I was doing some math behind the EV of jamming, c/c, and c/f river. This spot comes up often when we are OOP and have a hand we're not sure if we can vbet river profitably. I've seen many posters say we can vbet if we have >50% equity against villain's calling range, but I believe this is incorrect, as the results will show.

The usefulness of the results are a bit unclear at the moment, and I was hoping others can help me brainstorm how we can better make use of these results in real time during a session.

Anyways, here it is:

Suppose the pot is p and effective stacks are x * p on the river, where x is usually between 0.5 and 1.
We have a hand we aren't sure if it's strong enough to vbet river profitably.

Villain's range here will always consist of 4 different ranges:
(a) hands that will call and will jam river (value hands)
(b) hands that will call but won't jam river (bluffcatcher/marginal hands)
(c) hands that won't call and won't jam river (there is usually nothing in this range...but it doesn't matter as this won't affect the results since they get cancelled out everywhere)
(d) hands that won't call but will jam river (air/hands he will turn into bluffs)

We denote each range respectively by probabilities a, b, c, and d, where a+b+c+d=1. Let e be the equity our hand has against his range that calls AND jams river (note this is different from the equity against his entire calling range, ae+b, assuming we have 100% equity against his calling range that doesn't jam river).

We will make three assumptions, which I think are valid most of the time:

1. Our hand is strong enough such that we always win against his range that doesn't jam river

If this was not the case, then it's usually obvious river is going to be a check.

2. e < 0.5

If it was 50% or more, then we often have a hand good enough to vbet or c/c river.

3. ae+b < 0.5

Note ae+b is our equity against villain's entire calling range. If it was 50% or more, then we usually have a hand good enough to vbet or c/c river, though not as good as when e >= 0.5.

The EV of jamming is (c+d)p + b(1+x)p + a[e(1+x)p + (1-e)(-xp)]
explanation: we always win the pot against his range that doesn't call, we always win p + xp against his range that calls but doesn't jam river, and we expect to win e(1+x)p + (1-e)(-xp) against his range that calls and jams river

The EV of c/f is (b+c)p
explanation: we always win the pot against his range that doesn't jam river

The EV of c/c is (b+c)p + d(xp) + a[e(1+x)p + (1-e)(-xp)]
explanation: we always win the pot against his range that doesn't jam river, we always win p + xp against his range that does not call river but jams river (air), and we expect to win e(1+x)p + (1-e)(-xp) against his range that calls and jams river

Here is where things start to get interesting, when we compare the EV.

First, the EV comparison for c/c and c/f is pretty intuitive. c/c is going to be better than c/f when d(xp) + a[e(xp) + (1-e)(-xp)] > 0. This is the same thing as figuring out whether we have odds such that calling the river bet is +EV.

If we compare jamming and c/c, we get that jamming is better than c/c when b > d. In other words, we can easily tell jamming is going to be better than c/c when villain's calling range that doesn't jam river is greater than villain's air range that won't call river, but will jam river.

If we compare jamming and c/f, we get that jamming is better than c/f when d + bx + a[e(x) + (1-e)(-x)] > 0, which is the same as d + [(ae+b) - (1-e)]x > 0. Recall we assumed e < 0.5, so 1-e > 0.5. We also assumed ae+b < 0.5. Hence, (ae+b) - (1-e) < 0. Thus, we see that jamming is going to be better than c/f only if d, his air range, is large enough. This makes sense intuitively because if he is going to bluff a lot, then we're going to get owned by c/f. If we face a villain who literally never bluffs, then d = 0, and betting is profitable when (ae+b) - (1-e) > 0, which I'm guessing it won't be the case very often. I would have to look at a bunch of hands and play around w/ stove for a long time...

I have a hunch the underlined parts can be useful, since by the time we get to river, we usually have a pretty good idea of villain's range. But it's not clear to me the best way these results can be used while we play. Please let me know if you come up with something good to build off this theory, or if you think it' s useless.

okay, i've thought about this more and I believe most of this is going to be useless

i can't edit my post anymore, but just a warning to those who plan to read it.

So.. is it worth the read or not?

As far as I can tell, you are saying that its better to check/call the river when he's going to be bluffing a lot of bad hands, and its better to check/fold the river when he's never bluffing and your hand sucks, and its better to jam the river when you are ahead or he will fold. AKA: play good. But how to know what he's going to do? Be Phil Ivey?

This. And how do you account for what I'd call "ranges within ranges" for lack of a better term, and it hinges more on mood/emotion. Say villain has 2nd or 3rd pair with a hand like 89s. Sometimes he'll fold it, if he's feeling cautious. Sometimes he'll call with it, if he's feeling loose or suspicious. and sometimes he'll raise you with it, if he's feeling ballsy and confident. I'm not a math guy, but the Sklansky-esque EV calculations always seem to be lacking in this regard.

Just FYI I believe "merged range" is the term, as opposed to a "polarised range", containing only air and value hands.