Full pay Jacks or Better gives an Expected Value (EV) of 99.544% when played optimally. Actually, this is one of the simplest games, so optimal play is truly within reach of anyone who wants to work at it for a while. The strategy given below is a complete, optimal strategy. The bulk of the strategy is "basic", meaning that it does not require consideration of penalty cards. The five additional rules to get optimal EV are given following the basic strategy, but they add only about 0.001% to the basic strategy, and so can be ignored unless you demand perfection.
A "basic" strategy for video poker only ranks the competing cards you might hold (the "holds"), while a "mixed" strategy considers the particular discards when deciding what to hold. For example, the correct basic strategy for 9/6 Jacks or Better is to hold JTs (Jack and Ten in the same suit) in preference to KJ (King and Jack of different suits). This is derived from considering all possible flops (initial hands) containing KsJhTh that do *not* contain something better than KJ or JTs (for example KK or KQJT) and computing which holding has the better expectation over those flops taken as a whole. A mixed strategy will sometimes hold KJ and other times hold JTs, depending on what the extra cards are. For example, if one of the extra cards is in the same suit with JTs, the KJ is a better hold.
Now, what I call a "best basic" strategy is the best possible basic strategy for a particular game. This is to contrast it with the optimal strategy, which is the best mixed strategy. Best basic strategies are important because we can actually play a perfect strategy in the casino (playing an optimal strategy without the aid of a computer is just about impossible unless you have some very unusual abilities). In any case, we can often find a perfect strategy that's withing a hair's breadth of optimal. My Best Basic strategy for the popular Full Pay (9/6) Jacks or Better game. It's only 0.00097% below optimal strategy in expectation. This is the strategy that appears on my video poker strategy cards, available on my VP page.
[BTW, I think it's a lot easier to play the strategy from my cards than from the typical linear listing used to give strategies, such as the one at the end of this note.]
As far as I know, I'm the only one that has a (self-written) program to evaluate a strategy (rather than a game -- there are several commercial programs for calculating the expectation for the optimal strategy for a given game). [Sorry, I'm not planing to turn my program into a commercial product.] I suspect that many of the published strategies are claiming higher expectations than they really deliver. For example, Lenny Frome's 9/6 strategy as published in _Winning Strategies for Video Poker_ has a claimed expectation of 99.5%, but I calculate its return as only 99.365%. If you play 600 hands per hour on a $.25 machine, this return is $1.33/hour worse than my strategy. (Not much perhaps, but it adds up.)
After the Best Basic strategy, I give the variations (using penalty cards) that turn it into an optimal strategy.
|Pay table: 1 2 3 4 6 9 25 50 800|
|Basic Strategy Expectation:||99.542939%|
|Off optimal by||0.000965%|
The EV numbers given below are derived from computing the actual EV that the given hold has over the hands in which the strategy plays it. The "**"s indicate inversions --- places where the EV value would misorder the hands. In all cases, go by the hand order rather than the EV value.
Note: For those of you that have my strategy cards, you may notice the numbers given here do not match those on the cards. The cards present the information in a simpler way, but the strategy is exactly equivalent.
Some abbreviations are used in the listing. In many cases, an explicit list of hands (possibly elided) is also given. "4cd Flush" means that you hold a four card flush. "3cd Ins Str Flush", or three card inside straight flush, refers to a straight with one gap such as Th8h7h. A "Dbl Ins Str" is a double inside straight, or a straight with two gaps. Note that A234 is an inside straight draw and A23 is a double inside straight draw. Finally, "n high cards" indicates the hold has n cards Jack or higher. Also, suited holds are shown in boldface to make them stand out.
|3||.555126||4cd Str Flush|
|2||.382535||4cd Ins Str Flush|
|1||.536540||Big Pair (AA,KK,QQ,JJ)|
|0||.872340||4cd Str, 3 high cards (KQJT)|
|0||.823682||Small pair (22,...,TT)|
|0||.808511||4cd Str, 2 high cards (QJT9)|
|0||.744681||4cd Str, 1 high card (JT98)|
|0||.731319||3cd Ins Str Flush, 2 high cards (QJ9s)|
|0||.726642||3cd Str Flush, 1 high card (JT9s)|
|0||.680851||4cd Str, 0 high cards (T987,...,5432)|
|0||.636597||3cd Dbl Ins Str Flush, 2 high cards (KQ9s,KJ9s,QJ8s)|
|0||.631241||3cd Ins Str Flush, 1 high card (QT9s,JT8s,J98s)|
|0||.625463||3cd Str Flush, 0 high cards (T98,...,543s)|
|0||.595745||4cd Ins Str, 4 high cards (AKQJ)|
|0||.581734||2cd Royal, 2 high cards (KQs,KJs,AQs,AKs,AJs)|
|0||.536012||3cd Dbl Ins Str Flush, 1HC, no K,Q (JT7s,J97s,J87s,A45s,...,A23s)|
|0||.531915||4cd Ins Str, 3 high cards (KQJ9,AKJT,AQJT) [*but not AKQT*]|
|0||.530375||3cd Ins Str Flush, 0 high cards (T97s,...,532s,432s)|
|0||.515264||3cd Ins Str, 3 high cards (KQJ)|
|0||.434920||3cd Dbl Ins Str Flush, 0 high (T96s,...632s)|
|0||.359792||Garbage (Toss all)|
Play variations from the basic strategy below to get optimal play. In each case, I give the basic play and the conditions under which it should be reversed. The "x" and "y"s stand for penalty cards, which must have the values shown. The boldfaced cards with "s" after them imply the cards are of the same suit.
|JTs > KJ||unless hand is K(JTxs) with x=2s-6s||QTs > AQ||unless hand is A(QTxs) with x=2s-7s||KTs > K||unless hand is (KTxs)9 with x=2s-8s||QJs > AKQJ||unless hand is AK(QJsx with x=9 or 2s-7s||AJTs > 4 Flush||unless hand is AJTxs,y with x=2s-9s and y=T,Q,K||AQTs > 4 Flush||unless hand is AQTxs,y with x=2s-9s and y=T,J,K||AKTs > 4 Flush||unless hand is AKTxs,y with x=2s-9s and y=T,J,Q|
The following table shows the frequency results for each possible final paying hand. This can be used to compute the EV of the strategy for games with similar, but different pay schedules.