A Deuce-to-Seven Puzzle for the New Year (low content)

[MarkGritter]

Happy New Year!

Can you come up with a nontrivial 2-7 lowball situation where the correct play, in terms of showdown equity, is to stand pat with an ace?

By nontrivial I mean that your opponent is not keeping a pair or a worse ace-high hand. A hot and cold analysis, like pokenum, should report a higher 'EV' for standing pat instead of drawing--- no folding equity, additional bets on the river, etc. matter here.

What is the fewest number of known cards (dead, discarded, or in a player's hand) needed to produce this situation?

[MarkGritter]

No takers, huh? Here is the situation I came up with:

Hero: 2/images/graemlins/club.gif 4/images/graemlins/club.gif 6/images/graemlins/club.gif 8/images/graemlins/club.gif A/images/graemlins/diamond.gif
Villian: 3/images/graemlins/heart.gif 4/images/graemlins/heart.gif 5/images/graemlins/heart.gif 6/images/graemlins/heart.gif
Dead: 5/images/graemlins/diamond.gif 3/images/graemlins/diamond.gif 3/images/graemlins/spade.gif K/images/graemlins/diamond.gif Q/images/graemlins/diamond.gif (14 cards total)

Keeping the ace:
<font class="small">Code:</font><hr /><pre>
5-card Draw 2-7 Lowball: 37 enumerated outcomes
cards win %win lose %lose tie %tie EV
8c 6c 4c 2c Ad 20 54.05 17 45.95 0 0.00 0.541
6h 5h 4h 3h 17 45.95 20 54.05 0 0.00 0.459
</pre><hr />

Dumping the ace:
<font class="small">Code:</font><hr /><pre>
5-card Draw 2-7 Lowball: 1332 enumerated outcomes
cards win %win lose %lose tie %tie EV
8c 6c 4c 2c 702 52.70 630 47.30 0 0.00 0.527
6h 5h 4h 3h 630 47.30 702 52.70 0 0.00 0.473
</pre><hr />

I'm somewhat unhappy with this situation because Villian is better off keeping either the Kd/Qd if he had those, or the non-flush card if he had paired. So Villian has made an error in his discard.

In fact, Hero is better off dumping both ace and eight! His equity by doing so increases to 0.573.

Obviously, if we kill off everything but clubs then Hero is strictly better not drawing because Hero can only make a flush by doing so. So there are (heavily artificial) situations where Hero is actually better off standing pat with an ace.

Challenge 1: I will give $5 Stars for a proof specifying the lowest number of cards needed to engineer such a situation (a nontrivial example where standing pat with an ace has superior equity to drawing). An example with just ten cards will be sufficient proof.

Challenge 2: For an additional $5, solve the problem above given the constraint that Villian makes his best discard. Alternately, provide a proof that Hero must discard an ace whenever Villian makes his optimal move.

Relevance: Is there any admissible heuristic for discarding high cards? (i.e., a rule of thumb that always works.) A likely admissible heuristic for keeping low cards is that you need never discard a deuce and an unsuited wheel card. In contrast, you might have to discard a 6, 8, or a suited wheel card to win. But if there are situations in which keeping an ace is correct, then there cannot be any rule which says it is unconditionally OK to dump a high card.

[MarkGritter]

Tapirboy suggested 8c 7c 6c 5c As as a hand where it was preferable to keep the ace against an opponent drawing two. While this does provide 10-card example where standing pat is preferable to dumping the ace, that is not Hero's best draw in the cases I looked at--- he is better off drawing two (discarding A7.) So I wasn't able to turn this suggestion into a solution for #1, although picking the right Villian cards might tilt things sufficiently.

<font class="small">Code:</font><hr /><pre>
pokenum -l27 8c 7c 6c 5c as - td 9d 8d / kd ks
5-card Draw 2-7 Lowball: 1722 enumerated outcomes
cards win %win lose %lose tie %tie EV
As 8c 7c 6c 5c 996 57.84 726 42.16 0 0.00 0.578
Td 9d 8d 726 42.16 996 57.84 0 0.00 0.422

pokenum -l27 8c 7c 6c 5c / as - td 9d 8d / kd ks
5-card Draw 2-7 Lowball: 68880 enumerated outcomes
cards win %win lose %lose tie %tie EV
8c 7c 6c 5c 32684 47.45 36196 52.55 0 0.00 0.475
Td 9d 8d 36196 52.55 32684 47.45 0 0.00 0.525

pokenum -mc 500000 -l27 8c 6c 5c / 7c as - td 9d 8d / kd ks
5-card Draw 2-7 Lowball: 500000 sampled outcomes
cards win %win lose %lose tie %tie EV
8c 6c 5c 290034 58.01 209919 41.98 47 0.01 0.580
Td 9d 8d 209919 41.98 290034 58.01 47 0.01 0.420
</pre><hr />

[MarkGritter]

OK, so Tapirboy finally set me straight. It does matter what the quality of Villian's draw is here. If we give Villian 23377 (so he has to draw two) and Hero 8c7c6c5cA, then we have a situation where Hero is better off standing pat even against Villian's best draw, satisfying both #1 and #2.

I like this example, because Villian is better off drawing anything but one, even four to a 7 is better than drawing to the OESFD.

<font class="small">Code:</font><hr /><pre>
pokenum -l27 8c 5c 6c 7c as - 2s 3d 7h / 3s 7d
5-card Draw 2-7 Lowball: 1722 enumerated outcomes
cards win %win lose %lose tie %tie EV
As 8c 7c 6c 5c 678 39.37 1044 60.63 0 0.00 0.394
2s 3d 7h 1044 60.63 678 39.37 0 0.00 0.606

pokenum -l27 8c 5c 6c 7c / as - 2s 3d 7h / 3s 7d
5-card Draw 2-7 Lowball: 68880 enumerated outcomes
cards win %win lose %lose tie %tie EV
8c 7c 6c 5c 20368 29.57 48512 70.43 0 0.00 0.296
2s 3d 7h 48512 70.43 20368 29.57 0 0.00 0.704

pokenum -mc 500000 -l27 8c 5c 6c / 7c as - 2s 3d 7h / 3s 7d
5-card Draw 2-7 Lowball: 500000 sampled outcomes
cards win %win lose %lose tie %tie EV
8c 6c 5c 176738 35.35 323262 64.65 0 0.00 0.353
2s 3d 7h 323262 64.65 176738 35.35 0 0.00 0.647

pokenum -mc 500000 -l27 7c 5c 6c / 8c as - 2s 3d 7h / 3s 7d
5-card Draw 2-7 Lowball: 500000 sampled outcomes
cards win %win lose %lose tie %tie EV
7c 6c 5c 191485 38.30 308483 61.70 32 0.01 0.383
2s 3d 7h 308483 61.70 191485 38.30 32 0.01 0.617

pokenum -mc 500000 -l27 7c 5c 8c / 6c as - 2s 3d 7h / 3s 7d
5-card Draw 2-7 Lowball: 500000 sampled outcomes
cards win %win lose %lose tie %tie EV
8c 7c 5c 191031 38.21 308937 61.79 32 0.01 0.382
2s 3d 7h 308937 61.79 191031 38.21 32 0.01 0.618

pokenum -mc 500000 -l27 7c 5c / 8c 6c as - 2s 3d 7h / 3s 7d
5-card Draw 2-7 Lowball: 500000 sampled outcomes
cards win %win lose %lose tie %tie EV
7c 5c 180259 36.05 319655 63.93 86 0.02 0.361
2s 3d 7h 319655 63.93 180259 36.05 86 0.02 0.639

pokenum -mc 500000 -l27 7c / 5c 6c 8c as - 2s 3d 7h / 3s 7d
5-card Draw 2-7 Lowball: 500000 sampled outcomes
cards win %win lose %lose tie %tie EV
7c 168921 33.78 330914 66.18 165 0.03 0.338
2s 3d 7h 330914 66.18 168921 33.78 165 0.03 0.662
</pre><hr />

I did verify that Villian is not better off keeping a pair, too. /images/graemlins/wink.gif

Tapirboy gets the prizes.

Unfortunately this means there is no easy way to shortcut the decision about whether to keep an ace (or K, or Q) or not. I would still be interested in finding an admissible heuristic for thse cases (doing so could eliminate 50% of the subcases that need to be searched), but it will not be a simple one based on the number of cards remaining, as I had hoped.