I decided to mess around with the maths of this as an exercise…

Let’s start with some simplifying assumptions.

Plucking figures out of thin air, suppose BU is opening with the top 40% of hands, SB is 3Betting the top 10%, BB will always fold, and BU will not 4bet.

[Per Equilab,
40% range:
44+, A2s+, K2s+, Q4s+, J7s+, T7s+, 97s+, 87s, A3o+, K7o+, Q8o+, J8o+, T8o+
10% range:
77+,A9s+,KTs+,QTs+,AJo+,KQo]

Suppose also that everyone’s strategy is public.

BU opens to 4. SB raises to (4+x), and BB folds, leaving 1 in the pot. Pot is 9+x, and SB must contribute x to call. Setting aside positional advantage, SB has pot-odds of 9+x:x.

Currently, BU’s range has ~37% equity, so calling is trivial until x~5.5. (This isn’t terribly exciting to learn, perhaps.)

x    BU’s EQ    Range achieving this EQ    Get fold
6    0.4          29%                                27.5%
7    0.43        19%                                52.5%
8    0.47        13%                                67.5%
9    0.5          10%                                75%
10   0.53        8%                                 80%
11   0.55        7.5%                              81.2%
12   0.57        5.3%                              86.8%

x    Win from immediate fold   Equity when called                         EV
6    27.5%*5=1.375              .6*(5+6)-.4*(4+6)=2.6                   3.9
7    52.5%*5=2.625              .57*(5+7)-.43*(4+7)=2.11              4.7
8    67.5%*5=3.375              .53*(5+8)-.47*(4+8)=1.25              4.6
9    75%*5=3.75                   .5*(5+9)-.5*(4+9)=0.5                   4.3
10   80%*5=4                       .47*(5+10)-.53*(4+10)=-0.37         3.6

Which is nice, because 2.75x to 3x is looking optimal, and this doesn’t feel too far off a normal heuristic approach. This post is too small to contain a mathematical model including the possibility of BB calling, BU re-raising, or the effect of being OOP on realising our equity post-flop, except to say that betting any more than 3x is leaving SB in deep and second-favourite when called.

(This post has certainly made me reconsider my habit of 3.5x-ing OOP…)

Reading Andrew’s post, I guess I’m talking about much wider ranges than he has in mind. My guess is the EV curve is the same shape, but it’d be fun to re-run the numbers with some different assumptions.

3for3:

Thanks – that’s really interesting. I can get my head around numbers, but I’m hoping to build my understanding of the more heuristic component.

For instance, mathematically it feels straightforward that villain’s calling frequency should be pretty elastic, but if that’s never true at the table, that’s important to know. I guess that this deep a big blind here or there might feel trivial, even though that difference will get inflated if the pot gets big later in the hand.

I’m wondering, therefore, what the value of 3-betting is here. If villain will call whatever, hero is OOP post-flop, facing a broad – and therefore harder to predict – range. Why not just call, so as not to grow the pot in a volatile situation?

For instance, if hero is 3-betting with the sort of narrow range Andrew talks about, he’s likely to lead the flop. Shouldn’t villain raise with any two? Even on a dry board, he can convincingly represent two pair… Haven’t we turned our top 4% hand into an (overcards/top pair/overpair) bluff catcher most of the time? (With the occasional joy of set-mining with aces…) I guess that’s still true if hero calls pre, but at least the pot is smaller, and hero hasn’t taken the betting lead.

Best outcome I can see from (big, tight) 3-betting is inducing a light 4-bet. Which I suppose is not undesirable – a big pot with a hand that’s often second-best.

Just thinking out loud, really. Here to learn.