Polarization and bluffing w/ small value on showdown last to act? Ft. Full Tilt replay

I am very a polarized player near the showdown, essentially bluffing my worse hands, and betting my strongest hands.

A maxim that holds true to me is that if I am bluffing with X hand on the showdown, I am going to be 100% bluffing (bluffing every time).

In terms of exploitability, polarization is not exploitable, as an opponent doesn't know whether my raise is a bluff or high value. And of course this allows us to get showdown value from our middling hands.

In essence, in mixed strategy scenarios where we bluff 50% or 30% of the time, I let my hand strength be my randomizer, instead of the hands of my watch, or the suits of my hole cards.

Regarding exploitability in terms of range prediction (for example our opponent might now that it's more likely we hold 23 instead of 78 when betting the river). I guess this is marginally exploitable, but not to the extent that I would care, it would require someone knowing for a fact what my strategy is, would require the opponent to make themselves very exploitable, and in a situation where someone can exploit that, we are at best losing to rake anyways.

Let's take a look at a replay:

https://www.youtube.com/watch?v=mmiZencJ...

Heads up 200bb deep.
Isildur (SB) raises x3
Ivey (BB) flats.

Flop is a dry x9T,
Ivey check
Isildur's almost pot bets 5bb
Ivey check raises to 15bb
Call

Turn is a wet backdoor 8
Ivey bets 77% pot
Isildur calls.

River is another wet 6 completing any backdoor flushes.

Ivey Checks
Isildur shoves, overbetting 2x the pot.

Isildur shows Q9o

Now without considering exactly whether this would be a nice spot to bluff or not, without considering the complexities of the backdoor flush that I could represent, the range that Ivey has, and the odds he gets to call different types of bet sizes. I would just check for value here.

The idea is that since we have a card of middling strength, bluffing here means that we would have to bluff with ALL of our weaker hands 100% of the time. (Or conversely shove all of our higher hands if we were betting for value, but this is not the case).

The only scenarios where we would bet here are if:

- The hand combo is not actually small/middle value, but is actually the lowest end of our showdown range. I don't think this is the case, I can see lower pairs and high cards above T here (JK, QK, AK, AQ, AJ).
- Hero is not using a mixed strategy, but rather has a hard read and wants to exploit. Which would be sensible in some scenarios, except against a megalodon shark like Phil Ivey.
- Some combination of our hand being middle-low value, the bluffing frequency of our range being so big that we ARE bluffing all of our lower spots, and the frequency of lower hands in our range being lower than expected (maybe we folded a lot of high cards on the flop, etc).
- A distribution strategy with more than 2 modes? Maybe hero bets straights and flushes. Bets his bottom range, but also bets his non straight non flush top range? I don't know this would still be a bimodal distribution. Unless there is a third mode in the center, (That is we bet high cards, check low pairs, bet high pairs, check two pairs and sets, and bet straights and flushes.) I'm pretty sure that a bimodal distribution outperforms.

Just as a quick actual calc, let's enumerate all of our combos to see what our bluffing range needs to be to include up to middle pair.

we'll assume we are in flop and ignore river showdown value.

Flop is 39To, we cbet and call the checkraise, we have:

Overcards AK,AQ,AJ: 16*3 = 48
A8: 16
Suited Ax with backdoor: 21 -1 (As3s)
pocket pairs 8 or below (minus 33): 6*6 = 36
Mid pair Q9,K9,A9,J9,98,97,96: 12*7= 84
High pair: QT,KT,AT,JT,8T,7T,6T: 12*7= 84
Two pair: 39,3T,9T: 9*3 = 27
Sets: 33,99,TT: 3*3 = 9
Overpair: JJ,QQ,KK,AA: 6*4= 24
Open ended: QJ, J8,78: 16*3 = 48
Gutshot with overcard: J7, Q8, JK,KQ : 16*2 = 48

Now there's a total of 444 combos in here, and our mid pair is at this point above average in value.

Now reranking all of these on the river is hard, but the idea would be that the shoving range (7x+ flushes) needs to be as frequent as all of the cards worse than mid pair.

So worse cards
missed Suited Ax are the worst: 13
A8, except suited: 16-1
Overcards are still trash, except JQ and flushes (AK, AQ, AJ,JK,KQ): 80 - 4 = 72
Low pocket pairs (2,4,5), pretty bad: 18
Low pairs turn and river pairs (J8,Q8)= 12*2
Mid pair (Q9,K9,A9, J9): 12*4 = 48

Skipping all higher pairs and sets

77 = 3
97= 12
78= 12
j7=12
Ax Flushes = 11
JQ = 16
other overcard flushes(AK, AQ,AJ,KQ,JK) = 5

So we have:

16 flushes
16 nut straights
39 mid straights

The next best hand is like TT or 99, which is a huge jump in the hand strength distribution.

Now however often we bet will be a function of how often we bet the top mode for value, whether it's half, 150% or double bluffs.

So we have a range X (with less than 444 combos (calling flop raise and betting turn further reduce it).) and we are betting the top of that range for value which are 63 combos.

The range of worse cards than Q9o is 142, and 180 including such pairs.
In order to bet Q9o while also betting all worse hands we need to have a range of 2 or even 3 times as many bluffs as we have actual straights and flushes.

One thing we can say for sure is that if the maxim is followed, Hero has a HUGE bluff range (2 to 3 times as many bluffs), to the point where Q9 actually has value that might defend against bluff catchers. Which might lend credence to this trimodal theory.