Flop Donkbet OOP
Say you call from the BB against a button raise. The flop is scary and drawless, for example A72 rainbow. Your opponent is a TAG.
Because of the dangerous board, the weak spot of the opponent is far in front. Basically, a donkbet on the flop will often cause him to fold if he didn't hit. Furthermore, a donkbet costs us half as much as a check-raise. Thus, we can get the best ratio of cost to fold equity with a simple donkbet in this situation. We should often bet with nothing. If the opponent calls we play check-fold on the turn unimproved. The opposition must fold in only 18.2% of cases to make this bluff +EV.
Now assume the same situation with A3o, that is, top pair no kicker. If we play check-raise here, we will almost always cause the opposition to fold a worse hand, yielding little value. We don't really want to bring him to a fold, since freecards are hardly a danger to us.
A standard play here would be check/call flop, check/call turn, check/call river or bet river if he checks the turn. In this way, however, the opponent will be able to easily fold on the river in the higher limits. This manner of play is only advisable against opponents who are loose-aggressive or have such a small range that we can seriously believe to be ahead with our hand.
So what is the alternative? Donkbet flop! This will balance our bluffs and we'll get a lot of value on average. If the opponent does not fold directly to the donkbet, we'll usually have him on the angel. If he calls our bet, we'll check/raise on the turn. Thus, against a bluff call we'll get a further BB and against a made hand between 1 and 3 extra BB. Against better hands we'll lose 3 BB, but usually not more.
If the opposition raises our flop donkbet we should switch to calldown mode. If he actually has a better hand, we'll save money. If he's bluffing, he'll bet the turn with a high probability after his flop raise. If he has a marginal made hand, he'll probably check the turn, but call a river bet from us. This then is the optimal play for us.
We could even go so far as to say that we either donk the flop or check/fold on drawless, scary boards. With strong hands we play donk flop, c/r turn, with average hands donk flop, c/c turn and with hopeless hands donk flop, c/f turn. This manner of play sounds plausible, however I have not yet tested it.