In another thread, there is a discussion about opening ranges from the SB given population tendencies in a certain pool. I think the discussion is independently interesting, so I'm making a new post. The scenario is the following

SB opens 3x. BB folds 51% of the time and 3-bets 9.5% of the time. Should SB be opening 100% of hands?

Let's do the calculations based on various assumptions. SB always has the option of folding preflop. So we have to compare the above strategy to folding every time.

We always look at the EV for the SB. Let's start from the point before SB posts the blinds.

Then EV of folding = -0.5 (or -50bb/100)

Scenario 1: SB is relying on pure fold equity. Namely: SB folds to every 3-bet preflop, and check/folds every flop.

EV = prob. BB folds * EV(BB folds) + prob. BB calls * EV(BB calls)
= 0.51 * 1.5 - 0.49 * 3
= -0.705

So, at first glance SB should not be opening 100% of hands, because always folding is better. However, let's go one step deeper.

Any hand in NLHE has some amount of equity postflop. SB has a weak range and is out of position, so they will not realize their full equity. Let's assume that they realize 70% of their equity postflop. To make calculations easier, we will ignore the 3-betting from BB preflop, and assume that they call with their whole continuing range.

Scenario 2: SB raises 3x. BB calls with 49% of hands and folds 51%. SB realizes 70% of their equity postflop.

Equilab says that SB (100% of hands) has about 42% equity against BB's continuing range (49% of hands).

EV = 0.51 * EV(BB folds) + 0.49 * EV(BB calls)

EV(BB folds) = 1.5

When BB calls, the pot is 6. SB "owns" 42% of the pot, so 2.52. SB only "realizes" 70% of this equity, so 1.76.

SB had to put in 3 to realize 1.76.

So EV(BB calls) = 1.72 - 3 = -1.24

EV = 0.51 * 1.5 - 0.49 * 1.24
= 0.15

The strategy of opening 100% of hands now looks pretty good.

Scenario 3: Now we incorporate 3-betting. We first take a naive approach. BB 3-bets with top 9.5% of hands and calls with the rest of their range. SB will fold always to a 3-bet preflop.

EV = 0.51 * EV(BB folds) + 0.095* EV(BB 3-bets) + 0.395 * EV(BB calls)

EV(BB folds) = 1.5

EV(BB 3-bets) = -3

Against BB's calling range, SB now has 44.75% equity.

So EV (BB calls) = 6 * 0.4475 * 0.7 - 3 = -1.12

EV = 0.51 * 1.5 - 0.095 * 3 - 1.12 * 0.395

= 0.0376

We can probably make more sophisticated calculations, but that's all for now. We should also include rake.

Opening 100% of hands from the SB is likely to make a small amount of money. It's much better than folding preflop.