I just had what feels like a revelation on this article. The article states that
Auto-profit is an exploit which takes advantage of your opponent folding too often. Starting at a certain fold equity, you can bet and raise with every hand profitably.
The article states that a half pot bet is automatically profitable if the opponent folds more than 1 out of 3 hands. This is a totally misleading conclusion, and here is why.
Suppose the pot is 100$ and my opponent's range consists of two hands. Suppose I beat hand 1 and lose to hand 2. Now lets say I can either check or bet, and villain can call or fold, after which the pot is payed out to the winner.
If I check the hands pay out like this: half the time I receive 100$, the other half 0$. My EV is 50$.
If I bet half pot, lets say villain folds hand 1 and calls hand 2. Villain folds half the time, so my bet must be profitable right? The hands pay out like this: With hand 1 villain folds and I receive 100$. With hand 2 I lose my 50$ investment. My EV is 25$, which is 25$ lower than checking. Betting lost money.
The reason that there is no "auto-profit" in this spot is because the "auto-profit" profit calculation does not take my equity share of the pot into account. I will take the money in the pot from hands that I beat regardless of whether I check or bet, but I pay off the hands that beat me.