Just some stream of consciousness related to bluff catching.
We’ll start by pretending we’re playing some kind of super AI that always bluffs at a perfect ratio, and we’ll imagine we’ve faced prior bets on the flop and turn, and are now facing a polarized range, and have to decide on whether or not to call off our remaining chips. Since our initial premise is that our opponent plays perfect poker, and that our hand beats all of their bluffs and loses to all of their value, our entire range by definition is indifferent to calling or folding; the EV of our range is zero for both actions.
While that is true of our range as a whole, blockers will often move us from a position of perfect indifference, to either a mandatory call or a mandatory fold, as now the range we are facing is no longer perfectly balanced between bluffs and value. So if our hand decreases the combinations of value hands, without a proportionately large reduction to the number of bluffing combinations (which are usually harder to identify when faced with a very strong opponent), then we should call at a 100% frequency. Likewise, if we significantly unblock value and block bluffs, we have a 100% fold.
Stepping away from this rare case where the bettor’s range is perfectly polarized, what do we do in a case where we think our hand beats a small number of value combinations, or loses to a small number of the bluffs? Again, assuming our opponent’s ratio of value to bluffs is set by the ratio of the bet to the pot, then if our hand beats even 1 “value” combination that our opponent has, then we have a mandatory call, and losing to even 1 bluff gives us a mandatory fold.
But have any of you yet faced an opponent that you think had a perfect balance between value and bluffs? I suspect that is not true of any human player, and so as we identify frequencies that appear to be bluff heavy or value heavy, how do we respond? The degree to which various hands switch from calls to folds or vice versa can be pretty extreme with even small levels of imbalance. Imagine someone that never bluffs… your bluff catchers are now completely worthless, and most of your “value” hands have to fold also. Conversely, if someone is bluffing a lot more than they should be, all of a sudden we are not only calling with all of our bluff catchers at a 100% rate, but now some of our bluffs can often have positive EV and should call.
In practice, how can you decide how wide to call? Everyone will probably have different types of strategies that might be best suited in combination given the limits of their notes or observational powers. I used to play with a HUD all the time over 10 years ago, and that to gives you an incredibly deep pool of statistics to draw from to identify imbalances, and very few humans are probably able to get such accurate statistics without something like that. But I like to take notes that simulate a HUD. I want to know if bets of various sizes represent strong value, marginal value, or bluffs, and for that to be numeric data that can be easily converted into percentages. What percent of the time am I facing a bluff, and what percent thin value, and what percent strong value? Approximately answering that question is my goal. If the range is tilted too far to strong value, all of the bluff catchers go in the muck. If the range is tilted too strongly to bluffs, all of a sudden, now you will be sometimes making queen high calls.
