I watched this really interesting hand on YouTube. The guy made a fold that I would probably consider standard. A lot of people in the comment section seemed to trash this guy's fold to a min-check/raise on the river arguing that you should never fold a value hand to a min-check/raise on the river. I was thinking that you should probably fold your bluffs and the bottom part of your value range.

I decided to do the math for a perfectly balanced spot since I wasn't sure. I arrived at the following equation for determining what portion of your value range you should fold to a min-raise on the river:

1 - [1 - (2b / (3b + p)) / (1 - (b / 2b + p))]

p stands for pot size in the middle
b stands for the amount that the bettor bet

In the denominator "(1 - (b / 2b + p))" gives the bettor's value percentage.
In the numerator "1 - (2b / (3b + p))" gives the MDF for the bettor.
Dividing those two numbers gives the percent of value hands that are defending. Subtract that all by 1 to get the amount of value hands that are being folded to the min-raise.

Notice as the bet size gets smaller the percent of value hands you should theoretically fold to a min-raise gets closer to 0, but never actually reaches it. The larger the bet size the more hands you should theoretically fold.


Here is a quick summary for the percent of hands you should fold to a min-raise after making x bet size on the river:

3x pot: 30%
pot: 25%
1/2: 20%
1/3: 17%
1/4: 14%
1/98: 1%


Here is the hand that inspired me to look into this (Sorry, don't know the stack sizes, just assume 200BBs effective) :
Blinds are $25/$50
HJ opens $175
CO calls with 9 8
BB calls too

Flop($550) 4 5 8
BB checks
HJ checks
CO bets $525
BB folds
HJ raises to $1375
CO calls

Turn($3300) 4 5 8 K
HJ checks
CO bets $1750
HJ calls

River($6800) 4 5 8 K K
HJ checks
CO bets $3100
HJ raises to $6200
CO folded

Now perhaps CO shouldn't bet the river given that the board paired. If you think your opponent is very loose passive it could make sense if you are targeting 76 and your opponent is heavily weighted toward that. Given the check/raise though it seems like a fold is in order. Theoretically you should be folding around the bottom 20% of your value range. Given that you might not be bluffing versus a loose caller you will need to fold even more of your value range. Exploitatively you may even overfold.

Thought I would share this because I thought it was cool. I did not consider 3-betting on the river since that doesn't happen too terribly often and shouldn't affect the percentages too much.

What do you think? Would you fold this river?

HJ can have 88, 55, 44 and KK as value hands (8 combos). How many bluffs can he have with that sizing, and what would they be?

The pot odds being offered to the original bettor is 3100/(3100 + 3100 + 6200 + 6800) = 16.15% which should equal the percentage of bluffs in the check/raiser's range. Roughly he'll be able to bluff 1 and a third of a combo. I think the best candidate for check/raising would be counterfeited two pairs. So 85s if HJ plays it. If not, then probably 54s.

Interesting.

Question:

If every value-bet from CO can beat the counterfeit 2-pairs, then can CO properly defend against the minraise by randomly calling all his hands with the frequency 1-x where x is the portion of value to be folded?

(Disregard blockers)

I suppose you could do that. I'm not a fan of it in practice since it could be hard to randomize correctly. Also, in spots your opponent could be incorrectly value betting too thinly. Like if the opponent thought Q T was good.

Nice. If you decide to play a pure strategy and locate your MDF combo with which to defend, how would you account for the misplayed QTss ?

Thanks.

I think I understand what you are asking, but correct me if I'm wrong.

Let's say we bet the river with KK, 88, 55, 44, and flushes: AQs-A2s, QJs, QTs, JTs, J9s, T9s, T8s, and 98s. This would be 26 value combos. Let's also assume we aren't bluffing since our opponent calls down too much. Now our opponent surprises us with a min-raise and we aren't sure if he is only doing this with the nuts, over valuing his holding, or is making a goofy move. We decide to play balanced with the range we arrived here with.

MDF = 60%, 15.6 combos
We continue with KK, 88, 55, 44, ATs-A8s, A3s-A2s, and A7s half of the time. We fold everything else.

Now QTs wins 0% of the time when we continue. If we were to call with each of our flushes at a specific frequency we would lose some of the time when we call with lower flushes such as 98s.

I’d bet 1/2 pot on the flop. Changes potential turn and river profitability.

Lol YouTube’s comment section. I wouldn’t consider those comments as having any value worth your time to read, except for laughs.

If we have a balanced betting range, I believe MDF will always have us folding some value hands to a raise on the river no matter how small (even less than a legal min-raise), except maybe when we have some bluffs which somehow make better bluffcatchers than our value hands.

At equilibrium, I believe the range of hands which win when we bet river and get called has to be larger than the range of hands which win when we bet and get raised.

Some solid analysis, thanks TheGodson.

As you probably know, accounting for possible mistakes is a quirk of GTO when multiple solutions are technically optimal.

I had not noticed how far down the 98s is in your range.

-Rob

I almost included 87s and 76s, but those block the straight and the other lower combos are already on the board.

As it turns out, the other guy had J 4.

In the video, the 98s guy asks the other guy if he has 8s full. Since he holds an 8 in his hand 55 or 44 would be much more likely versus a typical opponent. This leads me to believe that he thinks the other guy is super passive. If the guy with 98s would have been overfolding then he would have been exploited by an ambitious J 4. 98s guy assumed the other guy was a loose passive for whatever reason and that cost him.

Since the guy had J 4 there is a good chance he is over bluffing when check/raising the river. It is easy to say 9 8 is a call though when you can see the hole cards as an outside observer.

His value range probably should only be boats, so 98s is a better defend than the nut flush. This isn't that low in our range when accounting for blocking effects.

That's some hardcore spew. I wish I could afford to play 25|50 with these guys.