charliepatrick
Joined: Jun 17, 2011
• Posts: 1321
May 31st, 2017 at 3:18:26 PM permalink
Quote: gordonm888

...I'm not sure that Charlie considered the cases where the dealer keeps discarding and has only 2-4 cards with which to make a poker hand...

Sorry if it wasn't clear, the LHS of the table was for dealer having 6-card hands and the RHS where the dealer threw away the first card.

In the second case the dealer could land up with the last card, so can play with 5 or all the way down to 1. This is why you see hands such as K2 (which requires four high cards then the 5th card to be a 2 and the 6th card to be a K) and "K" (which requires five high cards and the 6th card to be a K) appearing in the right hand half of the table.

All I've done is go through all the permutations of 5 4 3 2 and 1 card hands and rank the 5-card straights, 5-card flushes (estimated), 2P or more, pairs, High cards (highest two), one High card only.

You may have noticed that some hands, such as "J5", which the dealer can make in various ways J5432 J543 J542 J532 J532 J54 J53 J52 J5, cannot be made by the player. For instance (cf below) "J5" or higher is made 80.04% when the dealer has a Q up-card.

My earlier estimates ignored your own cards or the dealer's first card. A quick look, now taking into account the dealer's up-card, is you need J9 vs T or 9, but any Jack-high vs J or Q (and I'm guessing A and K). Thus there may be some weird borderline cases where higher hands fold (as they leave more low cards for the dealer thus increasing the chances of 5-card hands etc) and lower hands using up more low cards play.
gordonm888
Joined: Feb 18, 2015
• Posts: 613
June 2nd, 2017 at 12:50:03 PM permalink
Quote: charliepatrick

Sorry if it wasn't clear, the LHS of the table was for dealer having 6-card hands and the RHS where the dealer threw away the first card.

In the second case the dealer could land up with the last card, so can play with 5 or all the way down to 1. This is why you see hands such as K2 (which requires four high cards then the 5th card to be a 2 and the 6th card to be a K) and "K" (which requires five high cards and the 6th card to be a K) appearing in the right hand half of the table.

charlie, you are amazing. You make the rest of us look like we are asleep. I was grinding away at my calculations for a dealer upcard of 5 and when I saw you had already posted "power rankings" of poker hands for all the 9-A upcards, I was very surprised -I mean "jaw on floor" surprised. I think I know what approach you must have done - it took a lot of insight and work.

I've been using your results as an aid in writing my composition-dependent hand calculator. I learned from your work that, versus 9-Ace, all of the decision points involve high card hands, because with a pair or higher your results indicate that the player should Bet 2X. So, versus 9-Ace, I've only had to consider 6-card dealer hands that have a possibility of decomposing down to a High Card hand (i.e., hands such as 55-AKT4 cannot decompose to less than 55 pair). So, that has saved me a lot of time. Also, I'm using your results to error check my work. My answers are coming out slightly different than what you have posted -so I would appreciate any comments you have when I post my results.
gordonm888
Joined: Feb 18, 2015
• Posts: 613
June 2nd, 2017 at 2:05:55 PM permalink
Quote: beachbumbabs

Nice work, Gordon.

Re : 9 and up.

I'm pretty sure the dealer can get down to a single, unexposed card per the rules, not just 2. They expose all but the last in turn. If each is 9 or above, then for them to always qualify, they play the last card. I may have misunderstood, but that's how it reads to me.

After re-reading the (poorly-worded) rules again, I agree with you. The minimum dealer hand is ONE card.

And, LOL, I just finished an analysis of the case in which the dealer upcard is an Ace but I assumed the minimum length of the dealer hand was TWO cards. LOL.

I think I'll post my result anyway - and then work at modifying the calculator to allow a minimum length of 1 card for the dealer. I do this with the caveat that the dealer's hand will decompose down to one single card about 3.5% of the time when the upcard is 9 or higher. So I realize that my 2-card-minimum calculation results are going to be off by 1-3 %.

Composition dependent calculations versus Ace (upcard)

1. My calculator considers the rank and suit of all seven cards -6 player cards and one dealer card.
2. Versus Ace, the dealer hand is decomposed by discarding all cards that are turned over that are 9 or higher, until an 8 or lower is turned over, or until the dealer has only two cards left. (The minimum length of the dealer's hand should have been ONE card -so I will re-do the calculation this weekend, with that modification.)
3. I have just written this calculator for the "vs. Ace" scenario, so there may be errors in this.

Versus Ace
BET 2X with AJxxx (High Card hand) or higher.
BET 1X with Q9xxx (High Card hand) or higher.
Fold with Q8xxx or lower.

The AJ hands can illustrate an important point -the benefit of having low cards (2-8) in your hand. Low cards make it more probable that the dealer's hand will decompose to 4 cards or less, thus weakening the dealer's hand.

Player hand
EV for Bet 1x
EV for Bet 2x
AJT94-3
-0.4834
-0.4768
AJ643-2
-0.4638
-0.4493

Notice that the player hand with a lower poker value, AJ643-2 v A has a higher EV than AJT94-3 v A by about 0.028 for the BET 2x. The hands have 4 identical cards, but the higher-performing hand has a 62 for its last two cards while the last two cards of the lower-performing hand are a T9. The presence of the extra two low cards in the player's hand changes the probability of how the discard process will proceed and results in a higher frequency of shorter dealer hands (2-4 cards.)

When I include the possibility of the dealer's hand shrinking to 1 card this should become even more pronounced. For instance, AJT94-3 has only 2 low cards, so the dealer's hand should shrink to one card about 2.6% of the time. The hand AJ643-2 had 4 low cards, so the dealer's hand will shrivel to 1 card about 4.0% of the time!

Another point. When facing High Cards (9-A) it does not help your hand to hold a Match card (in this case, an Ace) or cards that are the same suit as the upcard. Indeed, your EV is slightly reduced when you hold cards that match the rank and suit of the upcard. That makes sense when you remember that the upcard will be discarded and that the rank (in this example, Aces) and suit are now less likely to pair or flush-up in the dealer's remaining 5-card hand.

Okay, back to the drawing board on my calculator.
Last edited by: gordonm888 on Jun 2, 2017
Joined: Feb 24, 2011
• Posts: 2052
June 2nd, 2017 at 2:20:10 PM permalink
Has anyone seen this game outside of Vancouver?
charliepatrick
Joined: Jun 17, 2011
• Posts: 1321
June 2nd, 2017 at 2:58:11 PM permalink
Interesting that your work confirms that sometimes having lower cards increases your chances. So, like original 5-card poker, there might be a cutoff where you always raise, and cutoff where you always fold and between depends on what you have.

I got calling with Jack-high as if the dealer gets to 1 or 2 cards he has to pull A K or Q (or make a pair) as the last card to beat you. Also remember with 4-card or fewer the Dealer can't make a Straight or Flush.
gordonm888
Joined: Feb 18, 2015
• Posts: 613
June 2nd, 2017 at 3:56:18 PM permalink
Quote: charliepatrick

Interesting that your work confirms that sometimes having lower cards increases your chances. So, like original 5-card poker, there might be a cutoff where you always raise, and cutoff where you always fold and between depends on what you have.

Yes. I've been expecting that to happen but haven't yet seen it.

Quote: charliepatrick

Also remember with 4-card or fewer the Dealer can't make a Straight or Flush.

Yep. No straights, flushes or Ties with 4 cards or less. I did manage to get some of the rules correct!

Quote: charliepatrick

I got calling with Jack-high as if the dealer gets to 1 or 2 cards he has to pull A K or Q (or make a pair) as the last card to beat you.

I think the dynamics of this discard progression are fascinating. First, it must be dramatic to sit at the table and see multiple cards being turned up and discarded -it sounds like fun. Multiple "reveals." But consider a hand with a 9-A upcard - over 10% of the time the remaining 5 cards will be 4 high cards and 1 low card. With 4 H+1 Low that are facedown, here is what could happen:

- 20% of the time that hand will decompose to ONE card -and a card that is in the range of 2-8. Whether its a 2 or 8, that dealer hand will literally have a 0% chance of winning -and it has nothing to do with the cards in your hand.
- 20% of the time that hand will decompose to TWO cards - one High and one low - so it will be a weak high card hand somewhere in the range of A-8 to 9-2.
- However, 20% of the time that hand will play as a five card hand -with 4 high cards. It will usually be a high pair that crushes your hand, or at least a very strong high card hand.

And some dealer hands like 444-AKJ (or any trips that are 8s or less) are "irreducible." The kickers are irrelevant because there cannot be a 444 vs 444 showdown. And no matter how the upcard progression goes it will always be a 444 hand. Whereas a trips hand like 999-AKJ is always going to dissolve down to a one-card hand.

Its fun to analyze.
gordonm888
Joined: Feb 18, 2015
• Posts: 613
June 3rd, 2017 at 12:30:09 PM permalink
I have revised my Discard inferno Calculator so that the dealer's hand may "de-compose" down to a single card. I also corrected two major errors I found in the calculator -each of which moves my answers closer to charliepatrick's results.

Charlie did an "infinite deck" calculation that indicated that, on the average, against any dealer upcard of 9-A that you should:
- BET 2x on A9xxx and higher
- BET 1X on J7xxx (any Jack-high hand) and higher

I have modeled the "vs. Ace" scenario which should be the most favorable to the player -who wouldn't like to see the Dealer discard an Ace from his hand rather than a jack or nine? Therefore, we would expect the "vs Ace" results to have the lowest thresholds for the player decisions.

This is what I now calculate with my composition-dependent calculator. The term "low card" refers to 2-8.

versus an Ace
Bet 1X on any Jack-high, J-xxxx and higher, otherwise fold.
Bet 2x on any Ace-high, A-xxxx and higher
Also, Bet 2X on a KQJxx-x with 3 low cards in your hand and Bet 1X with less than 2 low cards
Bet 1X on KQJTx-x and KQJ9x-x
Bet 2X on any KQ8 to KQ5; that is, any KQxxx-x with 4 low cards in your hand

Charliepatrick, or anyone else, does anyone have a clue what the house edge is on this game?
charliepatrick
Joined: Jun 17, 2011
• Posts: 1321
June 3rd, 2017 at 12:46:37 PM permalink
Quote: gordonm888

...Charliepatrick, or anyone else, does anyone have a clue what the house edge is on this game?

Good work, I'm glad you found an interesting quirk of raising with a lower hand while calling with a higher hand. Sorry I have no idea what the House Edge is which is why when I see a new game I tend to use infinite deck to get a rough strategy. As you've shown when the dealer throws away an Ace this seriously reduces the chance of them making Ace-high.

btw J7xxxx is the same as any J-high hand (since J65432 is a straight) and beats a Dealer 2-card J7 or lower. I'm guessing the reason for the cutoff there, is a hand such as T98754 leaves four Jacks whereas having J7xxxx uses one up.
gordonm888
Joined: Feb 18, 2015
• Posts: 613
June 3rd, 2017 at 4:26:32 PM permalink
Quote: charliepatrick

Good work, . . . . . I'm guessing the reason for the cutoff there, is a hand such as T98754 leaves four Jacks whereas having J7xxxx uses one up.

Thank you. And, yes, I notice a significant incremental change in EV whenever I move up a rank in the highest card of the HC hands. I think its for exactly the reason you state.
gordonm888
Joined: Feb 18, 2015