TD 2-7: 23456 on the last draw

[MarkGritter]

I am somewhat surprised to discover that, when both players are drawing one, it is usually better to draw to 2346 on the last draw rather than to 2345.

For example,
<font class="small">Code:</font><hr /><pre>
pokenum -l27 2c 3d 4h 5s / 6c - 2d 3h 4s 8d / kc
5-card Draw 2-7 Lowball: 1722 enumerated outcomes
cards win %win lose %lose tie %tie EV
5s 2c 3d 4h 768 44.60 945 54.88 9 0.52 0.449
4s 8d 2d 3h 945 54.88 768 44.60 9 0.52 0.551

pokenum -l27 2c 3d 4h 6c / 5s - 2d 3h 4s 8d / kc
5-card Draw 2-7 Lowball: 1722 enumerated outcomes
cards win %win lose %lose tie %tie EV
6c 2c 3d 4h 807 46.86 906 52.61 9 0.52 0.471
4s 8d 2d 3h 906 52.61 807 46.86 9 0.52 0.529

pokenum -l27 2c 3d 4h 5s / 6c - 2d 3h 6s 7d / kc
5-card Draw 2-7 Lowball: 1722 enumerated outcomes
cards win %win lose %lose tie %tie EV
5s 2c 3d 4h 766 44.48 956 55.52 0 0.00 0.445
6s 7d 2d 3h 956 55.52 766 44.48 0 0.00 0.555

pokenum -l27 2c 3d 4h 6c / 5s - 2d 3h 6s 7d / kc
5-card Draw 2-7 Lowball: 1722 enumerated outcomes
cards win %win lose %lose tie %tie EV
6c 2c 3d 4h 798 46.34 915 53.14 9 0.52 0.466
6s 7d 2d 3h 915 53.14 798 46.34 9 0.52 0.534
</pre><hr />

Even killing off one or both of the remaining 6 does not change the superiority of 2346:

2345 / 66 vs. 2367 / K: 0.445/0.555
2346 / 56 vs. 2367 / K: 0.464/0.536

2345 / 666 vs. 2367 / K is 0.445/0.555
2346 / 566 vs. 2367 / K is 0.462/0.538

So the difference is not due to the straight possibilities; after all, in the last example there are more ways to make a straight with 2346 than with 2345. (And in the very first example there are just as many outs to the 6-high straight whether the 6 or the 5 is discarded.) Perhaps it is something to do with pair vs. pair showdowns? What's going on here?

[SumZero]

Note pokenum has a bug in its 2-7 evaluation function where A2345 loses to A2346 because it considers the first one a straight and not a smooth A. I don't know if this is enough to make a difference in some of those close situations.

[randomstumbl]

Yeah, that pretty well explains it. Thanks for letting us know. This had been rolling around in my head for awhile.

[MarkGritter]

[ QUOTE ]
Note pokenum has a bug in its 2-7 evaluation function where A2345 loses to A2346 because it considers the first one a straight and not a smooth A. I don't know if this is enough to make a difference in some of those close situations.

[/ QUOTE ]

Yup, that would explain it. I put together a spreadsheet to verify the values by hand.

2345/6 vs. 2367/K with faulty 'A' evaluation: EV = 0.445
2345/6 vs. 2367/K with correct 'A' evaluation: EV = 0.473
2346/5 vs. 2367/K: EV = 0.466

Now the results make more sense. It is only worth discarding the 5 instead of the 6 if all the 5s are dead, and only marginally:

2345/6555 vs. 2367/K: EV = 0.5331
2346/5555 vs. 2367/K: EV = 0.5354

The only cases where the switch from 2345 to 2346 matters are the draws:

5 6, (win-&gt;loss)
5 7, (win-&gt;loss)
6 6, (loss-&gt;win)
6 7, (loss-&gt;win) and
7 4, (win-&gt;tie)

In this situation, normally the losing cases and decreased equity from a tie significantly outweigh the winning ones.