So, I hit on the Shufflemaster Ultimate Texas Hold 'Em machine at my local casino as a decent way to grind out comps in preparation for the poker room opening up, and maybe learn something about short stack AIPF situations/hot and cold equities by accident.

I found the basic strategy table available at Wizard of Odds what says to raise with 22+,A2s+,K2s+,Q6s+,J8s+,A2o+,K5o+,Q8o+,JTo. I've been keeping to that (and not betting trips, much to the consternation of my tablemates), even in situations where one or potentially more of my "outs" are dead.

Feeling an obligation to somehow utilize the exposure of 8 dead cards at a full table, I've decided to analyze how close some of these decisions are, and how they might be swung in either direction (i.e. K4o potentially becoming a raise or 33 becoming a check).

Please feel free to comment on my analysis/conclusions; I haven't really done anything like this before so I'm open to being corrected if I'm looking at things wrong. I make no claim that this produces a +EV game, but merely create this in the interest that I and other UTHE players realize the reduced house edge afforded by knowledge of dead cards.

I'm doing this all by hand too, in the hopes that I might again learn something by accident in the process, but if there's someone out there who can program (unlike me) and wants to actually codify this, by all means go for it. I'm feeling the limitations of stove a bit as well since the dead cards can't be randomized that I know of, and certainly not to any parameters (i.e. "4 cards J or lower not including any 5").



Outline
-Part 1: Establishing minimum equity vs. 2 random cards
-Part 2: Dead overs for borderline K, Q and J-high offsuit hands
-Part 3: Dead set cards for pocket pairs
-Part 4: ????
-Part 5: Profit!

(just kidding, I'm done with 1-3 so far and I'll decide how I want to approach the rest of the hands soon enough)



Part 1: Establishing minimum equity vs. 2 random cards

As we are aware, UTHE is played against a dealer with a random hand that (for better or worse) must call down to the river and takes our blind + ante bets if we check the river. Therefore, we should theoretically want 50.1% equity against a random hand before we start shoveling money in.

However, we must also try to beat the mandatory push (excepting wins on straight+ boards) for the blind bet and occasional push (dealer hand < one pair) on the ante bet, and this is part of why I suspect most basic strategies do not advocate raising hands like Q5o.

equity win tie pots won pots tied
Hand 0: 50.120% 47.96% 02.16% 12071696952 543963330.00 { Q5o }
Hand 1: 49.880% 47.72% 02.16% 12011245188 543963330.00 { random }

Therefore, in order to establish what basic strategy would suggest is the minimum equity required to make the 4x raise, I provide the following stove results for random hands vs. 33, A2s, A2o, K2s, K5o, Q6s, Q8o, J8s, and JTo (the weakest pair, suited and offsuit hands recommended for each high card by basic strategy).

equity win tie pots won pots tied
Hand 0: 53.693% 52.84% 00.85% 6650047464 107459442.00 { 33 }
Hand 1: 46.307% 45.45% 00.85% 5720468052 107459442.00 { random }

equity win tie pots won pots tied
Hand 0: 57.379% 55.51% 01.87% 4657136004 157119696.00 { A2s }
Hand 1: 42.621% 40.75% 01.87% 3418914204 157119696.00 { random }

equity win tie pots won pots tied
Hand 0: 54.929% 52.95% 01.98% 13327296792 498698646.00 { A2o }
Hand 1: 45.071% 43.09% 01.98% 10846174716 498698646.00 { random }

equity win tie pots won pots tied
Hand 0: 53.212% 51.24% 01.97% 4299180380 165437728.00 { K2s }
Hand 1: 46.788% 44.82% 01.97% 3760233764 165437728.00 { random }

equity win tie pots won pots tied
Hand 0: 53.314% 51.25% 02.06% 12901056312 518533842.00 { K5o }
Hand 1: 46.686% 44.63% 02.06% 11232744804 518533842.00 { random }

equity win tie pots won pots tied
Hand 0: 53.613% 51.68% 01.93% 4336038248 162211338.00 { Q6s }
Hand 1: 46.387% 44.45% 01.93% 3729828676 162211338.00 { random }

equity win tie pots won pots tied
Hand 0: 53.600% 51.93% 01.67% 13071363540 420169800.00 { Q8o }
Hand 1: 46.400% 44.73% 01.67% 11259165660 420169800.00 { random }

equity win tie pots won pots tied
Hand 0: 54.016% 52.31% 01.70% 4389099188 142969788.00 { J8s }
Hand 1: 45.984% 44.28% 01.70% 3715250836 142969788.00 { random }

equity win tie pots won pots tied
Hand 0: 55.248% 53.83% 01.42% 13548473136 357853722.00 { JTo }
Hand 1: 44.752% 43.33% 01.42% 10906688220 357853722.00 { random }

A cursory analysis of hands just below the "cutoff" (22, J9o, J7s, Q7o, Q5s, K4o) indicates equities below the low point we see here from 53.212% for K2s. Therefore, we will proceed as if 53.212% is our minimum for making a 4x raise in light of stove analysis of all basic strategy hands (and all borderline hands), taking into account one or more potentially relevant dead cards.



Part 2: Dead Overs For Borderline K, Q and J-high Offsuit hands

I decided to start with this category since it seems less of a grind than the others. Here, we will begin with K4o, Q7o and see what we learn about their equities when one or more overcards are dead.

K-high Offsuit:

equity win tie pots won pots tied
Hand 0: 52.343% 50.25% 02.10% 2997910 125288.00 { K4o }
Hand 1: 47.657% 45.56% 02.10% 2718378 125289.50 { random }

Dead: As

equity win tie pots won pots tied
Hand 0: 53.159% 51.03% 02.13% 2928461 122199.00 { K4o }
Hand 1: 46.841% 44.71% 02.13% 2565969 122200.50 { random }

Dead: As Ac

equity win tie pots won pots tied
Hand 0: 53.874% 51.73% 02.15% 2959728 123051.50 { K4o }
Hand 1: 46.126% 43.98% 02.15% 2516417 123054.00 { random }

As we can see for K4o, killing one ace gets it pretty close to our 53.212% mark, and killing two pushes it over by a decent margin. HOWEVER, this does not apply if even one of our Ks or 4s is dead:

Dead: As Ac Ah Ad Kh

equity win tie pots won pots tied
Hand 0: 51.922% 50.30% 01.62% 2968991 95727.00 { K4o }
Hand 1: 48.078% 46.46% 01.62% 2742164 95727.50 { random }

Dead: As Ac Ah Ad 4h

equity win tie pots won pots tied
Hand 0: 52.335% 50.10% 02.23% 2979853 132865.00 { K4o }
Hand 1: 47.665% 45.43% 02.23% 2702084 132868.00 { random }

K3o is sufficiently weaker than K4o to require 3 dead Aces to barely get over 53.212%:

Dead: Ac Ad Ah

equity win tie pots won pots tied
Hand 0: 53.303% 51.16% 02.15% 3259242 137053.50 { K3o }
Hand 1: 46.697% 44.55% 02.15% 2838385 137056.00 { random }

And K2o can't quite make it, even with 4 dead aces.

-K-high Offsuit Conclusions:
--K4o becomes a raise with two dead Aces, K3o becomes a raise with three dead aces, but ONLY if both pair cards are live.

---

Q-high Offsuit:

equity win tie pots won pots tied
Hand 0: 51.746% 49.88% 01.87% 2982848 111741.00 { Q7o }
Hand 1: 48.254% 46.39% 01.87% 2774081 111742.00 { random }

Dead: Kc Kd

equity win tie pots won pots tied
Hand 0: 53.512% 51.63% 01.89% 2962174 108441.00 { Q7o }
Hand 1: 46.488% 44.60% 01.89% 2559145 108442.50 { random }

Dead: Ac Kd

equity win tie pots won pots tied
Hand 0: 53.827% 51.94% 01.89% 2887058 105311.00 { Q7o }
Hand 1: 46.173% 44.28% 01.89% 2461565 105313.50 { random }

Q7o steps over the line when we remove at least 2 Ax/Kx from play.

Dead: Kh Kd Kc

equity win tie pots won pots tied
Hand 0: 53.249% 51.16% 02.09% 2923032 119701.00 { Q6o }
Hand 1: 46.751% 44.66% 02.09% 2551784 119704.50 { random }

Q6o barely makes it with 3 Ax/Kx gone.

Dead: Kc Kd Kh Ks

equity win tie pots won pots tied
Hand 0: 52.376% 50.10% 02.28% 2844560 129365.50 { Q5o }
Hand 1: 47.624% 45.35% 02.28% 2574823 129367.50 { random }

Dead: Kc Kd Kh As

equity win tie pots won pots tied
Hand 0: 53.230% 50.95% 02.28% 2830192 126672.50 { Q5o }
Hand 1: 46.770% 44.49% 02.28% 2471334 126673.00 { random }

Q5o requires a minimum of 4 dead overs, one of which must be an ace.

Dead: Kc Kd Kh As Ah Ks Ad Ac

equity win tie pots won pots tied
Hand 0: 53.161% 50.71% 02.46% 3074434 149029.00 { Q4o }
Hand 1: 46.839% 44.38% 02.46% 2691103 149032.00 { random }

and Q4 can't quite get there, even with every over in the deck gone.

---

Dead: Qc Kc Kd Kh Ac Ad Ks

equity win tie pots won pots tied
Hand 0: 53.411% 51.86% 01.55% 2930373 87818.00 { Q7o }
Hand 1: 46.589% 45.04% 01.55% 2544931 87819.00 { random }

Dead: 7c Kc Kd Kh Ac Ad

equity win tie pots won pots tied
Hand 0: 53.600% 51.71% 01.90% 2964228 108687.00 { Q7o }
Hand 1: 46.400% 44.51% 01.90% 2551490 108687.50 { random }

Similar to what we saw with the K-high hands above, having dead pair cards is devastating for the offsuit Q-high hands, requiring 5-6 overs be dead for Q7o to be playable.

Dead: Qc Kc Kd Kh Ac Ad Ks Ah

equity win tie pots won pots tied
Hand 0: 53.226% 51.46% 01.76% 2837400 97255.00 { Q6o }
Hand 1: 46.774% 45.01% 01.76% 2481677 97257.00 { random }

Dead: Kc Kd Kh Ac Ad 6c Ah

equity win tie pots won pots tied
Hand 0: 53.595% 51.44% 02.16% 2892016 121281.00 { Q6o }
Hand 1: 46.405% 44.25% 02.16% 2487826 121283.00 { random

Q6o requires 6-7 dead overs with just one missing pair card.

Dead: Kc Kd Kh Ks Ac Ah As Ad Qc

equity win tie pots won pots tied
Hand 0: 51.795% 49.95% 01.85% 2708587 100331.00 { Q5o }
Hand 1: 48.205% 46.36% 01.85% 2513912 100332.50 { random }

Dead: Kc Kd Kh Ks Ac Ah As Ad 5c

equity win tie pots won pots tied
Hand 0: 52.505% 50.10% 02.41% 2707693 130125.00 { Q5o }
Hand 1: 47.495% 45.09% 02.41% 2436916 130125.50 { random }

and Q5o remains unplayable minus either pair card, even with all 8 overs dead. Also, all Q7o-Q5o hands with two missing pair cards are unplayable, even minus every over in the deck. NOTE: I am aware discussion of dead overs/pair cards in excess of 8 is academic given that there are 5 total positions at the table.

-Q-high Offsuit Conclusions:
--Raise Q7o with at least 2 dead overs (5 if a 7 is also dead, 6 if a Q is also dead)
--Raise Q6o with at least 3 dead overs (6 if a 6 is also dead, 7 if a Q is also dead)
--Raise Q5o with at least 4 dead overs that include an A (ONLY if all pair cards are live).

---

J-high Offsuit:

equity win tie pots won pots tied
Hand 0: 53.264% 51.65% 01.62% 3026856 94777.00 { J9o }
Hand 1: 46.736% 45.12% 01.62% 2644267 94779.50 { random }

equity win tie pots won pots tied
Hand 0: 51.502% 49.73% 01.78% 3076632 109909.50 { J8o }
Hand 1: 48.498% 46.73% 01.78% 2890837 109911.00 { random }

While J8o is, as expected, below the semi-arbitrary 53.212% benchmark we have set, J9o is interestingly just above it. I suspect its exclusion from the BS table for automatic 4x hands involves the reduced frequency at which it pushes with a random hand vs. Q-high and better hands.

As such, we will continue on the assumption that J9o becomes a raise with just one dead overcard (ideally the A so as to interfere with less straights).

Dead: Qc Kd

equity win tie pots won pots tied
Hand 0: 53.529% 51.74% 01.79% 2960502 102577.50 { J8o }
Hand 1: 46.471% 44.68% 01.79% 2556617 102577.50 { random }

J8o becomes a raise with 2 dead overs (minimum one K or A).

Dead: Qd Qc Ks Qh

equity win tie pots won pots tied
Hand 0: 53.459% 51.47% 02.00% 2835057 109911.00 { J7o }
Hand 1: 46.541% 44.55% 02.00% 2453945 109913.00 { random }

J7o requires 4 dead overs with 1 K or better.

Dead: Ac Kd Qh Ad Kh Qs

equity win tie pots won pots tied
Hand 0: 53.370% 51.07% 02.30% 3254788 146447.50 { J6o }
Hand 1: 46.631% 44.34% 02.30% 2825354 146449.00 { random }

J6o requires 6 of the 12 overcards be dead, with a minimum of one A. As one might expect, J5o doesn't make it even with all Ax/Kx dead.

---

The exact combinations are a more involved than I care to map out manually, but based on casual observation regarding JXo with dead jacks/pair cards:

J9o - 4-5 dead overs
J8o - 5-6 dead overs
J7o - 6-7 dead overs

Exactly which overs are dead also has a pretty significant impact, considering that QQQQ kills all straights above J-high and that J9o actually only needs 3 overs dead to beat a missing J, but they have to be the right combination (i.e. AKQ does it, AKK doesn't) to minimize interference with potential straights or otherwise more dead overs are needed -- I presume to improve the high-card equity of J9o vs. random hands.

Also, for J8o and J7o, having AAAA or KKKK among the dead overs chops a solid couple of % off your equity I guess because it kills 4-card broadway. I feel like in thin spots with J high and a dead pair card + handful of dead overs, if the last card or couple of cards interfere with our straight possibilities, that could be enough to take it from barely above 53.212% to just below.

-J-high Offsuit Conclusions:
--Raise J9o with one dead over (4-5 with dead pair cards)
--Raise J8o with two dead overs (5-6 with dead pair cards)
--Raise J7o with four dead overs (6-7 with dead pair cards)
--Raise J6o with six dead overs (unplayable with dead pair cards)
---Dead overs must be a mix of A/K/Q, preferrably more A and K than Q but 4x of any rank has significant negative impact on our equity



Part 3: Dead set cards for pocket pairs

Dead: 6d 6h

equity win tie pots won pots tied
Hand 0: 55.233% 54.76% 00.48% 2963086 25729.00 { 66 }
Hand 1: 44.767% 44.29% 00.48% 2396723 25729.50 { random }


Dead: 5h

equity win tie pots won pots tied
Hand 0: 56.219% 55.60% 00.62% 3244884 36233.50 { 55 }
Hand 1: 43.781% 43.16% 00.62% 2518999 36233.50 { random }

Board:
Dead: 4s 2d 7c 8d Jh Qs

equity win tie pots won pots tied
Hand 0: 53.250% 52.53% 00.72% 3075184 42336.00 { 44 }
Hand 1: 46.750% 46.03% 00.72% 2694626 42336.50 { random }


Dead: 3h Ac Kd Qh Js Tc 9d 8h

equity win tie pots won pots tied
Hand 0: 51.939% 51.15% 00.79% 2802475 43547.00 { 33 }
Hand 1: 48.061% 47.27% 00.79% 2589997 43547.50 { random }

Dead: Ac Kd Qh Js Tc 9d 8h 7s

equity win tie pots won pots tied
Hand 0: 53.651% 52.70% 00.96% 3468995 62952.50 { 22 }
Hand 1: 46.349% 45.40% 00.96% 2988325 62953.50 { random }

---

AA-66: playable with 2 dead set cards.
55: playable with 1 dead set card.
44: playable with 1 dead set card if at least 5 other exposed cards are not a 4.
33: must be live, unplayable with 1 dead set card.
22: barely playable if set outs are fully live and 1-card straights are mostly live



Initial reflections

A lot of these hands that go from check to play are close enough to the fulcrum that playing them all will be like giving an eightball to the carnie operating the variance rollercoaster.

I think, however, that examining the rest of the hands in the basic strategy chart taking into consideration dead pair cards/flush possibilities/etc. will do a lot to even things out. -- Maybe not; just checked and it looks like some Tx, 9x and even 8x hands may become raises with the right combination of overcards.

It's not that that keeps you from raising hands with less than 53% equity, it's that by raising 4x with a small edge you forgo the chance to raise 2x or 1x with a significantly higher edge, and not raise when you are less than 50% on a given board.

Anyways, not to discourage you, but I think most of this has been done before (and much better), and it only shaves 1%ish off the house edge, which is not very significant on a game with such significant variance (a decent amount of the return is in the straight flush/royal flush blind bets). If you are really looking to play it for better comps, then the actual play of the game is very minor compared to when you play, where you play, and how often you play. As a general hint, if you have never asked "What did you have my average bet for this session as?", then you should do it at least once at every casino you play at.

If you're looking for a way to reduce the house edge in table games using known hole cards, there are much better games to analyze.

This a lot of work to do when the game has been examined by an up and coming gaming mathematician with neighbor collusion. He's shown the game to be still -EV even with all hole info of your fellow players. http://discountgambling.net/2010/01/...-texas-holdem/

He didn't discuss the effects of overs in opponents' hands, however. But I would assume the effect isn't enough to fully eliminate the house edge.

It's an electronic game, this is why the entire table save the dealer plays face-up. No pit boss to talk to, no dealer making mistakes in my favor. There are probably better games, but none that offer practical experience that can be applied to heads up/BvB play in NLHE, or evaluating changes in equity with exposed cards (which will be useful if/when I ever decide to take up stud).

Good point regarding when we get our money in and at what edge; I hadn't realized the effect that had on the marginal hands preflop -- instead, I had mostly been applying that to flop play where I was getting smacked betting my weak bottom pairs, so I decided to start checking more of them (mostly the ones with no high card potential or redraws) and placing the river bet instead.

Originally, I thought since I would never c/f a pair on the river, I might as well get the money in on the flop and hope it either holds up or improves to the best hand. Thinking about it more in depth now, my weak bottom pairs were likely flipping, even against a 100% range. So, it makes sense to get the money in at a higher edge (especially being offered 4 to 1 when the board pairs), even if the amount bet is less overall.

Thanks for the link, good stuff -- Am aware this still produces an -EV game. If I'm going to be robbed and the guy with the gun offers to only take half of what's in my wallet, I don't mind taking him up on the offer.

I feel like I should read up on propokertools (besides using the Omaha simulator to show friends who call off with QQ98r pre how badly I have them crushed with KKQJ double-suited), since it seems like they would have some way to randomize the dead cards or maybe assign them paramaters like I mentioned above (i.e. 4 random cards between J and 4 that are not clubs or the like).

I also thought briefly that the most comprehensive way to go about this would be to take all 169 starting hands and then run them individually with each of the remaining 50 dead cards and assign some sort of point value + or - for each card. This would nominally be 8450 runs (something of a project, but doable), except for the complicating factor that the effect of dead cards on a given hand's equity isn't necessarily linear, as shown:

Dead: Ac

equity win tie pots won pots tied
Hand 0: 64.583% 63.91% 00.67% 1152882998 12130167.50 { AhKh }
Hand 1: 35.417% 34.74% 00.67% 626768931 12130167.50 { random }

Dead: Ac Ad

equity win tie pots won pots tied
Hand 0: 61.657% 61.15% 00.50% 945581952 7769911.50 { AhKh }
Hand 1: 38.343% 37.84% 00.50% 585088737 7769911.50 { random }

Dead: Ac Ad As

equity win tie pots won pots tied
Hand 0: 58.199% 57.90% 00.30% 764652552 3988923.00 { AhKh }
Hand 1: 41.801% 41.50% 00.30% 548091081 3988923.00 { random }

From 67.045 to 64.583, 61.657, then 58.199, the drops go from 2.462% to 2.926% and finally 3.458%.

This teaches me a couple things I had never really thought of regarding hand equity; among them, that a decent chunk of it comes from a given holding's ability to make strong 5-card hands as opposed to simply having big cards and expecting them to hold up just because they're "supposed to".

So, while I'm checking out PQL, I think I'm going to start looking at the less marginal hands in the BS table and work my way down, keeping an eye on the effect of copies/overs/flush cards and all.

I didn't read the link posted, but before you put a ton of work into this, you need to calculate what's called the Effect of Removal and make sure that the EoR is large compared with the house edge.

For instance, baccarat has a non-zero EoR and you could theoretically count baccarrat. But the EoR is so small that you're looking for the blackjack equivalents of TC +17 before you have an edge.