[W. Deranged]

All right... let's think about it this way.

Let's make the dodgy assumption that for some reason our average equity here is only 40%. Let's say that the check-raiser has equity equal to about 50%, and the player behind us 10% (with like a gutshot or something... I'm not so interest in a realistic distribution of hands but in the theory at play; and I like easy numbers).

Let's imagine that we raise and we force out the player behind us. Let's say our equity increases to 45% and check-raising villain's increases to 55% (this'll be about right if there aren't weird cross effects and yada yada yada...)

By folding out the villain behind us, we increase our equity by about 5% of the pot, or somewhere around half a big bet. Considering that we will get back on average about 90% of our additional bet anyway (we are only a 55-45 underdog), we notice that the bet which "costs" about .1 BB or so will generate a positive effect of about .5 BB. We benefit about .4 BB or so.

Let's consider the other option, namely villain calls two cold. We take 40% of all the extra money he puts into the pot, or .4 BB when he calls two cold (we're considering the effect of him calling that extra bet, not the first one). We are going to lose .2BB from our big bet to the field, but, again, we get more than that back from villain because he is calling as a significant underdog. We net like .2 BB or something.

This is a VERY simplified model, but you get the idea. Basically, EVEN if we are an underdog to the check-raiser here on average, because of the third player in the pot betting will generally be positive EV whether he calls (because he'll be calling as an underdog) or folds (because we gain equity). My point is that we don't have to be ahead for it to be right to raise.

True, if we honestly think our average equity is tiny here (like <30% or whatever) raising becomes foolish because the cost of our bet is large compared to the benefits of a raise. But we will be ahead more often than that in my opinion so raising is better.