We see a flop of Qs8s2d, giving us the nut flush draw.  PIO wouldn't lead here, but I did force us to check.  Now Villain bets 1/3 of the pot.  

It seemed to me this was either a call or a jam.  PIO does in fact prefer call with this exact hand, though it does check raise with A9s and below.  What is most interesting to me is the sizing it chooses.  I gave it a 3x (about the original pot size) or a Jam.  Given those 2 raise sizes, it almost always goes small.  This is counter intuitive to me, and not the way most people play.  I wondered, what it hands it was check raising with, that it would then fold if we were jammed on...

It turns out that PIO balances its range with a lot of back door flush draws, with the Ace of spades, and its weaker Ax diamonds draws (it flats AJ/ATdd to the first bet).

If we do check raise small, and get called, it jams turn with a flush draw (our actual hand, or when we pick up the BDFD) and gives up on blank turns with AsX...

The optimal strategy for 87 is to either bet zero, or get all in.  Obviously we will never know this in real life, but I do have a scenario where I like to use this strategy:

We have raised preflop, and get called by in position.  We hold JTs and the flop is something like Q63, rainbow one of our suit.  We bet the flop, leaving something like 2-2.5 SPR behind.  The turn is one of our best cards, one that gives us a flush draw, and either a gutter or OESD.  This could be the A/K/9/8 of our suit.  If we barrel the turn, and get jammed we will be in a spot where our call will need about ~30% to break even.  This is about what we will have, so the value of our hand is close to zero.

What if we check the turn? This goes counter to everything most of us do; flop some back doors/cbet/turn equity/fire again.  I am not saying this bluff never works, but it will leave us in an awkward spot when we are jammed.  

How does checking fair?  If Villain checks behind, that is fine for us.  We will realize all of our equity, with a free card, and still be in a good bluffing spot when we miss.  If Villain does something other than go all in, we can then jam, have fold equity, and still have the right price when he calls.  

Now it is Hero action, what's the best play and why?

Flop is A87 with a flush draw; hero has none of either suit.  Villain bets 1/3 of pot; hero calls.  Turn is an 8, putting a second flush draw on board.  Hero bets 70% pot.  River is a 7  (not a flush card), hero bets 1.5x pot.

What does PIO do here?  Hero never folds flop, but does a fair bit of check raising; 14.4% of range, and 26% with his actual hand.  

On the turn, PIO basically never donks.  However, I reran the sim with hero not allowed to check raise preflop, now PIO donks about 25% of the time.  PIO doesn't like it when you limit its options, like no check raises on flop.  Making up for some lost time, now it bets.

On the river, as played, I gave PIO multiple bet sizes.  Interestingly, it prefers either a smallish bet say 1/3 of pot, to get folds from Villains missed flushes (KQ,KJ, etc.) that we are chopping with.  Or, it uses a large size as hero did, targeting Villains Ax hands.  

This is something I have seen a lot; not sure if I posted this here before.  PIO tends to avoid medium sizing on the river.  Either polarize, or go smallish seems to be its preferred sizing.  

Just as calling is not a good compromise (Thanks, Andrew), using a middling bet size isn't either.  You want to have an idea of your bluff targets and value targets and size appropriately.  One caveat, this assumes we are playing PIO.  Many Villains will look at your bet in absolute (dollar, or chip) amounts and NOT in fraction of the pot terms, as we know they should.  In that case, something like 2/3 pot might seem 'big/polarizing' when it probably shouldn't.

I've been meaning to form some minimal mantra-like questions in to my game to avoid lazy-thinking-syndrome. 

You can have 2 possibilities here: 1) Shove the strongest hands in your range, call the middle portion of your range and shove the best hands that cannot profitably call; or 2) Shove or fold here with no calling range.  

(There are more than these 2 options here as exploitative adjustments, but if we are discussing purely mathematically balanced ranges, this is how I view it.)  (Also, I'm sure solvers will spit out more complicated split ranges but they are hard for humans to develop on the spot without lots of study time with solvers.)

The answer really lies in whether you can have a call range here.  And that does depend on the 2 limpers, since you are not closing the action.  Are they straight forward limpers, where they would have already raised their strongest hands and will either only call or fold.  Or are they tricky limpers and will back-raise if you call.  In today's game, many of the loose limpers tend to be more of the straight forward kind of limper.  And if that is the case, I think you can have a middle calling range.  Which would mean shoving the top of your range and then the best hands that cannot profitably call here to balance the shoving range.  

Preflop, obviously all hands have some equity, and they are mostly closer together.  

My main question was 'do you have hands in between that are calls', and hands that are pure value shoves, and semi bluff shoves.

I consider the AWs stuff to be a good semi bluff shove; we have modest equity vs calling ranges and sufficient FE to make them profitable.  Of course, our biggest hands are pure value; we hope to get called.  There are hands that are 'in between', where we are good enough to call, but do not play well as shoves.

That was what I was trying to get to..

This brings us to the question in the title.  What should our 3 bet shove range look like?  My thoughts are:

1. I do not have a 3 bet size that is less than AI.  I guess we could make it something like 13.5BB, but I prefer to shove my 3 bet range.

2. I do think we can be polarized here.  

Now, Villain can call flop with his pairs 99-QQ about 1/4 of the time, and get to the river about 10% of the time, since hero now splits his turn betting frequencies.

Keeping AA (and to a lesser extent) KK in your range does a lot to help the weaker hands here.  Personally, when the SPR is going to be so small, it seems fine to slowplay AA, because we can always get all in by the river, especially in position.

I have about 60K, Villain has 30K.  I open A5cc to 3500 (at 1600BB), Villain calls button.

Flop is JcJh2s.  I bet 4K; Villain calls.  Turn is Qc.  I bet 6K; Villain shoves, I fold (Villain showed AJ)

What does PIO think of this spot?   I ran this simulation twice.  First time, I allowed Villain to raise, second time I did not.

When I allowed Villain to raise, my range (about 22%) bets about 56% of the time, splitting evenly between 1/3 and 2/3.  My exact hand also splits that way.  What is most interesting to me is that Villain almost never calls!  PIO has Villain raising 65% and calling just 8%.  This seems a bit counter intuitive to me; Villain has position, and does not need 4 bets to go into the pot to get all in.  Why is he raising so much?  Looking at the hand hands PIO strictly prefers to raise gives me a clue.  It is almost always raising 77-TT, I guess as protection from my overcards.  It also raises some complete airballs some of the time, like A6s no BDFD.  Of course it raises a bunch of Jx as well, though it does slow play Jx some of the time.

In reality, I think most players literally have no raising range here.  I must admit that is how I'd play this spot in Villain's shoes as well.  How can we exploit this?  One obvious exploit is to bet our entire range here, and bet on the small side (as I did, although I think I can/should go even smaller).  When we take away Villains raising range here, we now bet 90% of the time. The only hands it checks (as hero) with reasonable frequency are JJ, and the Axs with no BDFD.   

I think the key exploit on this hand is that we can bet range, we can bet it small.  Villain is likely going to be inelastic with his calls, although I can't say for sure if he will play his pocket pairs the same to small/large sizing.

On the turn, when we pick up our Flush draw, this seems like an obvious bet/fold.  PIO does bet my exact hand most of the time, and does call the jam.  Here, I think this is an easy exploit to fold.  PIO thinks this is a small +Cev call for about 6% of the pot, but I think this is not close.  Villain is never bluffing here, so we have at most 8 outs (or 7 if he has KxJ, or none if he is already full).  Add in this is a tournament (and about 15% from the bubble) and this is an even easier fold.