Set mining and the 3-8 rule (long with math)

[Jouster777]

I have never seen real justification of the 5-10 rule (or more recently the 3-8 rule) so I thought I’d play with the numbers…I know tldr

I’m looking at a small pocket pair with respect to a standard PFR not a re-raise though that calculation wouldn’t be all that different. This is different than D.Bitel’s much quoted calculation D.Bitel's post because he was looking at situations where you knew you were against a higher PP whereas I assume we are against a top 10% PFR range.

Overall villain’s equity with a 10% PFR range when we flop a set: ~11% but we need to separate equity and actions based on villain’s holding:

Assuming villain raises PF with the top 10% range his down cards will consist of both pocket pairs and high cards. Assume a high card hitting any piece as “TP” or better and whiffing flop as “OC’s”. PPs will be analyzed as QQ+ acts as an high PP and 77-JJ acts as mid PP. S = effective stack

High PP
- Frequency of facing 14.6%
- Chances of lose with a set to an overpair (sees all 5 cards): 18%
- % of villains stack we win (on average): 90%
- EV vs hPP = .82(.9S)-.18(S) = .56S
Mid PP
- Frequency of facing: 19.4%
- Lose to an underpair (w/o set sees 3.5 cards, fold to flop 50%, turn o/w): 10.5%
- % of villains stack we win (on average): 35%
- EV vs mPP = .895(.35S)-.105(S) = .21S
TP+
- Frequency of facing: 22%
- Villain’s equity/actual win rate* with TP+: 6% (actual 4%)
- % of villains stack we win (on average): 60%
- EV vs TP+ = 96(.6S)-.04(S) = .54S
OC’s
- Frequency of facing: 44%
- Villain’s equity/actual win rate* with “OC’s”: 2% (actual ~0%, fold to flop reraise)
- We win one cbet and villain folds to RR with rare calls: 16BB
- EV vs. OC’s = 16BB = ~.16S
* actual rate is an estimation because a significant part of the equity villain has on the flop comes from seeing the river card…he will be priced out of many draws and thus his actual win % is lower than his equity. OC’s include some flopped flushes and straights but we’ll call win rate 0% compensated by villain drawing expensively other times (not as many outs vs. sets as is expected).

EV calculation becomes:
EV (hit set) = .146(EV vs hPP) + .194(EV vs. mPP) + .22(EV vs TP) + .44(EV vs. OC’s)
= .146(.56S) + .194(.21S) + .22(.54S) + .44(.16S)
= .312S
= expect to win 31.2% of the effective stack when we hit a set

We will hit a set on 1/8.5 flops so...
EV(overall)= 1/8.5(.312S)-7.5/8.5(xS) = 0
x=4.16%
Where x = the % of the stack we must call to make this EV neutral

This seems rather low but within the 3-8% rule that many now use. Anyone want to fix math or estimations?...that is if you made it this far.\

How does this change as we vary villain’s PFR range?…don’t know and I’m not doing any more calculations for a while. [img]/images/graemlins/tongue.gif[/img]



Text results appended to pokerstove.txt

47,520 games 0.016 secs 2,970,000 games/sec

Board: 5h 7c Qs
Dead:

equity win tie pots won pots tied
Hand 0: 02.128% 02.13% 00.00% 1011 0.00 { AKs, AKo }
Hand 1: 97.872% 97.87% 00.00% 46509 0.00 { 55 }


---

546,480 games 0.125 secs 4,371,840 games/sec

Board: 5h Ad
Dead:

equity win tie pots won pots tied
Hand 0: 05.758% 05.55% 00.20% 30354 1113.00 { AKs, AKo }
Hand 1: 94.242% 94.04% 00.20% 513900 1113.00 { 55 }


---


16,052,850 games 2.766 secs 5,803,633 games/sec

Board: 5h
Dead:

equity win tie pots won pots tied
Hand 0: 18.456% 18.06% 00.39% 2899368 63396.00 { TT+ }
Hand 1: 81.544% 81.15% 00.39% 13026690 63396.00 { 55 }


---

79,194,060 games 14.859 secs 5,329,703 games/sec

Board: 5h
Dead:

equity win tie pots won pots tied
Hand 0: 10.370% 09.99% 00.38% 7914288 298158.00 { 77+, A9s+, KTs+, QTs+, ATo+, KQo }
Hand 1: 89.630% 89.25% 00.38% 70683456 298158.00 { 55 }


---

20,333,610 games 3.812 secs 5,334,105 games/sec

Board: 5h
Dead:

equity win tie pots won pots tied
Hand 0: 11.888% 11.48% 00.40% 2335233 82060.50 { QQ+, AQs+, AKo }
Hand 1: 88.112% 87.71% 00.40% 17834256 82060.50 { 55 }

[Angrymoog]

Excellent post! I am moving away from set mining, because as your math shows, it is not as profitable as it seems intuitively. Typically now I will consider an approximation of FE to go along with my pp. Especially OOP, I can count on donking certain flops for some EV.

Without being able to outplay your opponent, with the typical raising ranges of most players becoming wider, implied odds for just set mining have drastically reduced.

[Jouster777]

Thanks Angrymooq

Bump