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?