In river spots where the bettor has a polarized range (only nuts and bluffs) and the caller only has a range of bluff catchers (can only beat a bluff), you can use GTO frequencies to maximize your EV, if you can't exploit or don't know Villain's tendencies.

Using these frequencies,

- the polarized range always wins the Pot, EV = 1P

- the bluffcatching range always has an EV = 0

I have this simple table memorized, so I don't need to solve equations in game :

- Optimal Calling Frequency : used on the river to prevent the bettor to make a profit on his bluffs and thus, to make him indifferent to betting or checking his bluffs; EV(river bet) = 1 P

- Minimum Defense Frequency : used on the flop or turn, to prevent the bettor to make an immediate profit on his bluffs; same as above, except that it is used as a minimum threshold and is not static (more streets to play, other factors relevant)

F = vc / D - vc : Number of Bluff Combos to Bet given a known number of value combos and the optimal value betting frequency

G = bc / C - bc : Number of Value Combos to Bet
given a known number of bluff combos and the optimal bluffing frequency

These shortcuts are much easier for me than solving the equivalent equations.

The 2 sets of frequencies (A and B) are the most important to remember, because they are the GTO target frequencies to make the opponent indifferent :

The polarized range is always betting and sizes his bets to target column B for his optimal bluffing frequency to make the bluff-catching range indifferent to calling or folding.

The bluff-catching range targets column A to determine his optimal calling frequency to make the polarized range indifferent to betting or checking.

Example (from the "Value Betting" video by Felipe Boianovsky, but also elsewhere like in Lefort's series) :

You have 7 bluff combos in your river range and decide to make a 2/3 pot sized bet. So you need to calculate the GTO number of corresponding value combos to bet.

1. deriving the Optimal Bluffing Frequency : C = B : 29%

2. deriving the required number of value combos : G = bc / C  - bc : 7 / 29% - 7 = 18

Now you have your GTO river betting range of { 18 value combos : 7 bluffing combos}.