I came across an interesting scenario while working with PokerSnowie today.

## Scenario

6max, 100bb effective

• Preflop:
• LJ opens for 2.5bb
• Action folds to BB who calls
• Flop:
• Flop comes 8d8c7s (rainbow)

I’m interested the following questions:

1. does BB ever donk here?
2. what does LJ continue with?

## Preflop Charts

Here are the preflop charts from PS (these are approximate, and doesn’t differentiate between some mixed 3betting in the BB):

• LJ: raise 15% of hands: 66+, A2s+, K9s+, QTs+, JTs, ATo+, KJo+

• BB: call with 18% of hands: 22-TT, A2s-A9s, K4s, K6s+, Q8s+, J8s+, T8s+, 97s+, 86s+, 75s+, 64s+, 53s+, ATo-AQo, KJo+, QJo

## Question 1: Does BB Donk?

BB and LJ have about the same number of combos of the nuts in their range (actually, BB 3bets 88 every once and a while), and since BB’s range is larger than LJ’s range, 88 and 77 make up a larger proportion of LJ’s range than BB’s range.

However, BB also has a 98s-A8s, for an extra 12x trips, and 87s,97s,K7s,and A7s, for an extra 12x second pair. While this doesn’t give BB the equity advantage, it does mean that BB is connecting with this flop in relatively nutty ways that LJ can’t, and that means it might make sense to develop a donking range:

Hypothesis: Given that BB has a nuts advantage, BB will donk bet on this flop with 88, 77, some 8x, as well as some bluffs.

### What hands will BB donk with?

#### My guess

I think that BB will donk with 88, 87, 77, and maybe a couple other 8x and 7x hands. I also think that T9 and 65 might bet here. I’m not sure what bluffs BB would add.

#### PokerSnowie

PokerSnowie advocates that BB donk 1/2 pot about 7% of the time:

There are some surprising things here:

1. 88 is only bet 25% of the time

Since we will mainly be checking this keeps our checking range strong, allowing us to x/r.

2. 77 is almost never bet (about 7% of the time)

Again, allows us to x/r effectively. I’m surprised we aren’t betting out here more though.

3. No 8x are bet.

This is surprising because flopping trips here is pretty nutty. However, given the ability to x/r, this actually makes sense.

For instance, given Q8h, what turn cards are we afraid of? JTs gets there on a 9, and even then we have around 14% equity. Obviously overpairs might make a boat, but this also shouldn’t be too likely. With no flush draws on the board, we are pretty secure against any turn cards, and giving LJ a chance to cbet is probably higher EV.

4. A bunch of suited queens are bet: QJs, QTs, and Q9s w/ a BDFD are all bet here 100%.

This surprised me, and is actually why I’m writing this. I want to figure out what these QXs are doing in the donking range.

I also noticed that these hands are being cbet by LJ, so there is something going on with these hands.

5. BB’s donk range has more bluffs than value bets.

There are about 8.5 combos of the Qx bluffs and only 3-4 combos of ‘made hands’ (including a 7 or an 8). PS is betting 1/2 pot here, so I guess by combining fold equity along with potential to improve and getting paid when we have the nuts, this might make sense? But it’s really weird that we have more bluffs here than value bets.

### Why donk with Qx?

Here are the possible reasons I can think of:

#### Blockers

The Qx hands are all blocking over-pairs 99-QQ

#### Fold Equity

These hands benefit from folds from hands like Ax and Qx, and maybe hands like 66. Hands like KTs-KQs don’t benefit from fold equity as much because there are fewer overcards that beat us. QT getting KJ to fold is much better than vice versa.

#### Potential To Improve

First, all of these hands are BDFDs and can possibly make nut straights.

### Revising Hypothesis

I’m revising my hypothesis:

Hypothesis 2.0: On MMM paired flop (i.e., three medium cards from 6-9 with a pair) the BB has the nuts advantage and will donk bet on this flop with a small percentage of nutty made hands like 88, 77, and some weaker made hands, as well as around 1.5x-2x as many bluffs that (a) have good blocking potential, (b) benefit from folds, and (c) have a potential to improve.

I can test this on other flops to see if the pattern continues.

## Question 2: What does LJ continue with here?

### My guess

LJ has an equity advantage but not the nuts advantage. This is a relatively dynamic board: any 8x is clearly nutted and probably can bet to the river for value. However, a paired 7 is vulnerable. BB has some OESDs (65s, T9s) that probably won’t fold, but we still have good equity against those.

Because of the nuts advantage we don’t want to bet too wide because of the risk of x/r. Therefore I think we will want to bet about half of our range about 1/2 pot, including:

• A8, 88
• A7
• 99+
• Some bluffs:
• Question: what hands make the best bluffs?
• Answer: we kinda already answered this above (and I also already saw this from PokerSnowie, so I cheated): the QTs+ hands w/ a BDFD make very good bluffing candidates because they (a) benefit from folding out Kx and Ax hands, (b) have potential to improve on later streets (straights, flushes, pairing the turn/river), and (c) block some overpairs that BB might have.

PokerSnowie suggests cbetting 1/2 pot with only 23% of hands:

Lojack's CBetting range: The lojack continues with any 7 or 8, as well as most Qx and Kx holdings. Ax doesn't cbet, presumably because they aren't as vulnerable to an A on the turn/river

This is a polarized range. Overpairs and A-high don’t bet, but nutted hands like 88, 77, A8 and A7 continue.

## Experiment design

Unfortunately I don’t have access to a solver, so all experiments will be by hand using PokerSnowie (i.e., I can’t script PioSOLVER).

To test my hypothesis I want to

1. Try a number of paired MMM flops
• Does rainbow matter?
• Does adding a gap matter?
• Does making the unpaired higher than the paired cards matter?
2. Try some MML paired flops
• Does changing the unpaired card to a low card matter?

To test this, I am going to investigate two of each of the following flop textures:

1. MMM paired/rainbow/no gaps (e.g., 887r)
2. MMM paired/rainbow/1 gap (e.g., 886r)
3. MMM paired/rainbow/unpaired>paired (e.g., 877r)
4. MMM paired/two-tone (e.g., 8s8d7s)
5. MML paired/rainbow (e.g., 883r)

I will randomly generate the ranks (when there are more than two possible flops). The suits will be chosen by hand since they don’t matter.

import random
l = [2,3,4,5]
m = [6,7,8,9]
h = [10,11,12,13]
def make_paired_flop(ranks='mm', paired_is_higher=True, gap=False):
while True:
paired_card = random.choice(eval(ranks[0]))
unpaired_card = random.choice(eval(ranks[1]))
if paired_card == unpaired_card: continue
if paired_is_higher and paired_card < unpaired_card: continue
if gap ^ ((paired_card - unpaired_card) ** 2 == 1):
return "{}{}{}".format(paired_card, paired_card, unpaired_card)


I explore these: