hes an expert poker writer as well i see? can he write me a detailed analysis of wa/wb situations in headsup hands with optimal ranges on flush boards?
Assuming NLHE, except in weird situations where we've checked down the flop and turn after limps or a single raise pre-flop, or in situations of a monotone flush, a WA/WB when a flush completes typically implies that Hero was the aggressor on the flop and/or turn with a hand that was at or near the top of his flop range (typically sets or top two). The two main cases are whether Hero is in or out of position, but you can break down further cases based on how many bets are left. Let's narrow down one situation to cover instead of looking at them all for the sake of brevity. Assume no rake for the sake of simplicity.
Suppose we're out of position with a bet and a raise left behind with an SPR of something like 2.5, and we're going to have our first bet be two-thirds of the pot. Let's also assume that we have a fair idea of Villian's range so that we don't have to establish Villain's GTO range after four streets of play before we tackle the actual situation.
First we have to establish the structure of Hero's range. We'll start by noting that Hero's optimal bluffing frequency is 28.57 percent (conveniently exactly 2/7), a well-established value for a two-thirds pot-sized bet. Hero's bluff:value frequency should be 2:5 along these lines.
Note: Villain's optimal fold percentage is 40 percent along similar lines. We'll need this later.
When Villain shoves over, he needs Hero to fold 30.56 percent of the time for the bluff to be break even, so that's Hero's folding frequency after betting. So here's what Hero's range looks like so far with the strongest hands to the left (with slight variations possible based on issues of combinatorics thanks to hold'em being a cunt):
<--check/raise (value)--|--bet/call (value)--|--bet/fold (value)--|--check/call (inducing etc)--|--check/raise (bluff)--|--check/fold--|--bet/fold--|--check/raise (bluff)>
To sum up the ratios for the betting ranges, there's a 5:2 ratio here of the right-most bet/fold range to the sum of the left-most bet/call and bet/fold ranges. Additionally, the two bet/fold ranges will be 30.56 percent of the sum of the bet/call and both bet/fold ranges.
This is getting long enough as it is, but the next part is to establish the ratios that happen when Hero check/raises and check/folds along similar lines. It's lots of well-established but boring (for most people) math that I don't expect anyone to follow.
Once you have these ratios in place, you get to the crux of the WA/WB situation which is that our profitable value leading range is virtually non-existent (or very thin at best). While all of the frequencies found for Hero's ranges in the above are based on ratios, they all take their specific frequencies from maximizing the number of +EV value bets there are. Every other range is based on this frequency, like from the ratios we looked at above.
Along these lines, the hands available to Hero for a profitable value bet are minimal outside of barreled flush draws that hit. This means we're also bluffing a very small percentage of the time, and we'll be checking a lot more. All of the exact GTO percentages for the range fall out of some simple algebra that comes from knowing what percentage of Hero's range will be profitable to value bet.
TL;DR: Check/raise and check/call more OOP.
