I don't use mixed strategies in multiway pots. I think that the addition of the third player results in pure strategies all around in real poker.

This is for a few reasons:

slowplaying goes way down in value, thus there's no need to mix with strong hands.

bluffing goes way down in value, thus only the strongest semibluffs available in the ranges involved will be 100% bet. The rest of the potential semibluffs available will be checked 100% of the time.

That's just my loose theory though. I may be wrong.

And you may be right! Thanks for your input.

If this one hand has ~ 33% equity on the flop then you are going to arrive at the river with 1/3 of a hand with 100% equity and 2/3 of a hand with 0% equity So you have 1 value hand for every 2 bluffs.

(This assumes 76s only has value when the flush hits)

Say you want to bet pot on the river. You would then need 2 value hands for every bluff to keep villain indifferent to calling with his bluff catcher. But you have 1 value hand for every 2 bluffs. So you need to randomize your 76s whenever it misses and only bluff with it 1/4 of a time.

I think you can work backwards to figure out bluffing frequencies on the turn and flop....but yes you can "bluff" with 76s.

Too tired to check the math of the equation, but basically the idea u wrote seems correct.

This obv only applies to multistreet games. But the basic idea is that lets say when u have a valuehand OTT, but it isn't always guaranteed river valuebet, we treat it not as a pure 2 street value hand but a fractional value.

Like it's 0.8 combos of value, which means 80% of the time it is still a valuebet OTR, but 20% it's not.

Same idea applies to bluffs with equity. A bluffhand OTT with 10% equity to turn into valuehand can be considered 0.1 valuecombos.


Obviously this is pretty simplified, but the idea is still good to understand and before solvers this was how you calculated ranges. Pretty sure this was covered in either Janda's first book or Will Tiptons books. Haven't done the math on this in years but IIRC it's pretty simple adjustment from multistreet polar vs bluffcatching range.





PS. Not sure you realized this or not, but OTR it doesn't matter if ur valuebet has 60% eq vs calldown range or 100%, it's still considered a full valuecombo, because when comparing the 60% and 100% hands to the part of villains range that is supposed to be breaking even, they both have 100%.


PPS. Decided to check some of my old notes from Janda's book and its covered in there.

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Basic example would be.

We are IP and can bet 100%PSB on every street. OTR we wanna have an 33% bluffing frequency. Let's say that OTF our bluffs have an average eq of 10% to turn into hands that can valuebet OTR, and our valuehands have 80% to stay as a hands that can valuebet OTR. To decide how much value we have to have OTF->

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If the toygame was normal polar vs bluff we would calculate it just simply: we bet POT otf so we need to bet turn 66% time, and river 66% and have the 66% valuehands OTR, so our flop value% would have to be 0.66*0.66*0.66 = 0.29

But when we deal with hands that can turn into valuehands or non valuehands we solve it by

0.8*X + 0.1*X = 0.29 --- > X = 0.322

So with the non perfectly polar range we can bluff OTF with freq of 1-X= 0.678

This. Though in your example Janda is multiplying the bluff equity by (1-X) which results in having a higher bluff frequency when compared to the normal polar vs bluff. Tipton also looks at the case where hero’s bluffs lose equity when called and shows that villain needs to fold at a higher frequency to maintain indifference.