Hey all, I am looking to learn some of the logic/mathematics that go into building preflop raise ranges.

Lets look at a scenario 6-max, 100bb effective. UTG vs. CO.


UTG opens around 18% of hands for 2.5bb.

CO responds with a linear 3 betting range, raising to 7.5bb, about 9% of the time.


How exactly is this range being constructed. Are we looking to have a certain equity percent range v. range?

When I run the ranges in an equity calculator it looks like CO has around 55% equity with their range. Shouldn't we be able to open wider since their is dead money in the pot we don't even need 50% equity range v. range.

Someone with more expertise about how the preflop solver algorithms work will have to help explain that part.

I just wanted to chime in to say in the example you gave the CO has to play tighter because the remaining players yet to act can still raise and push them off their equity

The short answer is that preflop mathematics aren't well known. We calculate optimal ranges using an iterative process where algorithms play against themselves.

Part of the problem is that equity is almost useless preflop, due to implied odds and betting on future streets. For example, 72o has equity to call a standard 2.5x BTN open. But it's still a losing call because it won't realize that equity. You can use EV instead to gauge the true value of a hand, but that requires solving the optimal strategy in the first place. Gauging the true value of a hand is no small task.

GreatWhiteFish is right about this; the more players yet to act, the tighter you have to play.

Here are some general heuristics:

• In a vacuum, the optimal opening frequency is approximately: 1/(number of players still in the hand). So in 9max, UTG opens about 1/9 of hands, 11% or so. In 6max, UTG opens 1/6 of hands, 18% or so. There's lots of stuff that can skew this. Like if there are antes in play everyone opens wider. If you're on the BTN and have a stack deep enough to exercise a positional advantage then you can open quite a bit wider than 1/3 of hands.
  
• As a rule of thumb, 3-betting and 4-betting ranges should be (at most) half as wide as the previous raiser, usually less, never more. So if someone is opening 20% of hands, you should not 3-bet more than 10% of hands.
  
• The frequency that you steal the blinds, get called, or face a 3-bet, depends on your opening size rather than your position. For example, UTG RFI expects to steal the blinds almost exactly as often as CO RFI. It's remarkably similar. However, positional advantage can skew this. Hence why LP tends to size up, while EP tends to size down.
  
• RFI% by seat is almost identical in NLHE/PLO/PLO5, indicating a deeper connection between how often you open, and how many players are yet to act.
  
• The collective defense for all players yet to act should be less than the opener's alpha. For example, if HJ opens 2x, they are risking 2bb to win 1.5bb, so their alpha is 2/(2 + 1.5) = 57%. Therefor, the product of CO-BB's folding% should multiply to less than 57%. If not, HJ can open any two. Actually, they should collectively defend quite a bit more than this, because HJ's bluffs have equity postflop.