I've heard Phil Galfond in several videos say, when his hand is very weak on the river, that according to game theory he should be betting. What is the rationale behind that assertion?

A strong hand would bet for value and a medium strength may win by checking or calling a bluff, but the very weak hands can only win by betting or raising.

(Because of card removal effects, it may actually not be theoretically correct to always bluff with the very bottom of our range. Say the board is 23KQ8 and we have 54. Villain c-bet the flop and we called. Turn went check-check. It may not be correct to bluff with our nut low 100% of the time, because the opponent may be able to exploit it by calling with 84 or 85 while folding other 8x.)

If you have one of your absolutely worst hands at the river and you're not thinking about bluffing, then which hands are you bluffing? Perhaps you aren't bluffing at all, which, according to game theory, cannot be right.

There's more to it (number of combos, pot-odds, card removal, whether or not you are trying to exploit your opponent), but whenever you are at the river with the very bottom of your range, you should at least be thinking about bluffing.

As Amy said, you can only win with air by betting or raising.

Whether or not to actually execute the bluff depends on a few things. We must gauge our opponent's range, as well as his propensity to call. We only complete one part of the puzzle by determining our opponent's hand strength.

If he's a station and will call down with any pair, then we may have to reconsider bluffing.

If he's a scared nit, then we can bluff frequently.

Another thing to consider is how well we can represent a strong hand OTR. Bet sizing, along with the line we've taken are both quite important for bluffing.

Conceptually, if your equity in the hand is absolute, the GTO decision can be figured out by taking what data is affected by it and taking the absolute limit of it.

As hand equity approaches 0%(impossible to know with absolute certainty as the nut low can always split the pot with an identical hand. Given the only way to have the nut low is to have an unpaired hand at showdown, its always possible for the same unpaired hand of different suits to exist, but regardless...) your equity bluffing with it approaches infinity, therefore, bluffing has to beb the GTO play.

I like to do it w card removal instead, but sometimes that isn't possible and I have to do it, but then any hand that can hardly beat a bluff is a good hand to bluff with, not just the bottom of my range.

this is pretty much the right answer. but to follow up on it, that assertion basically comes from looking at solutions to simplified river situations where checking has at least a good chance of getting you to showdown. there's plenty of other spots where it's best to give up with air and bluff (or semi-bluff if its before the river) with stronger holdings.

The worse the hand you bluff with the better the effect when caught in meta game value to get paid later big time. The more of a joke they think you are and they remember it longer.

Also you are best served in real life, that people are nowhere near game theory optimal balancing, to not bluff vs calling station idiots that cannot understand risk and will call with garbage vs potential top 5 nuts in the board lol. Exploit them with value and never pay them with nothing.

And also best served when playing vs thinking opponents to bluff with hands that involve blockers of nuts or with whatever you have that is worthless but exploits the board consistently with how you have bet so far representing a ton of good hands that the opponent based on preflop action cannot typically have and be forced to fold, even dropping sets, 2p etc (and where due to preflop action/position cant have the nuts on these nasty boards all that often eg flush or straights). Basically bluff not when game theory wants it vs real life weaker players or in tournaments that the same spot cant be replicated all that often if at all, so dont go for the game theory proper mix (only in very advanced deep cash games vs expert players), the hell with it, bluff instead when it makes sense as part of the story of action so far to exploit the statistical difficulty the opponents find themselves at when their range cant be nut range connected all that often and they are careful thinking conservative players that can recognize the way you played your range fits more like the bad thing they dont want to see.

Always absolutely always bluff with hands that a check or fold have worse EV of course. I mean cases that you check and the opponent will bet or check back and you lose super often to always are examples that you must always consider if a bluff however expensive is the best option. Dont think about how expensive it looks, think about the EV of the bold move. If you think their range is folding 60% of the time why not bet pot/2 with nothing even in a big pot if your hand has no chance in hell otherwise?

Also if you have bluffed them before successfully very often lately, even if they dont know it (not having shown anything), its better to cut it down for a while because you have started to tilt them and they will call you light, try to pull some hero call stunts of their own and that will boost their confidence nicely when they get you, which you do not want. Its better to have the goods after repeated successful bluffs so that they will be eager to call you and become even more tilted that you have it!

One last thing. When you bluff it must always be a happy moment, a decision that you back 100% as profitable and therefore no different than value betting with a great hand. It doesnt matter if the potential opponent call is devastating in cost, all that matters is that the decision itself was a great choice. If you focus on the choice being great and not the risk it comes with vs the value bet alternative, you will look unreadable to them.

Always remember that even in a value bet from a super strong position you are in serious risk. What risk? The risk of not getting paid and losing value (or of getting reraised out of the blue). Its always about value so either you feel both at value bets and at bluffs equally happy because they are both good choices to begin with or equally anxious because in both cases you risk losing something. See is as a difference from the base line. In both cases a good decision will be plus EV always so you must go into it either confident both times or anxious both times. So choose which it is. To be happy on value bets and unahappy when you bluff is ridiculous. You must be happy always and therefore unreadable. Do not focus on the prospect of receiving chips vs not. Focus on the difference between this decision and the alternatives. This makes the risk irrelevant and the only material emotional state is that of confirmation of a good decision leaving you absolutely happy after both kinds of bets.

If you can convince yourself you are always at a happy state of mind nobody will be able to read you because what they will be reading is permanent conviction. To confuse others make sure you either see it the right way yourself through cold logic or if you cant rise to it, try to confuse yourself as well and act delusionally happy if you have to. Its better than giving off all kinds of signals because the brain focuses wrongly on the receiving chips vs not result instead of the decision part.

so you're suggesting he should call with bad 8s when you're less likely to have the nut low, but call with good 8s when you're more likely to have the nut low. i'm not quite sure you have this right.

Yeah, sorry, it should be the other way around; us holding a 4 or 5 makes it less likely the opponent is bluffing. I must have been tired when I wrote that. Anyway, the point was that card removal interferes with "pure" game theory bluffing regions.

I thought about this a bit more today and got pretty confused, anyone able to help me out with logic fail?

Given that the 2 reasons for betting is as a bluff or for value, as our value increases, we should bet more frequently. However as our equity decreases we should also bet more frequently.(simplified: as our equity gets further away from 50%, betting is more likely the best play)

What if we can range villain to a single hand, which is the absolute nuts, then we can assume his calling range is 100%, and bluffing has infinite negative value

As equity decreases past 50%, bluffing value increases, 0% equity = infinite value

How can something have both infinite value, and infinitely negative value?

Betting with the nut low when you range a villain to the nuts is exactly neutral EV?

This also makes semi-bluffing extremely confusing, if semi-bluffing is bluffing with equity, it means we think we behind but not behind by much, so our equity is closer to 50% and we should be less willing to bet and more willing to call/fold.

I agree with AmyIsNo1, yaqh and stinkypete.

You have to bluff with some hands, or you will not get value from your best hands.

If you bet, you give up the value you have from winning the pot after two checks. The weaker your hand, the less that value is, so the less you give up by bluffing.

So if you bluff with your medium-bad hands instead of your worst hands, you throw away value needlessly.

Of course, real poker situations are more complicated, but the general principle still holds.

masque de Z is describing a popular use of the word "bluff" which is not the game theory sense. If you are betting because you think it is +EV, because you think that other player is likely enough to fold (that his, his chance of folding is greater than B / (P + B) where B is the amount of the bet and P is the pot before your bet) then it is a bet for value (for fold equity), not a game theory bluff. In pure game theory, you don't care if your bluff gets called or not. You make equal money if the other player calls all your river bets or none of them. Of course, if he can read you and calls your bluffs but not your strong hands, then you're in trouble. But that's outside of game theory.

The assertion as it is without caveats, he's incorrect. If he meant that's the spots where you should bluff but that doesn't mean you should do it all the time, he's correct.

It's easier if you put numbers on it. We care about the probability the other player will call (c) and win (w). Assume there is P in the pot and you are considering betting B. Ignore raises to keep it simple.

If you check, your equity is (1 - w)*P

If you bet, your equity is (1 - c)*P + c*(1 - w)*(P + B) - c*w*B

The difference, bet minus check, is w*P*(1 - c) + c*B*(1 - 2*w)

If the other player has the nuts and knows it, then c = w = 1 and the difference is -B, which is always negative, so don't bet.

If the other player is sure to lose and knows it, then c = w = 0 and the difference is 0, so it doesn't matter what you do.

In between, increasing w makes the difference more positive due to the first term, but more negative due to the second term. If P / (2*B + P) > c, then the weaker your hand, the more likely you are to bet for value.

But none of this matters for game theory bluffing.

Doesn't this depend on your definition of the "game" in question? If you define poker by its rules alone, then game theory doesn't take into account any "villain specific" characteristics. In this case, GTO is Nash.

However if its possible to quantify any given reaction(i.e. statistics on how often a villain does something) then it should also be possible to find the "true" GTO of any given scenario by recognising that the player specific data we have is usable in a GTO equation merely by widening the area of the system(or definition of the game) studied to include them.


This is only based on my understanding of game theory, which admittedly is a very brief overview and my own logical conclusions.

This is great stuff, especially bolded. You should post more strat and stop dicking around in SMP.

What you say makes logical sense, but is not the way the terms are generally used.

There are two general theories for decision-making under uncertainty. In probability theory, we treat the uncertainty as something random. In game theory we treat the uncertainty as the behavior of a rational actor.

Say the betting suggests that you are working on a flush draw and the other player has a strong made hand that's only likely to be beaten by a flush. A third suited card appears on the river, so it appears you have made your flush.

As it happens, you have nothing. You want to know if it makes sense to bet and hope the other player will fold. One approach would be to ask how often he folds to a bet in this situation in the past. This is treating him like a random number generator. The game theory approach is to assume he knows how often you have the flush, and how often you bluff without it. Then ask what bluffing frequency gives you the largest EV. In simple poker situations, that often turns out to be the frequency that equalizes your profit if he always calls your river bets in that situation or always folds to them.

In real poker, both statistical tendencies and game theory considerations are useful. Generally, the better the player, the less help you can get from statistical observation, and the more it pays to rely on game theory.

starcraft semesa?

In cash game if you bluff $X then your bluff has -$X value.

If you have 0% equity, then your bluff has a value modeled by an integral over some appropriate probability mass function.

Define Ω as the sample space/hand range with |Q| as its cardinality.

Define f as the probability mass function that maps an element of the range, ω in Ω, to some 0 ≤ (1-p(ω))q(ω) ≤ 1, where p(ω) is defined as the probability that villain will call with holding ω and q(ω) is the probability that villain is actually holding ω. Constraints are \sum_{i=1}^{|Q|} (q(ω_i)) = 1. So we define f: Ω --> [0,1].

Then the value with 0% equity is the lebesgue integral over Ω that integrates (P+X)*f(ω)*dP - X.

$X is, again, defined as the bet size. $P is the pot size before you bluff.

The solution is bounded in the interval over the reals [0,P].

It'll be easier this way. Consider vectors I and O that have the number of elements equal to the number of ω in Ω. Let each i_ω ∈ I be the chance of folding holding ω, and each o_ω ∈ O be the chance that ω ∈ Ω is the actual holding. Then the E[V] is (P+X)I'O - X.

This could all be wrong (the lebesgue integral, not the matrix algebra). I'm incredibly tired.

I have some ideas but they're too early to post.

This advice can be a little misleading. Game theory only suggests bluffing with the very bottom of your range in certain situations. Specifically situations where we have the option of checking back. If we are facing a bet, for instance, we're better off just bluffing with the strongest of the hands we must fold instead of the weakest, since we must fold them anyway and can no longer realize the little equity that they have.

If this is the case then Villain is playing in an exploitable way (his range is not balanced at all) and we can exploit this accordingly. (by c/f unless we also have the absolute nuts)

Even if we aren't making any exploitative adjustments, our strategy as a whole will still win against Villains overall strategy because he won't end up with the absolute nuts every hand, and as already pointed out, his strategy as a whole is likely flawed.

How is it that you are giving up any value from winning the pot from bluffing? Don't you keep your showdown value from checking and just add value from your opponent folding to your bluff? Or is your statement based on the fact that we could fold if rebluffed?

Very interesting. Am I correct is concluding that most poker books and discussions center around probability theory as opposed to game theory? If so are there any good books that treat poker from a game theory POV?

It makes some sense but there are too many other things to think about that maybe one should all forget about it and anyway the hand you are going on the river with is usually either like nothing or you think it will be no good anymore once you have got there and in that case it's pretty strong actually but one should be thinking about bluffing maybe even more than in any other case, even when the draw misses as the opponent isn't exactly blind even if he is a fish s he sees no danger while if you have got all the way to the last card, it's more likely that the opponent has something. Maybe it's a -EV bet to think about the bluff with the lower end of the range as it simply doesn't happen in situations that are the best for bluffs asnd even if it sometimes is, the pot is rather nothing and the main rule is generally to just let it go until known otherwise but then too it's about the cards out there and about the opponent and it just might be better to bluff even with a something hand also, and a something hand does often have cards that are good to be off the opponent's hand.

You're right of course. The "bottom of the range" rule of thumb really means to bluff raise with hands you would otherwise fold, rather than hands you would otherwise call. If you are facing a bet, raise with your strongest hands, call with the hands not good enough to raise, bluff raise with the hands just too weak to call, fold the rest.

However, in real poker, there are too many other considerations in every hand to stick to a strict rule.

Yes, it is based on the assumption that you will fold if your bluff is raised.

That was true years ago when most poker books were written by bridge players. However most poker books today have at least some game theory background, although the authors may not understand it. But you can find books on 2+2 that cover the game theory in any level of rigor that you like.

Personally, I don't think knowing game theory helps with poker. You need to understand some concepts that can be demonstrated with game theory, but that's like saying a baseball outfielder has to understand physics to catch a fly ball. As the saying goes, if poker were math, mathematicians would all be rich.

Game theory can also blind you to some important aspects of poker, because it makes very restrictive assumptions. I think the most interesting work, and the most useful for poker, is done by people who do experiments in game theory rather than pure mathematicians.

In general, wouldn't it be better to bluff raise the turn instead of the river? My reasoning is you may have better FE on the turn instead of getting into a situation on the river where you are unclear where you are at and end up getting called or reraised only by hands that beat you...