Not sure if should be in theory forum or probability forum:

Working on a Nash equilibrium problem. Let's say you raise preflop and get one caller who has position on you. You flop what I will stipulate for the sake of this example are nine clean outs to the best hand. The SPR is 12.5 on the flop before you c-bet. So you can 3 barrel pot, pot, pot and be all in on the river giving villain 2:1 on a call on the river. Game theory would suggest that given this line, you will need to be value betting the river >= 30% of the time [(2/3) x (2/3) x (2/3) = 0.3]

Here is my questions:

1) Since you will hit your draw by the river 35% of the time (9 outs twice), and this is > 30%, then can't you actually 3 barrel bluff 100% of the time?

2) If the answer to #1 is yes, then what are the implications of the turn where if you miss your draw on the turn, your chances of hitting by the river drop to 20% but when you pot the turn, you should be able to value bet the river pot 44% of the time [(2/3 x (2/3) = .44]? Now it seems the turn to river logic is conflicting with the flop to river logic.

Hopefully this makes sense. Thanks!!

I'm not sure what the "(2/3)*(2/3)*(2/3) = .3" represents, but if you are making a pot-size all-in on the river, GTO would be value-betting 22% of your range.

Ok, from a theoretical point of view opponent is going to call 2/3 of the times each time we bet the pot.
Also he should be indifferent to calling or folding with a bluff catcher on every street, so to calculate the ev of the play we may imagine he has a bluff catcher and he calls us down.
So, if he calls us down, 70% of the times we lose 12.5 "flop pot units", 30% of the times we gain 13.5 units. This is by far a -ev play.
Even if the play calculated this way was even money, we should consider that the cards falling down on the turn and on the river will help villain making a correct call or fold if our range is biased towards draws, so that he will not just blindly call down both when we close our draw and we don't.

So
1) no

Moreover the 2/3*2/3*2/3=times you need to be value betting doesn't make any sense to me... how does theory suggest that?