This should be easy for those who did the other questions.

Since I am going to analogize the question into tournament play when facing someone who hates risking his sure min cash, it means that when the possible blind pusher has a big stack he isn't harmed much beyond his chip loss.

Thus I would like to change the question to one where the pusher is not raked even if he is called. Only the caller is. The previous two answers were about 21 % rake and about 17% rake if I remember correctly. So the winner of the pot has his profit raked either 42% or 34% But if the pusher isn't ever raked he can push even if the caller gets raked a smaller amount than the two previous solutions.

Precisely the new questions are:

One dollar in the pot provided by George Soros. A big stack can bet ten dollars putting the caller all in preflop. If the caller calls and wins he is raked big so he doesn't call much. But the pusher is not raked if he wins.

At what rake does every hand that the pusher could have show a profit, (given that the caller sees that he is pushing blind)?

At what rake does the pusher show a profit if he pushes without even looking at his hand and the caller sees that?

The pusher shows profit with every hand once the caller starts folding A7o, which happens at 19,07% rake. At any rake above 19,07%, 27o is a +EV push.


EV of 27o for the pusher if A7o calls:


EV of 27o for the pusher if A7o folds:


EV of A7o for calling a push at 19,07% rake:


The caller's calling range at 19,08% rake (when he starts folding A7o):

He shows profit when the rake is above 10,58%. If the rake is under 10,58%, it is more profitable for him to just fold any two cards.

EV of blind pushing when J9o calls:


EV of blind pushing when J9o folds:


EV of calling J9o when the rake is 10,58%:


The caller's calling range when the rake is 10,57% (J9o still calls):



As you can notice there's a bit more difference from the original result when we're calculating the answer to your second question (21% --> 19,07% . . . 17% --> 10,58%). This happens because the pusher has much more equity on average in the second question and therefore benefits more from the lowered rake on his side.

Thanks again.