pastis

I'll leave the following that I wrote before doing the math. I've changed my conclusion based on math alone here to go from disliking your 79 into 262 bet with the nines hand otr to strongly preferring it to checking. I think a lot can be extrapolated from this, and the spot certainly caused me to think deeper about thin valuebets in river spots where villain can xr wider than usual (such as here). More specifically, smaller bet sizes forces villains to defend very wide, and typically (and perhaps especially on the texture in this particular example) the strong parts of their ranges won't be wide enough to take advantage (by xr) of us betting wide and small.

Again, I went from strongly disliking the river bet to strongly preferring it, but I still don't understand why you'd prefer to call the xr with 8x over 99. Most villains shouldn't have 88 in their range (3-bets preflop) and I view it as quite likely they wouldn't turn 8x into a bluff at any sort of high frequency. Thus decreasing villains number of 8x combos doesn't achieve much against a polarized range that normally doesn't contain 8x (apart from 85, but that's 6 combos and 4 combos if we block 8x, and some can probably be discounted since he didn't xr turn).

On to what I initially wrote:
Nines hand I was saying "overbet" to the screen ott. Was cool that you mentioned this possibility right after as it played out. Otr I don't think I'm a huge fan of using this small sizing unless we have some sort of read that villain is check-raising rarely or xcalls too wide on similar runouts. For this bet to be higher EV than checking we have to calculate the following:

First of all, let's assume he never xcalls better.
EV of betting relative to checking = (betsize)(freq villain calls with worse) - (bet size)(frequency villain xr better) - (bet size + pot size)(freq villain xr bluffs)

We can plugin this, using actual $ amounts.
Let the pot size be 262 and our bet size be 79.
Let freq villain calls with worse be X.
Let's assume villain checkraises better hands 5% of the time, and balances this with an additional 2.1% of bluffs.
This leaves us with the following equation: 79X - 79*0,05 - (79+262)0,021 ---> 79X - 3,95 -7,161.
Now, we can set this equation equal to 0 to be indifferent between checking or betting this amount on the river, which yields -> 79x - 3,95 - 7,161 = 0 --> 79x = 11,111
X = 0,14

So we need him to call with a worse hand more than 14% of the time in order for betting to be better than checking.

However, I suspect that our small river bet makes villains more inclined to xr wider as a bluff and also XR wider for value. Let's assume he now check raises hands that beat us 10% of the time and throws in an additional 5% of bluffs. I don't think this is very unreasonable, seeing as he should have pretty much all 5x he plays preflop in his turn xcalling range against the size we used (85% pot).

The equation now becomes: 79X - 79*0,10 - (79+262)0,05 = 0
--> 79X = 24,95 ---> X = 0,315

So we now need villain to call our bet with a worse hand than ours more than 31,5% of the time in order to prefer betting to checking.
Now, we're laying him a price of 341/79 = 4.32 : 1, meaning he needs to win slightly more than 1 / 5.32 = 18,7% of the time in order to call profitably. Furthermore, we can profitably bet any two cards if he folds more than 79 / 341 = 23,16%. So 1-A = 76,84% in this spot, which he'll have to defend if he wants to keep us indifferent to bluffing.
Going by my assumptions of his high xr percentage (15%) he'll have to defend another ~62% by calling here in theory, so our bet should still be way superior to checking, and by some distance.

My assumptions regarding frequencies could of course be way off, but for the most part I think
at least the math should check out.
If I've overlooked anything, please feel free to correct me.

TL;DR - betting wide using a small size against weak(ish) ranges otr opens us up to being xr bluffed, but most villains don't do this nearly often enough, or if they do they aren't close to balanced (we can counter-exploit by either calling all of our bluff catchers or folding them all).

Great video. This format is excellent.

+1 for Pio, and also just wanted to chime in that theory based videos are more than welcome. Hope I don't sound like a douchebag or anything, but to me it seems like a lot of the pro videos that get posted are like the 5th attempt of recording, and the pro finally catches a good run of cards. If there was a simple way of verifying, I'd easily take a bet that instructors on here have >25bb/100 in their aggregated winrate for the videos posted.

With theory there's no variance involved, just plain and simple facts, delivered by some of the best in the business. To me that's often more interesting than watching 40 minutes of rungood.

Happy new year, may we both crush 2016!