The “best” basic strategy for 9/7 Triple Double Bonus

A few days ago, I wrote about the complexities of trying to figure out the best basic strategy for 9/7 Triple Double Bonus Poker. The software that was calculating that basic strategy finally finished its job and reported back. So I can report its findings.

My software took 28 hours and 24 minutes to run on a single CPU. (If I had spent the time to change my software and make it process data in parallel, which would distribute the workload between the four CPUs the server had, it would have only taken about seven hours. But I was in no rush, and I preferred that the machine does some extra work instead of me.) For the CPU time on their servers, the cloud service Google Compute Engine charged me $7.59 from the $300 free credit it had given me when I signed up for a trial. So I didn’t have to pay anything out of pocket.

Now I have the best order for the fifteen options I talked about. It is:

456, 567, 678, 789, 9TQ > 77 > 345, 89J, 8TJ > 66 > 89T > 55, 88, 99, TT

(For comparison, the Wizard of Odds Strategy Calculator, WOOSC, orders these hands as:

789 > 88 > 456, 9TQ > 77 > 345, 89J, 8TJ > TT > 678 > 66 > 89T > 99 > 567 > 55

And Video Poker for Winner, VPW, has “SF3+0 > low pair: 55–TT”, which translates to:

345, 456, 567, 678, 789, 89T, 89J, 8TJ, 9TQ > 55, 66, 77, 88, 99, TT)

I define a basic strategy as a strategy that just lists the options in some order, without other complexities such as references to other cards in the hand (aka penalty cards). Rules such as “A > AH when no other card suited with A” would break that definition because they would refer to other cards in the hand. So I don’t consider strategies that contain such rules a basic strategy. This is just a definition, of course. You can make your own definitions. But the useful thing about my definition is that it matches what WOOSC considers a basic strategy, so I can compare different strategies relatively fairly.

The WOOSC strategy returns 99.57613%, with an error of 0.001679%. (A strategy’s error is the difference between its return and computer perfect play.) The largest error it makes is in two hands: TT567 and 88345. In both, the SF3 is the better option to hold, while the WOOSC strategy recommends the low pair. Both are covered by the computationally-hard-to-figure-out fifteen-long list of options above.

I surgically went in and changed the order of the above fifteen hands in the WOOSC strategy to the best order (the order that my software had calculated). I changed nothing else. The strategy’s return improved to 99.57707%, with an error of 0.000737%. The error was cut to less than half!

But that’s not all the hands that WOOSC gets wrong. There are simpler options that WOOSC puts in the wrong order, hands that are not computationally difficult to figure out. Options that are unlike the fifteen options discussed above. I have no idea why WOOSC puts these in the wrong order. One example is QT8 vs AKQT. The 3-card straight flush is always better to hold than the 4-card straight, but the WOOSC strategy tells you to hold the 4-card straight.

Apart from the fifteen hands above, here is the list of all other option pairs where WOOSC’s strategy can be easily improved by putting the two options in the correct order:
QT8 > AKQT, AK
Q98 > AK
JT7 > AK, AQ
J97 > AK, AQ, KQ
J87 > AK, AQ, KQ
67T > QJT
569, 579 > QJ

With the above changes, the strategy would return 99.57713%, with an error of 0.000682%. I believe my software to be correct, and this number to be the smallest error possible with a basic strategy for 9/7 Triple Double Bonus.

As another point of comparison, VPW claims a return of 99.57702% for its strategy, for an error of 0.000793% (I have not verified its claim yet). But it may not be fair to compare these strategies: VPW has an intentionally simplified basic strategy that is much easier to learn than the WOOSC strategy, or my “best” basic strategy.

Anyway, the following strategy is the basic strategy with the highest return possible for 9/7 Triple Double Bonus:

RF5; SF5; 4-OF-A-KIND-WITH-KICKER
4-OF-A-KIND
3-of-a-kind with kicker: AAA–444 + A–4
AAA-444 > FULL HOUSE > 3-OF-A-KIND: others
RF4 > FL5; ST5 > SF4
AA > TWO PAIR
FL4
RF3: except AHT > KK, QQ, JJ > RF3: AHT
low pair: 22–44 > ST4
SF3 2hi; SF3: QT9, JT9, 456–789
77 > JT8, J98, 345 > 66 > 89T > 55, 88–TT
AKQJ
FL3 2hi
SF3: JT7, J97, J87
AQ, AJ, KH, QJ
AQJT, KQJ9
SF3: Q98, QT8
AK
SF3 0h 1i
AKHT
SF3: A-low, KT9
KQJ; ST4i 2hi
A
FL3 1hi: except K65, K64, K63, K62, K54, K53, K52, K43, K42, and K32
JT
FL3 1hi: K65, K64, K63, K62, K54, K53, K52, K43, K42, and K32
SF3−2: 569, 579, 67T
QJT > QJ
SF3−2: 458, 468, 478, 589
QT
SF3−2: others
KH
KT
K, Q, J
FL3 0hi
ST4i 0hi
SF2: 34
five new cards

That’s not the most beautiful or the simplest strategy, neither is it the most advanced strategy. But it’s the basic strategy with the highest return. No other basic strategy can have a higher return or a lower error!