Fwiw, I've tried to confirm some of my theories with little success. So for example, here's a GTO nash equilibrium to RI Poker which would seem to contradict alot of my post.

(Quick breakdown of RI Hold'em:
- 3 streets of poker counting preflop
- dealt a single hold card
- 3kind is better than striaghts which is better than flushes

PREFLOP:
SB raises with 2 BB calls.

FLOP: 9
SB bets. BB calls.

TURN: Q
SB Bets.

So I *know* this is a GTO strategy for the SB. It seems to contradict a couple of my points.


POINT 1: Your bluffs should never lose money, and your preflop range should be depolarized.

I'm confident that you should never lose money with any strategy as this is very basic game theory, so this must mean that in RI Hold'em it's +EV to raise the worse hand in your range pre flop with a raise.

POINT 2: You shouldn't bluff the bottom of your range.
As you can see in the above hand, the GTO strategy requires the Villain to bluff the bottom of his range.

I think this must mean either one of two things. You need to bluff with the bottom of your range some non zero % of the time, so you can have the nuts on the next street. Or I'm wrong.

This is not a strategy. This is one possible play that is part of a GTO strategy.

About polarisiation I have remarked before that your raising range should probably be polarised - meaning nothing else but consisting of a separate value and bluff range - if there is a passive option in between. Your range should look like Value raise - call - bluff raise - fold. I'm sure that a GTO strategy has a limping option. The fact that SB raises the nut low just tells me that there is no hand that he always folds preflop.

The fact that your bluffs shouldn't lose money isn't affected by this hand.

Maybe we have a simple logic problem here. The claim was that you should bluff the bottom of your range, and you said that this isn't generally true. Here we have a case where in fact you should bluff the bottom of your range; this instance doesn't make the general claim true.

Another thing to keep in mind is that a deuce in RIH is probaby a lot better than 72o in Texas Holdem. It's just as likely to flop a pair or flush draw as any other card, and flops straight draws just as well as a king does.

However, as long as you don't prove your claim you may well be wrong.

(I assume that trips > straight > flush > pair > high card)

yeah, I agree that it's not a strategy. But it is one of the correct plays -- not just possible -- part of the GTO strategy.

I agree with all this. I did like your point in a previous post about limping.


yeah, I've spent some time playing the solved game before, but it seems to me that all sorts of weird lines end up being part of GTO strategy. But I appreciate you taking the time to logically break down the points.

yeah, with straight flush being at the top.

Here's a link if you want to play it:
http://www.cs.cmu.edu/~gilpin/gsi.html

And if you want to feel dumb, type in Andrew Gilpin and game shrink into google, and you can read his papers.

It doesn't really matter where you place trips since they can never face a straight or a flush

well played.

I don't want to feel dumb, but I downloaded his thesis. Thanks for the pointer.

Can someone please explain this to me? So are ratio of value bet to bluffs should be 2.5/1 when we bet 2/3. So if we bet pot are ratio of value bets should be 1:1?

Also the last sentence, we should have AT MOST 15 VALUE hands (where is this number coming from?), which means you should have 6 hands (again have no idea what this means). Thank you![/QUOTE]

Let's say there are $30 in the pot. We bet $20. So our opponent has to call $20 to win $50. If he has a hand that does well against our value range he will always at least call. If he has a hand that doesn't beat our bluffs he will always fold. So it only gets interesting when he holds a bluff catcher, i.e., a hand that beats all our bluffs, but none of our value hands. If our value to bluff ratio is greater than 2.5:1 he can simply always fold. If it is smaller than 2.5:1 he should always call. Our result will be best when the ratio is exactly 2.5:1.

If we bet pot, i.e., $30, our opponent needs to call $30 to win $60, so he gets 2:1, so that should be our value to bluff ratio. In general, the bigger the bet is the more bluffs we should have.

Great Post. Although I am starting to come to a PERSONAL (not saying everyone should agree), that a pure Nash strategy is correct for Hold 'Em, and that a Prudent Decision Strategy may be more applicable.

Thx. Kinda curious what a Prudent Decusion Strategy would entail.

Thank you, I enjoy read this!

For multistreet bluffing, the main guideline is that on the flop, you have more bluffs than valuebets. On the turn, you have about equal for each, and on the river, you should value bet more than bluffing. The thinking is that the earlier you are, the more chances you have to improve and some of your flop bluffs can turn into turn bluffs. And you have multiple chances to get your opponent to fold.

What you describe is a guideline for a perfectly polarized range (either 100% equity or 0% equity) where the better makes 3/4 bet size on each street. Cangurino was asking about a hold'em game where equity changes over multiple streets.

I've never seen an equation which can accurately describe the ratio for each street and describe a nash equilibrium when all actions are available. Matthew Janda has described an equation for adjusting your flop ratio of value to bluff bets based on the equity of your value and bluff hands, but he makes an assumption which isn't quite accurate -- but he doesn't have a better option.


The logic behind the situation you describe doesn't have anything to do with your bluffs improving, since your bluffs have 0% equity. And I don't think I would characterize it as having multiple streets to get your opponent to fold either.

I think a better description is that you're trying to balance your value to bluff ratio on all streets so that the Villain is indifferent to calling. Since you have multiple streets to bet, you'll have multiple opportunities to drop some of your bluffs from your betting range. This means as you described the Hero has more bluffs than value hands on the flop, about equal on the turn, and more value than bluffs on the river.

Below is some quick math explaining it (you work backward from the river to the flop).

Hero bets 3/4 pot size bet on the river. Villain's getting 2.3 to 1 on the river. Therefore, you need to have a ratio of 70% value and 30% bluffs when you bet on the river.

So now on the turn, you want to make the Villain indifferent to calling on the turn, so this means that you want to have the right amount of value to bluffs on the river. When you make a 3/4 size bet on the turn, the Villain is going to be getting 2.3 to 1 on a call on the turn. This means that you need to follow through with a bet on the river 70% of the time, and check-fold the river the rest.

This same logic will apply to the flop. So we know that when we bet the river, we're going to have 70% value and 30% bluffs. When we bet the turn, we know that we're going to continue to bet the river 70% of the time. So on the turn, the % of our betting range which is value will be 70% * 70% = 49% So as Dawd described roughly 50% value on the turn.

On the flop the ratio of value to bluffs will be: 70% * 70% * 70% = 34% value.

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Also what I think DDawd was trying to describe about dynamic equity ranges is that since you have more bluffs than value hands on the flop that more of your bluffs will improve to value hands than value hands will not be able to bet all 3 streets. Therefore, you can have even MORE bluffs on the flop than when your range is perfectly polarized. (This assumes that the equity that your bluffs have is the same equity that your value hands don't have -- ie. your bluffs have 20% equity and your value hands have 80% equity. Also there's not really a good calculation for determining this exactly, and instead you need to make the assumption that when your value hand has 80% equity -- what we're saying is that it can bet all 3 streets for value 80% of the time -- which isn't exactly how the equity would work out).

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The Rhode Island Hold'em link that I posted to Cangurino not only perfectly describes value to bluffs over multiple streets -- but it also includes all actions by both opponents -- ie, that both players can raise and check-raise etc. and has Nash Equilibrium. It's a solved hold'em game. What DDAWD and I describe here is just a model to better understand how poker works.