^^ what Bob said.

So imagine we take the line BB with all our value and drawing hands for the sake of argument. Every time we have anything and every time our draw hits on the river we will bet (BBB); every time we miss we will check (BBX). We are hoping to make good hands on the river to enable us to bet the river, but when we do not we give up. So all our likely showdown-winning hands are going into BBB, whereas all our crap ends up with the river check in BBX. This would, for example, conceivably be an exploitative strategy against a very loose passive opponent. The overall strategy is +EV, but when we end up in the BBX line that particular hand will result in a smaller stack than we started with; that “leaf” of the tree is -EV, *when things end up there*. Do you get what I mean? It’s just like open-folding in the SB results in a lead which is -0.5bbs, but to find the overall EV of our strategy we need to find the EV of each leaf, multiply it by its % frequency, and then add up all the results for the total strat EV.

Maybe it’s important to distinguish the difference between plus EV plays *which are still of a negative expectation* and those which are actively positive. IE plays that remain a negative number in terms of their outcome vs starting stack, but are +EV vs taking another line. For example, if 72o did better than -0.5bbs we would open it from the SB, but against most opponents it is likely to do worse. Therefore, folding for -0.5bbs is a +EV play (in contrast to other actions) even though it is of a negative expectation. There is an ambiguity when people write “+EV” between “better for the size of your stack compared to other options” and “resulting in a larger stack than you started with”; really the formerly does not always entail the latter. Choosing our actions all the way to the river SB decision node will be +EV in terms of the former definition, as will checking our crap, even though *when we do check* the leaf might result in, on average, a smaller stack than we started with.

The hands most vulnerable to such -ev outcomes(if considering the hand as a whole) are those that derive most of their ev from fold equity on the flop and turn. When called on the flop and turn, these hands perform quite poorly if considering late street ev as a function of the entire hand.

Pure bluffcatchers on the river are also in this category of -ev outcomes. When you call a pure bluffcatcher on the river vs gto, you're essentially just getting your call back; the call is 0ev in isolation relative to folding, but the hand as a whole goes in the loss column.

There are also preflop raising hands, which are profitable open raises, yet these hands are -ev when called. The postflop decisions with such hands do not recoup the preflop investment, yet these hands are a part of correct preflop ranges thanks to preflop fold equity. Take away the preflop fold equity and these hands cease to be profitable open raises, as happens vs loose players.

I understand how you can:

1. Have -EV lines that are still the best response to an opponent

2. How you can have an -EV node in an overall +EV path

What I was specifically confused about because it made no sense was how we could check the river with equity and have that specific node be -EV since you will always have the equity from the pot making that node positive. Well assuming we see showdown with no additional betting which I was assuming we were.

I can see what you're saying with the bet/bet/x line, but in that instance your river node in isolation isn't -EV, your summation of the path to that node is -EV, no? It's just that the positive equity you have at the terminal node can't compensate for the investment you made so your "overall line" in that instance is negative, not specifically the river node.

I suppose if the EV of each node is just the summation of EV to that path, instead of the EV of the decision point at that node then the EV of your river node would be -EV.

I see now when you use the term "overall line" you are specifically talking about the path that leads up to a group of terminal nodes and then summing the final EV over all of those nodes. So I believe we are in agreement overall and I was specifically having some semantic issues.

This depends on what you call + or - EV. Sure you have some chance to win still if you check, but if you are likely to lose then your overall stack size will, in this event, be smaller than at the start of the hand, so I call that -EV.