Olliepa and Shuttle talk about game theory and how it applies to games in general and then poker specifically.
Have you ever found yourself in a spot where you know your opponent's range, but you still don't know the best play? If so, then this series is for you! We will show you how to determine the optimal lines in tough situations AFTER you have already assigned your opponent's range.
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Of COURSE I would like to see the math that makes Gambit work!!
It's released under the GNU General Public License, so you should be able to browse the code that it uses to calculate the equilibrium points.
I don't get that . For him to be break even he have to: 1win:2losses. So the best strategy for us is to shove value twice as much as air ? We don't want him to be breakeven. We want him to loose money against us. Why is this the best strategy?
Betting doesn't strictly dominate checking, only weakly dominate here, even with a nut hand.
We are using different definitions. The EV of betting is always higher than the EV of checking given this opponents strategy and range. which is why we are saying that this strategy dominates the other one.
(Also, in these videos we are working under the implicit assumption that the opponent is not always folding 100%.)
I really like your series, it's very good to hear a more mathematical approach to poker.
Can you explain better this concept:
"If you are getting good odds the value of GTO strategies goes up."
My understanding of the issue is that its absolute value goes up, but non-GTO strategies absolute value also goes up. So although you have extra absolute value, GTO strategies don't become necessarily better than non-GTO strategies.
If you have time to answer a couple of questions that would be great.
You give this example:
To make us unexploitable we have to make the EV of his call 0
His EV for a fold = 0
His EV for a call = 200(Pr(Win) - 100*Pr(Lose)
Substitute 0 for the Ev of him calling
His EV for a call = 200*Pr(Win) – 100*Pr(Lose) = 0
Pr(Lose)*100 = Pr(Win)*200
If we estimate we have 65% equity against his range for our river value bet then in the above equation is it correct that we can multiply his Pr(Win) by .35 and his Pr(Lose) by .65 so the calc would be Pr(Lose)*100*.65 = Pr(Win)*200*.35 giving a value bet to bluff ratio of 1.35?
Also, could you explain some of the maths of whether we base those value/bluff numbers on the hands we actually value bet the river with or the hands that can value bet the river. So for example say we have played our hand to the river so that it is either a strong flush draw or a set, we actually have the nut-flush draw. Before the river we had 9 outs to the nuts so at this point we either have say KJd, KQd, ATd, AJd, AQd, or we have sets of say 22, 55, QQ, KK. There are 5 combos of flush draws and 12 combos of sets. The river is a blank. Do we bluff a percentage of the hands that can value bet this river or a percentage of the cards that fall on the river?
So, we decide if we're going to bet then we'll bet pot-size, do we bluff 9/2 = 4.5 river cards (we could have bet 9 cards for value with our actual hand) or do we bluff 12/2 = 6 combos (we could have bet 12 combos at this point for value so we'll bluff 6)? Not sure how we'd figure out the latter but just a theoretical question.
I'm particularly curious as to why one is right rather than the other. I imagine the former method is correct otherwise we'd hardly ever bluff but I'd love to know why it's right.
Thanks very much, D
If we pot it we give our opponent 2:1 which means they will be indifferent to a call if they have 33% equity vs our range. So if we have 66% equity and we pot the river our opponent will be indifferent to calling or folding as you found from your equation.
In this case our opponent will win 33% and lose 66% which means that we have a ratio of 2:1 value:bluffs. (see how this works?)
So if we are trying to make our opponent indifferent to calling or folding we just need to get the ratio right for whatever our betsize is. (important: the ratio is different for different betsizes, exercise: change the equation by using some different pot sizes and bet sizes and see what happens.)
The general approach to constructing a river range is to bet all of our value hands, say this is n combos, and just enough bluffs, say this is y combos, so that our opponent is indifferent to calling or folding.
But we don't necessarily need to bet all value hands, If you decided to bet half of your value hands, then you'd be betting b/2 combos, in this case you'd need to bet half of the bulff combos from above or y/2 combos to remain balanced.
The reason we bet all value hands if we are in position is that it maximizes our ev in the spot. This is because checking back value hands is a worse strategy than betting them. (If we are OOP we might wish to check value hands but this is for a different reason)
So to answer your example from before, lets say we want to unexploitably pot bet the river all in and we have z value combos, then we should have z/2 bluff combos. I pretty much always use the #combos when doing these sort of analyses.
Thanks a lot, Shuttle.
By your first paragraph do I understand correctly that if we determine our value range v his calling range to be 66% that we shouldn't outright bluff but should just value bet all our value hands (with a pot-size bet anyway) as then we'll get the 66%/33% or 2/1 balance that we need?
On the other hand if our value range (i.e. those hands which each have over 50% equity vs his range) has, as a total range, 75% equity vs his range, then we have the choice of either betting half-pot to be balanced or betting full pot with all our value hands + bluffing enough from our bluff range to make up the 2:1 ratio, e.g. we get to the river with 8 value hands (>50% equity), with which range we judge we have 75% equity against his range. We also get to the river with 6 non-value (<50% equity) hands. We should bet all 8 value hands giving Villain 25%/75% win/lose (we judged that range had 75% vs his calling range), or 3 to 1 so that's 6 true value hands and 2 "bluff" hands from our value range so we need to add 1 more true bluff combo from our non-value range (perhaps a flush draw that missed) to bring us up to 2:1.
I'm not sure how to go about doing a range v range calc without Pokerstove but perhaps I'm not thinking about this correctly. If the above is correct I have a lot of work to do before being able to think about this stuff at the table!
For the purposes of calculating the balance ratio we don't need stove at all.
Here's the crucial point:
For our bet to be unexploitable all of our opponents options must have the same EV. Seeing as folding has 0EV then all of our opponents options must have 0ev. You just need to solve the EV equation for your opponent such that they have 0ev on a call.
The ratio of your bluffs/value hands are from the range you bet. You aren't quite getting this right, if we have 66% equity vs opponents calling range if we stop bluffing our opponent can exploit us by just folding every time. So we still need to bet some air in order for our opponent to call us.