River AI Spots HU In Position

Introduction

For something else that I was working on, I had to break down river spots where Hero is in position with one bet left behind. I considered putting this in one of my other threads with long ass EV equations and things like that, but it's a lot easier than most of that because I'm not including the derivations here. Since I didn't want it to just get buried, I figured I would make a new thread.

The situation is that Hero has been checked to on the river against a single Villain, and Hero has the option to shove or check. In this situation, we'll call A the value of bet/(bet+pot) where bet is how much Hero is betting and pot is the size of the pot. This is a value that comes up a lot in poker. I'm going to keep this short and sweet so that people can use it as a reference if need be.

Villain's Unexploitable Strategy

Villain will need to fold with his worst A percent of hands to be unexploitable to bluffs. For example, suppose that Hero bluffs $1 into a pot of $1.50. When Villain folds A = 40% of the time, Hero's bluffs have an EV of 0. Villain will fold the bottom A percent of his range and call with the rest. Villain's range will look like this (not to scale):

Worst Hands <--fold--|--call-->

Hero's Unexploitable Strategy

Hero's perfectly balanced (unexploitable/game theory optimal) strategy involves value betting all of the hands that have 50% or more equity against Villain's unexploitable calling range. Then, Hero will bluff enough to have a ratio of bluffs to value bets at a ratio of A:1. You'll bet an appropriate number of your best and worst hands for your value bets and bluffs, respectively, and your range will look like this (not to scale):

Worst Hands <--bluffs--|--check--|--value bets--> Best Hands

And that's really all there is to it.

If we were OOP would this have any effect on this?

Originally Posted by seven-deuce

If we were OOP would this have any effect on this?

Yeah because your checking range has to be broken down in a way to protect you against bluffs (ie: you have to fold A percent of your checking range and call 1-A percent of it). I'm pretty sure I've broken down a similar situation in another thread before. I might add on to this thread with the OOP version when I get time.

Hero's perfectly balanced (unexploitable/game theory optimal) strategy involves value betting all of the hands that have 50% or more equity.

Should this not be "...50% or more equity vs villain's calling range"?

Originally Posted by daviddem

Should this not be "...50% or more equity vs villain's calling range"?

If your shoving all in on the river it's obviously going to be vs his calling range, he can't raise you once your all in.

When villain checks the river he has got a range, which splits into a folding and a calling range once hero bets the river. I was just remarking that Spoon did not specify this when he talked about "50% equity". It may be obvious to you that 50% equity is needed vs villain's calling range for a value bet, but it may not be to everyone who reads his post.

I reworded that portion to make it clearer since neither of you seemed to know what I meant. That was my fault.