DBitel/Set implied odds revisited (Math calcs)

[floppy]

Hi Everyone,

I took a look at DBitel's post on the implied odds you need to call a raise for set value, and there's some adjustments that I'd like to propose.

First, a caveat: This is establishing a maximum bound, so it assumes that villain will always put all their money in when you hit a set or quads on the flop. Admittedly, this is unrealistic. For example, even if villain's range is only AA/KK, when villain has KK an A will come on the flop along with your set about 1/6th of the time. However, since the original post focused strictly on set and set over set situations, that's what we'll do here.

Primarily, the adjustment is this: Since we're assuming villain has an overpair to ours, we should be calculating the odds of hitting our set (and villain drawing out on us) based on 48 remaining cards, not 50.

So P(set or better on flop) = 1 - C(46,2)/C(48,2) = .12234, or 1/8.17 (7.17 to 1 odds)

If we assume the 18% figure of losing to a set is still correct, DBitel's formula changes this way:

0 = P(~set) * (-1) + Z * (P(set ^ W) - P(set ^ L))
Z = P(~set)/(P(set ^ W) - P(set ^ L))
Z = .8777/(.1223 * (.82 - .18))
Z = .8777/(.1223 * (.64))
Z = 11.21

However, this does not take dead money into account. If we include D, the dead money in the pot when we decide to call the raise, the formula changes to:

0 = P(~set) * (-1) + (Z + D) * P(set ^ W) - Z * P(set ^ L))

which becomes

Z = (P(~set) - D * P(set ^ W))/(P(set ^ W) - P(set ^ L))

(D is in units of the bet we need to call, so if there's $6 dead money in the pot, and we need to call a $6 bet, D = 1)

Typically, we'll be facing a pot-sized 3-bet, so if we make D = 1, here's what we get:

Z = (.8777 - 1 * (.1223 * .82))/(.1223 * (.82 - .18))
Z = (.8777 - .100286)/(.1223 * (.64))
Z = 0.777414/0.078272
Z = 9.93

If you do the same formula for a standard opening raise of 4BB (i.e., D = 1.5/4 = .375), Z = 10.72.

So it looks like the 10% maximum still applies.

As I stated at the top, this is assuming absolutely perfect conditions. Even adding just AK to a KK+ range cuts the chances villain has TP/TK or better on the flop by about 1/2, so it's very rare that this maximum bound should be considered.

There's one other question: I took all the possibilities of us hitting a set or better on the flop, and all the possibilities for villain, and then calculated all possibilities of the turn/river improving each. I came up with villain catching up to us 15.75% of the time, not 18%. How was the 18% figure derived? Pokerstove? My calculations only dealt with sets and quads, so I suppose straights and flushes could account for the difference, but it seems a little far-fetched that the difference could be that big.

If anyone knows of a way to upload files, I'd be happy to provide the Excel file I put together for these calculations.

[Jouster777]

Floppy...nice adjustment of 50 to 48 unknown cards...valid point. However, this really doesn't change the numbers dramatically. Dbitel got 11.7 for Z and you got 11.2 I think.

You can look at dead money formally as you did or incorporate it into the "Z" but finding that Z is <10 does not mean the 10% rule holds because that means that we get a full stack every time we hit. I guess its a question of whether you want the upper bound of your "rule" to reflect real villains or extreme/idealized ones.

The other part of these calculations is that we are not always against an overpair. I tried to make that calculation using a range for villain and I'll post that below so as to not bury it in this post.

[Jouster777]

This is an old post and I don't have the link readily available but I do have the text:

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 calculation 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 higher pairs and high cards. Assume a high card hitting any piece as "TP" or better and whiffing flop as "OC's". PPs will be either sets, overpairs, or underpairs…we'll assume 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 = .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

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.



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 }

[ValarMorghulis]

I'm not sure I follow about the dead money. 2/4. You raise to 16, BB 3bets to 48. You have to call 32. Then you use the 11.2 or 11.7 figure. 11.2*32=360. You have to win 360 every time to make your call profitable, no?

[floppy]

Yeah, it's meant as an absolute maximum, and therefore makes a number of unrealistic assumptions, but they're the same assumptions DBitel made (along with the addition of the dead money calcs).

So when I say "the 10% maximum still applies", it doesn't mean you should be calling 10% of your stack very often. Basically, you'd have to be up against the tightest nit in Nittingham preflop, and then someone reckless enough to continue with KK with an A on the flop.

Your calcs look good. I would guess you get a number on the low end because you're underestimating the % of villain's stack you're getting in some cases. Definitely a good sanity check, in any case.

[floppy]

[ QUOTE ]
I'm not sure I follow about the dead money. 2/4. You raise to 16, BB 3bets to 48. You have to call 32. Then you use the 11.2 or 11.7 figure. 11.2*32=360. You have to win 360 every time to make your call profitable, no?

[/ QUOTE ]

You're calling 32, but there's already 32 in the pot from your initial raise and villain's reraise, so your absolute minimum stack/bet ratio is 9.93 (let's just call it 10), which means the effective stacks before calling the reraise need to be at least 32 * 10 = 320.

So if you or villain have less than 320 before you call the reraise (in villain's case, it's their current stack + the raise amt), there's no way you can profit from playing exclusively from set value, even if villain gets it all in with you every time you flop a set or quads.

In reality, villain's stack should be considerably larger (about 640, maybe even 700; depends on individual villain) for you to consider a call purely on set value alone.