiple reasons. The next obvious one would be "just pretend we are the pfr and we have raised the range we called with in position and OOP has called our raise with the range he pfr's with from that position". This seems to hold more promise.
I'd like to reiterate what Ben said. I'm not endorsing checking all hands on 872 as GTO. It's just a way to not get destroyed.
GTO is the most misused term in poker right now. We don't have any GTO strategies!!! The University of Alberta no-limit bot is beat by a minimum of 110bb/100 by a GTO opponent. If you think you have a GTO strategy, then all I do is change the bet sizing and the baby gets thrown out with the bath water.
In those hands I played a strategy that maintains an average profit (not the highest possible average profit!) against a range of unknown strategies. Since it is an average profit using a mixed strategy, there is always possibility of defecting to higher ev strategy if we knew how are opponent played exactly (always bluff, never bluff strategy).
Wait, what? It loses almost 1.5x more than the foldeveryhandpre strat?
Yep, its actually worse. UofA Post (Midway down) They are saying 3347bb/1000.
Tyler,
I thought i did a careful job of not misquoting you. Sorry if it came off that way. Thanks for posting.
I think he says it loses at least 3347 milli bb/hand to the best response strategy, so thats ~300 bb/100. Thats a whole lot but its versus the max exploit, not versus the GTO. Impossible to tell how much it loses versus the GTO strat but it must be way less than this number. It wouldnt suprise me if most humans would lose even more if they get maximally exploited. Max exploit strats do really exploit you hard, especially if your strategy is fixed. They will bluff nothing or everything everytime your defending ranges are slightly off, use differrrent betsizings for valuehands/bluffs etc
@Nick No worries, this is an interesting thread. I was clarifying my thoughts (though I probably muddled them instead).
Schulti GTO is supposed to breakeven against max exploit. The difference between no limit and limit bots is stunning. Polaris (limit bot) loses roughly 1bb/100 to a max exploit bot compared to at least 300bb/100 for the 2011 no limit bot.
@Tyler no way that a somewhat decent strategy can be exploited for 300bb/100, that's 3bb/hand! Just folding whenever it has the chance would have it lose at most 0.75bb/hand.
I believe that it is GTO to check 100 percent of your range when being oop on a couple of flop textures that smashes opponent´s calling pre range much better than ours. What do you guys believe?
Yup, you have a very valid point there.
Usually when you're way behind checking with range is a good strategy. If you have a bad hand, you can xf it, and if you have a good hand then your opponent is going to bet a lot. And if you have a medium hand you're screwed either way but arguably less screwed when you check :)
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