Well put question Teddy.  I think you'll need to provided some sketches at answering it yourself though before people are going to volunteer this type of info. 

Thanks Ben! I´ll try to do some math and then get back with what I have.

I have my own sketches of how i would split my range in some type of textures. Id show you here if i see any commitment on the answers, and people could give me any usefull information.. ! But im not sure if this gonna happend.. 

For now i can say that i usually split my range in order to my opponents weakness. Would be nice if i could chose always the option 2), bet all my air getting profit, but against some thinking regs i will have to resign some part of my range in some board textures, having to chose option 1).

Would be nice could start to design some starting points of splitted ranges, and then start to deviate of them in order to our opponents weakness.

Lets start the discussion by giving the solution to the one street version of this game:

Pot P=2, there is one pot sized bet of size 2 left. 

Both players can check, bet, call and fold. Lowest number wins.

OOP will:

Bet: [0, 0.17] 

Check,call: [0.17, 0.50]

Check,fold: [0.50, 0.92]

Bet: [0.92, 1]

Thus the OOP player will bluff his ~8% worst hands, bet his strongest hands and check three quarters of his range.

BTN will:

Call: [0, 0.50] when facing a bet

Fold: [0.50, 1] when facing a bet

Bet: [0, 0.33] when facing a check

Check: [0.33, 0.83] when facing a check

Bet: [0.83, 1] when facing a check.

Hence, the button bets a polarized range. The cutoff value betting hand 0.33 is the mean of the check,call range [0.17, 0.50]. And the ratio between value bet and bluffs is 2 to 1, which offers exactly the right pot odds.

Note that all numbers are actually fractions of the form n/12, but are shown as percentiles.

If you look at the flop in isolation, you would need a hand in the top 25% to valuebet, as oop will fold bottom 50% to a potsized bet.

In this statement bot the premise and the conclusion are incorrect. Firstly, for OOP to be folding the bottom 50% of his range, he needs to make it a losing or break even play for the BTN to bluff with his dead hands, but there is not reason to believe that BTN will not bluff his worst hands. This was exactly what he was doing in the one street example, and what he should be doing. When you check down your worst hands, the expectation to win the pot is very low. And when they bluff successfully they win the entire pot. Therefore the worst hands gain the most by bluffing.

Secondly, in a multistreet game it is no longer enough that you have 50% against the continuing range of your opponent. Since by betting you blow up the pot by a factor of 3, and you will still need to defend the bottom of your range on future streets. Calling bets on future streets after betting is much more expensive.

If you have 51% equity on the flop and you bet flop, check turn and call a bet on the river, you put in 2+6=8 (flop+river). While you only tried to win a bet of 2 on the flop. And if your opponent starts betting turn and river this becomes 26 to 2. The 1% equity advantage is most likely not enough to warrant this.

Optimally protecting all your ranges can be very difficult, even in a simple static game like this.

I see a problem with your solution, GT.

When you bet [0, 0.17]  of your range and get called by [0, 0.50], you are assuming OOP is valuebetting and winning always the pot. So, if you have 2/3 valuebets, 1/3 bluffs, EV equation for IP player when calling, assuming PSB, is:

EV=%OOPvaluebets * (Eq*Pot-Pot)+%OOPbluffs * (Pot)

So you are doing

 0.66(-2)+0.33(4)=0

But the thing is, when [0, 0.17]  bets against a calling range of [0, 0.50], hes gonna win around 2.5/3 of the times (so it has 83.33% equity). Thats because 2/3 of the times the calling range has hands between [0, 0.50], and 1/3 of the time they are going to show up with the same range. So, the EV equation for IP player should be in this example:

0.66(0.166*2-2)+0.33(4)=-1.11111+1.33333=0.22222.

This number obv should be 0.

You are right, forgot to take into account that I shouldn't take a look at my [0, 0.17], I should take a look at my bluffcatching range. Thanks. 

Could someone briefly explain what [x.x, x.x] mean?

I was curious why BT doesn't bet more when OOP only calls (.50-.17)/(.92-.50+.50-.17)= 44%Fx. EV check .82 = lose 1 vs [.17-.82] win 1 vs [.82] win 2 vs [.82-92] = 2(.92-.82)+(.82)-(.82-.17)= .37EV Ev bet .82 = lose 2 vs 44%, win 2 vs 56% = .56*2-.44*2 =.24EV Ev check .83 = lose 1 vs [.17-.83] win 1 vs [.83] win 2 vs [.83-92] = 2(.92-.83)+(.83)-(.83-.17)= .37EVEv bet .83 = lose 2 vs 44%, win 2 vs 56% = .56*2-.44*2 = .35 EVDid I make some error when calculating [.83] since the EVs of betting and checking aren't equal or is it just rounding ?

The ante is 1, so by checking you either win or lose 1, never 2.

And the bet is 2, so by betting you either get a fold and win 1, or get a call and win or lose 3, never 2.

You could consider the ante of 1 sunk cost, but then you have to be consistent.

Thanks! I believe I found my error and this is correct:

As BTN:

If you hold 5/6 (=0.83) and you decide to check your EV is:

Lose 1  versus [2/12, 10/12], win 1 versus [10/12, 11/12]  (9 intervals)

EV = (8/9)*(-1) + (1/9)*(+1) = -7/9

And when betting:

Lose 3 versus [2/12, 6/12], win 1 versus [6/12, 11/12] (9 intervals)

EV = (4/9)*(-3) + (5/9)*(+1)  = -7/9