Poker Video: No Limit Hold'Em by shuttle (Micro/Small Stakes)

Would it be possible to use game theory to come up with a set of optimal frequencies and bet sizes given preflop ranges and assuming good opponents that you should be check-raising the flop, 2 barreling, 3 barreling, c-betting, calling multiple streets, etc... over multiple streets? It seems like that would solve at least a generic optimal tree. Then you could "chop down the tree" by seeing where villains seem to deviate. Their deviation from the optimal assuming large sample sizes would be a proxy for evaluating the relative EV of different strategies. Obviously this would not be perfect as there are so many variables besides generic frequencies. But it seems like it might still be useful. Not sure if this makes sense.

You had me at hello.

Great episode.

As far as the "quiz hand" goes, I had my own thought process which I would love to get some feedback from everyone.

Based on PF bet of 3x bb from the BTN (villain calls) and then a bet of 5bb into a 6bb pot on the flop (Villain calls) and then the turn gives us the nut flush draw. We are in position, what is the best move?

Hero's hand:
A 5

board is:
10 8 6 K

now the pot is 16bb with the villain to act. The probability that the villain will check to us again is fairly high. The question becomes... what is the right play here and what gives us the highest EV?

If we decide that making a flush AND top pair makes us the best hand on the river, we have 12 outs. We are assuming that he has top pair or middle pair. Either way it plays the same against our hand. Giving us about 25% equity in the pot. Given that we fire again on the turn, making a bet equal to the pot of 16bb...

I would think that we have a 50/50 shot at getting the villain to fold because the K is not a good card for him as we were the raiser and he could of been floating the flop as well.

EV (fold) = .5 * 16bb = 8bb

If he calls our 16bb bet on the turn...

EV (call) = .5( .25 * 32bb) This is assuming that no more bets occur. - (.75 * 16bb)

= .5 (8 - 12)
= .5 * -4
= -2

EV = fold equity 8bb + call equity (-2)

EV= 6bb


So even if there is no more betting we still have a positive EV by betting the turn. However this does not take into consideration of IMPLIED ODDS. If we hit any of our 12 outs especially the 9 to a flush... we can expect to get paid a good amount on the river because the villain has a nice stack size and he has put so much into the pot already.

Take the EV that we came up with + The implied odds we have with our flush draw...

Making a pot size bet on the turn is the best move.



I would love feedback on this as I am still learning and somewhat of a noob. Thank you.

Great episode.

As far as the "quiz hand" goes, I had my own thought process which I would love to get some feedback from everyone.

Based on PF bet of 3x bb from the BTN (villain calls) and then a bet of 5bb into a 6bb pot on the flop (Villain calls) and then the turn gives us the nut flush draw. We are in position, what is the best move?

Hero's hand:
A 5

board is:
10 8 6 K

now the pot is 16bb with the villain to act. The probability that the villain will check to us again is fairly high. The question becomes... what is the right play here and what gives us the highest EV?

If we decide that making a flush AND top pair makes us the best hand on the river, we have 12 outs. We are assuming that he has top pair or middle pair. Either way it plays the same against our hand. Giving us about 25% equity in the pot. Given that we fire again on the turn, making a bet equal to the pot of 16bb...

I would think that we have a 50/50 shot at getting the villain to fold because the K is not a good card for him as we were the raiser and he could of been floating the flop as well.

EV (fold) = .5 * 16bb = 8bb

If he calls our 16bb bet on the turn...

EV (call) = .5( .25 * 32bb) This is assuming that no more bets occur. - (.75 * 16bb)

= .5 (8 - 12)
= .5 * -4
= -2

EV = fold equity 8bb + call equity (-2)

EV= 6bb


So even if there is no more betting we still have a positive EV by betting the turn. However this does not take into consideration of IMPLIED ODDS. If we hit any of our 12 outs especially the 9 to a flush... we can expect to get paid a good amount on the river because the villain has a nice stack size and he has put so much into the pot already.

Take the EV that we came up with + The implied odds we have with our flush draw...

Making a pot size bet on the turn is the best move.



I would love feedback on this as I am still learning and somewhat of a noob. Thank you.

What if he check-raises? How is this added into the equation? Do you call?

Would it be possible to use game theory to come up with a set of optimal frequencies and bet sizes given preflop ranges and assuming good opponents that you should be check-raising the flop, 2 barreling, 3 barreling, c-betting, calling multiple streets, etc... over multiple streets? It seems like that would solve at least a generic optimal tree. Then you could "chop down the tree" by seeing where villains seem to deviate. Their deviation from the optimal assuming large sample sizes would be a proxy for evaluating the relative EV of different strategies. Obviously this would not be perfect as there are so many variables besides generic frequencies. But it seems like it might still be useful. Not sure if this makes sense.

Magical post, i like it. Makes sense, and it's something I've thought a bit about before.

The big problem is that even when looking at "good opponents", there is a ton of deviation in people's ranges postflop, particularly in later streets, even if we're just looking at people who are 24/20 tags or whatever.

This is a problem because it makes it really hard to make a generic optimal tree, because what is game theoretically optimal is a function of people's ranges.

For example, if you're playing someone who only 4bets AA and KK, you're not going to have a balanced 5bet shoving range, because you can't make them indifferent between folding and calling.

In similar spirit, if we want to make a generic sort of tree, it would seem it has to be made to cater to quite a specific class of opponents, not a general one. It would be good to try and make a few trees that try to play optimally against a few opponents. It would illustrate how different attributes of an opponent influences how to play GTO against them.

Then we could make the maximally exploitative trees for those same opponents. This would also help get a good understanding of how other people's deviations from game theoretical optimal stuff should influence our decisions when we want to exploit them.

Hopefully we will get into this stuff in later episodes.

For example, if you're playing someone who only 4bets AA and KK, you're not going to have a balanced 5bet shoving range, because you can't make them indifferent between folding and calling.

It seems like if someone only 4bets AA and KK, it would be easy to balance your range vs that opponent. But the hands and frequencies you bluff and value 3-bet with would be different than vs another opponent who, for example, has a 4 bet bluffing range (ie. you would have a 3b/5b bluffing range vs someone w/ a 4b bluff range).

Great episode.

As far as the "quiz hand" goes, I had my own thought process which I would love to get some feedback from everyone.

Based on PF bet of 3x bb from the BTN (villain calls) and then a bet of 5bb into a 6bb pot on the flop (Villain calls) and then the turn gives us the nut flush draw. We are in position, what is the best move?

Hero's hand:
A 5

board is:
10 8 6 K

now the pot is 16bb with the villain to act. The probability that the villain will check to us again is fairly high. The question becomes... what is the right play here and what gives us the highest EV?

If we decide that making a flush AND top pair makes us the best hand on the river, we have 12 outs. We are assuming that he has top pair or middle pair. Either way it plays the same against our hand. Giving us about 25% equity in the pot. Given that we fire again on the turn, making a bet equal to the pot of 16bb...

I would think that we have a 50/50 shot at getting the villain to fold because the K is not a good card for him as we were the raiser and he could of been floating the flop as well.

EV (fold) = .5 * 16bb = 8bb

If he calls our 16bb bet on the turn...

EV (call) = .5( .25 * 32bb) This is assuming that no more bets occur. - (.75 * 16bb)

= .5 (8 - 12)
= .5 * -4
= -2

EV = fold equity 8bb + call equity (-2)

EV= 6bb


So even if there is no more betting we still have a positive EV by betting the turn. However this does not take into consideration of IMPLIED ODDS. If we hit any of our 12 outs especially the 9 to a flush... we can expect to get paid a good amount on the river because the villain has a nice stack size and he has put so much into the pot already.

Take the EV that we came up with + The implied odds we have with our flush draw...

Making a pot size bet on the turn is the best move.



I would love feedback on this as I am still learning and somewhat of a noob. Thank you.

Something that I think is interesting is assuming your opponent never folds and see what bet size gives us an break/even for EV(call).

EV(call) = (.25 * ( 16 + X) ) - .75(x)
0 = 4 + .25x -.75x
.5x = 4
x = 8

A 1/2 pot size bet is neutral EV, if our opponent never folds and never check/raises.

Hi Guys,

I want to try and use the information in these videos, and apply it to LIMIT Holdem. Are there any adjustments I need to make to the formulas, or other things I should consider?

Thanks very much!

This is the first time I've come across someone seriously tackling how to compute EVs correctly in poker - a great job! This is a topic I have also been working on for some time.

I do have two comments:
1) Hero's actions in future streets should be taken as the maximum EV for that street and applied in the recursive formula - whilst for villain we always take the weighted average of their options at each point (as specified in the second half of the video). The authors suggest at one point taking a balanced line for hero (which is only done as a defensive mechanism against a very strong opponent) but I think it would be clearer explaining to the audience that we always take the max of hero's available options.
2) The problem can be solved in a reasonable amount of time...just not deterministically. If one runs multiple monte carlo simulations - simulating future cards, villain actions, bet sizes etc a good approximation can be achieved off a reasonable PC in short period of time.

Loved the video - I hope these authors produce more work on this topic!

in response to olliepa's post regarding game theory and analysing villain's actions post flop...

Correct - we require an opponent model in order to assess EVs correctly, and the model needs to be specified in terms of how each villain acts on each street based on their relative hand ranking (relative to the board - not the absolute post flop hand ranking). Here I'm referring to categories such as overcards, overpairs, Top pair top kicker, gut shot draws etc. For the opponnet model was also need to consider, the current street, who the prior street aggressor was, the texture of the board, the bet actions based on whether villain is IP/OOP, and the Stack Pot Ratio (to list just a few of the key statistics that influence a player's actions). Ignoring multi-way post flop action for now, then we can deduce how big the opponent model matrix needs to be:

1) If we have a handful of common player types (say 12),
2) 3 streets,
3) 22 (say) relative hand group bins,
4) 3 prior street aggressor categories(as the prior street aggressor, vs, no prior street aggressor),
5) 19 possible bet actions based on position,
6) 3 board textures (wet, neutral,dry) and
7) 3 (say) SPR categories...we quickly see that the real problem isn't the EV calculation but deducing an appropriate opponent model. To make matters worse even if we have access to billions of hands, only about 5% go to showdown and these are very biased. Hence we would need to unpack the biases to get a reasonable representation of plausible player actions.

Of course the opponent model can be approximated.

One interesting finding I've come across is that approximations aren't too bad, partly because the difference in EV's across different options is actually often pretty small. Effectively there are quite a few spots where raising, calling or folding have very similar EVs of around zero...or at least where the 90% confidence interval of the EV for hero's action overlap with other possible options.

Hi Guys,

I want to try and use the information in these videos, and apply it to LIMIT Holdem. Are there any adjustments I need to make to the formulas, or other things I should consider?

Thanks very much!

So, I take it that is a "NO?!?"

Some examples of an EV calculation WITH these wonderful recursive functions, would have been nice! I realize guys like Voices above, who are already separating hands into "group bins" don't need this kind of thorough explanation, :rolleyes, but c'mon, some of us might! Other than that, I'm going to have to offer a site transfer to someone who would like to help me with this stuff!

And, 12 is more than a "handful," imo!

So, I take it that is a "NO?!?"

Some examples of an EV calculation WITH these wonderful recursive functions, would have been nice! I realize guys like Voices above, who are already separating hands into "group bins" don't need this kind of thorough explanation, :rolleyes, but c'mon, some of us might! Other than that, I'm going to have to offer a site transfer to someone who would like to help me with this stuff!

And, 12 is more than a "handful," imo!

With regards to your question regarding the applicability of these ideas to Limit, I see no reason why they wouldn't apply. I only play no-limit, but as Limit is a special case of no-limit (fixed bet-sizing) any general theory for no-limit should transfer easily to limit. However, the opposite is usually untrue. Take for example calling and relying on implied odds (e.g. calling a preflop raiser on the button with 22o and set-mining), whilst 3-betting may be a better line, calling in this spot against a deep stacked opponent who is aggressive is usually +EV in no-limit ... but may be -EV in limit. This idea explains why starting hands in no limit depend on the type of opponents you're up against as well as the size of the effective stacks.

Your request for examples is only fair...the problem is they are hard to produce. I haven't come across any software that does a decent job of doing so (StoxEV and PokerRazor are the best I've come across, but the EV trees grow too quickly to be practicable imo). Generating results by hand will drive anyone insane.

I have been working on developing software to find the "truth" in poker when analysing hands after a session, I'm happy to post an example output as a word document - unfortunately I'm not sure how this is done in this forum (I've just joined DC). My work has been focused on no-limit because I wanted to create a tool to check whether I acted correctly at differing spots, not based on villain's actual hand presented at showdown, but on a reasonable hand range pre-flop and how their hand range changed based on the board and their betting actions at each street.

Apologies about some of the statistical references in my post earlier...I was just excited to come across someone working on the same ideas that I have been thinking about for some time. 12 player types is actually not that many, the groups I use vary by preflop VPIP (tight, neutral or loose), Preflop aggression (aggressive or passive) and post flop aggression (again, aggressive or passive). My analysis has actually shown that players can be aggressive or passive (relative to the average) at different streets and depending on whether there's a prior street aggressor. Think of a player who is loose and passive at all streets until the river, and then shoves if a straight or flush scare card hits. Without realising that he's aggressive on the river its easy to place the player in the "calling station" category, whereas in truth, on the river he becomes a maniac...clearly we can call with a lot less against this player than a typical calling station where we would probably fold unless we have close to the nuts. As a result the 12 player types I use vary per post flop street (so there's actually 36 potential player types).

Some examples of an EV calculation WITH these wonderful recursive functions, would have been nice! I realize guys like Voices above, who are already separating hands into "group bins" don't need this kind of thorough explanation, :rolleyes, but c'mon, some of us might! Other than that, I'm going to have to offer a site transfer to someone who would like to help me with this stuff!

Perhaps that person could be me

Send me a PM and we'll talk on skype.

With regards to your question regarding the applicability of these ideas to Limit, I see no reason why they wouldn't apply.

The recursive EV formula does apply to all of these games. The only adjustment in the the form of the terms. In a limit game it will be simpler because we have known betsizes. Implied odds and pot odds are still the same but the individual hands will be different in different game types.

Your request for examples is only fair...the problem is they are hard to produce. I haven't come across any software that does a decent job of doing so (StoxEV and PokerRazor are the best I've come across, but the EV trees grow too quickly to be practicable imo). Generating results by hand will drive anyone insane.

It's particularly tedious, I've made some EV trees vs some opponents and it is very time consuming. There will be some of these appearing later on.