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.