Keeping double aces/deuces/threes in first draw at badugi?

[prinsrob]

I just play this game and have not thought about it a lot yet. Is it a common tactic to keep a low pair (a/2/3) and then see at the next draw which one you can drop?

If you have AA2X (2 of different suit than aces): is it better to drop only X (so you can draw a low card of 3 suits) or drop AX (so you have to catch low cards of 2 suits (not the suit of the ace and not the suit of the 2).

[Nomar]

My first instinct is to say no (better to draw 2 cards to try and make a 3 or possibly 4 card hand, vs drawing one, and roughly half the time you will still only have a 2 card hand if you match one of your 2 suits)

If I have no reason to keep one or the other, online I always keep the one on the left, live I always keep in suit order... eliminating any second guessing

[*TT*]

I don't understand your question.

TT [img]/images/graemlins/club.gif[/img]

[DeathDonkey]

He's saying if you start with AA2x, and just toss the x, then if you make a decent card unsuited to the 2, but suited to one of the aces, you will know which ace to drop. It's like if we instead dropped the 2 and one of the Aces, say the A[img]/images/graemlins/diamond.gif[/img] and hold the A[img]/images/graemlins/spade.gif[/img] and pick up the 3[img]/images/graemlins/spade.gif[/img] we are going to be pissed at ourself.

-DeathDonkey

[*TT*]

[ QUOTE ]
He's saying if you start with AA2x, and just toss the x, then if you make a decent card unsuited to the 2, but suited to one of the aces, you will know which ace to drop. It's like if we instead dropped the 2 and one of the Aces, say the A[img]/images/graemlins/diamond.gif[/img] and hold the A[img]/images/graemlins/spade.gif[/img] and pick up the 3[img]/images/graemlins/spade.gif[/img] we are going to be pissed at ourself.

-DeathDonkey

[/ QUOTE ]

Oh.... thats a good question. I don't know.

TT [img]/images/graemlins/club.gif[/img]

[MarkGritter]

[ QUOTE ]
I just play this game and have not thought about it a lot yet. Is it a common tactic to keep a low pair (a/2/3) and then see at the next draw which one you can drop?

If you have AA2X (2 of different suit than aces): is it better to drop only X (so you can draw a low card of 3 suits) or drop AX (so you have to catch low cards of 2 suits (not the suit of the ace and not the suit of the 2).

[/ QUOTE ]

I think this is an interesting question. Intuition says you should draw two, but intuition might be wrong. Can we come up with some quantification here?

The obvious goal is to avoid the "oh crap" reaction of drawing a low card matching the suit of the card you kept. The disadvantage is of course that you have one less card that can improve your hand. (I don't think there is any benefit to drawing one as a semisnow.)

So... one way to approach the problem is to try counting, for all the possibilities of what the next two cards are, whether you prefer taking two or one (where the choice of which of the pair to discard is fixed, of couse--- but if you discard two you have to pick one of the deueces to get rid of, and the argument is parallel whichever one it is.)

For example, with A[img]/images/graemlins/spade.gif[/img]2[img]/images/graemlins/heart.gif[/img]2[img]/images/graemlins/diamond.gif[/img]K[img]/images/graemlins/spade.gif[/img] (discard options are either 2dKs or Ks), you would prefer drawing one if the next two cards are 6[img]/images/graemlins/heart.gif[/img]Q[img]/images/graemlins/club.gif[/img], prefer drawing two if the cards are 6[img]/images/graemlins/spade.gif[/img]3[img]/images/graemlins/diamond.gif[/img], and be indifferent if the cards are 6[img]/images/graemlins/heart.gif[/img]6[img]/images/graemlins/diamond.gif[/img].

Unfortunately there are 2450 possibilities for the next two cards so exhaustive search is hard to do by hand. It's also not necessarily clear, with two rounds to go, whether you would prefer standing pat with a medium badugi or drawing.

But, if we can identify all the cases where we would prefer to draw one, and come up with a larger set where we would definitely prefer to draw two, then drawing two is correct.

If we know the next two cards (X and Y), and prefer to draw one, the result must be better than As 2h X Y. That means we want to use the 2d, so the very next card (X) must be a low heart, AND the card after that (Y) must be a higher rank of any suit--- so that we would have gotten rid of it anyway. In no other situation does keeping the extra deuce help us, because if we are going discard the 2d we would have been just as good throwing it away on the first draw,

If we're not drawing to a 9 then there are just 174 two-card permutations where we prefer drawing one. (There are 39 cards 4 and higher left to go with the 3h, 35 cards 5+ to go with the 4h, etc.)

We can come up with more cases where we prefer drawing two fairly easily without being exhaustive:

1st card a low diamond (3-8), 2nd card an unpaired low club giving us an 8 or better badugi: 56 possibilities
same but with diamonds and clubs reversed: 56
a high heart (9+) followed by a low diamond: 30 possibilities
a high heart followed by a low club: 30
8h followed by 3 through 7 of diamonds: 5
8h followed by 3 through 7 of clubs: 5
total = 182, good enough though we can come up with many more, like the first card being a spade and the next being a club or diamond.

So there are definitely more cases where we will have a better hand by drawing two than there are cases where we will end up with a better hand by drawing one. Since we can't tell in advance which is going to be the case, we should always break the pair and draw two.

(It is also worth pointing out that the best hand you can get from drawing two is much stronger than if you are drawing one, while the worst case is identical.)

[MarkGritter]

I thought of a little story to make the argument a bit clearer.

You have a demon sitting on your shoulder who can see the two cards you will get if you draw two. He can't see the rest of the deck or the other player's cards, so he's not an Oracle.

The only question you can ask the demon is whether it is better to draw to A[img]/images/graemlins/spade.gif[/img]2[img]/images/graemlins/heart.gif[/img] or A[img]/images/graemlins/spade.gif[/img]2[img]/images/graemlins/heart.gif[/img]2[img]/images/graemlins/diamond.gif[/img]. (You're not allowed to ask him and then decide to draw to A[img]/images/graemlins/spade.gif[/img]2[img]/images/graemlins/diamond.gif[/img] instead given the extra information.) The demon can answer "draw one", "draw two", or "it doesn't matter" based on the information available to him.

I showed that the demon answers "draw one" in just the cases where the next card is a low heart, and the card after that wouldn't give you a better draw by discarding that heart. This totals just 174 of the 2450 cases. But maybe "doesn't matter" is so large that the demon would say "draw two" even less than this.

However, by counting the cases we can see that there are more cases where the demon says "draw two" than "draw one", even if we don't figure out exactly how many. So if we're going without the demon's advice we should stick with "draw two" since that is more probable to be correct.

One possible hole in this argument is that for some "doesn't matter" hands we could face a similar decision the next round. Suppose the next two cards are spades or aces --- a "doesn't matter". But if it is advantageous to draw one to A22 next round then that might convince us to only draw one this round as well! However, we can argue (using the same reasoning) that drawing one to A22 is not advantageous on the last round, so it is not desirable to preserve our ability to do so on previous rounds (by picking A22 rather than A2 when the answer is "doesn't matter".)

[prinsrob]

Thanks Mark and others!, the exhaustive answer Mark gave makes the answer to the problem clear to me [img]/images/graemlins/smile.gif[/img].