Calling a RR for Set Odds?

In the war for information, it makes sense to analyze some of the more basic aspects of the game to ensure that we are making the correct decisions. One common situation is facing a RR with a small pair wondering what stack size is suitable to call solely for set odds. We will present three primary ranges with several subranges.

Primary Ranges:
-JJ+,AK
-TT+,AQ+
-99+,ATs+,AJo+,KQ

Subranges:
QJs-54s (SC)
QTs-75s (SG)
J9s-96s (S2G)
55-22 (Low Pair)

We assume that postflop villain stacks off with top pair+ and any draw that has at least 8 outs at the minimum on any flop. The following numbers represent the percentage of the following ranges that can stack off.

1.) JJ+,AK (44.7% + 1.4%) -> (46.1%)

with QJs-54s (37.7% + 9.9%) -> (47.6%)
with QTs-75s (31.9% + 12.1%) -> (44.0%)
with J8s-96s (29.9% + 12.5%) -> (42.4%)

with 55-22 (34.1% + 1.2%) -> (35.3%)

2.) TT+,AQ+ (48.1% + 1.9%) -> (50.0%)

with QJs-54s (36.5% + 8.3%) -> (44.8%)
with QTs-75s (32.0% + 8.3%) -> (40.3%)
with J8s-96s (30.4% + 10.9%) -> (41.3%)

with 55-22 ( 39.6% + 1.7%) -> (41.3%)

3.) 99+,ATs+,AJo+,KQ (40.3% + 3.1%) -> (43.4%)

with QJs-54s (34.1%+ 7.2%) -> (41.3%)
with QTs-75s (31.1% + 9.0%) -> (40.1%)
with J8s-96s (29.9% + 9.6%) -> (39.5%)

with 55-22 (35.9% + 2.9%) -> (38.8%)


We flop a set in 3a 11.2% of the time. 41.3% * 11.2% = the times I stack him = 4.6% of the time. We need 21:1 to achieve basic breakeven in this situation. If we need to call an additional 9bb (using a 3bb-12bb RR), we need to make sure we are deep enough (9bb+21) to sufficiently break even. This equates to ~200bb at the point of basic breakeven in situation 3A assuming, of course, no shenanigans.

4-Betting Small Pairs vs Aggressive 3-Bets

First we look at several CALL ranges of the 4bet. For simplicity, we assume that we 4bet all in and the opponent is simply presented with the choice of calling or folding.

The equity of {44} vs the following ranges:

KK+,AK (2.1%): 38.59%
QQ+,AK (2.6%): 35.14%
JJ+,AK (3.0%): 32.72%
TT+,AK (3.5%): 30.93%
TT+,AQ+ (4.7%): 36.64%
99+,AQ+ (5.1%): 35.05%
99+,AJ+ (6.3%): 38.41%
88+,AT+ (8.0%): 39.41%

As a result, of the equity distribution, we will (again for simplicity) use three different equities to analyze: 30%, 35% and 40%. We will analyze the stack sizes: 50bb, 75bb, 100bb, 150bb and 200bb from a risk point of view.

50bb

30%: .3(104bb)-47bb = -15.8bb
35%: .35(104bb)-47bb = -10.6bb
40%: .4(104bb)-47bb = -5.4bb

75bb

30%: .3(154bb)-72bb = -25.8bb
35%: .35(154bb)-72bb = -18.1bb
40%: .4(154bb)-72bb = -10.4bb

100bb

30%: .3(204bb)-97bb = -35.8bb
35%: .35(204bb)-97bb = -25.6bb
40%: .4(204bb)-97bb = -15.4bb

150bb

30%: .3(304bb)-147bb = -55.8bb
35%: .35(304bb)-147bb = -40.6bb
40%: .4(304bb)-147bb = -25.4bb

200bb

30%: .3(404bb)-197bb = -75.8bb
35%: .35(404bb)-197bb = -55.6bb
40%: .4(404bb)-197bb = -35.4bb

By looking at the sum of these results, we are able to gather some values previously hidden. At 35% at 100bb, we are risking 25.6bb by going all in as a response to a light 3-bet with a hand such as 44. The pot currently contains 21bb. This means that we must be right (and he must fold) 55.3% of the time.

To put this in perspective, if villain is calling AI with 4.7% (AQ+,TT+), then he must be 3-betting a total sum of 10.0% of hands and folding 5.3% of them. While possible, this might be tenuous and it helps immensely to know the portion that we are risking so that we can formulate our own pot odds for a variety of different equities in this situation that are closely grouped together.

Equity, Risk, Calculated Decisions