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Is the optimum strategy the same for 9-6 and 8-5 video poker?

20 February 2012

Video poker is a casino favorite. For many reasons. One of these is the participatory nature of the game. Players have to make decisions during the course of a round – which of the dealt cards to hold or dump. If folks were unfamiliar with poker, in general, this would be mind-boggling. Leading to another reason for the popularity of the game – most people who go to casinos know enough about poker to make intuitive decisions which turn out to be reasonably sensible, without first doing their homework. Reasonably sensible, although not optimum.

Sophisticated solid citizens also understand that video poker configurations often offer fairly robust return percentages or low values of house advantage – two sides of a single coin – better than almost all reel-type slots and superior to much table action as well. And particularly proficient video poker buffs know how to pick the specific machines with the best payoff schedules. The top of the skill ladder is mastery of the strategy. Some dealt hands are no-brainers like holding “made” triplets and dumping two unrelated cards hoping for quads or a full house. Others involve subtleties such as the choice between keeping a four-card possible flush or a “made” high pair (the latter has higher average return), or a low pair as opposed to a three-card possible non-royal straight flush (the former has greater expectation).

Assume an individual has learned the optimum strategy for a game on a machine with a certain set of payoffs. The casino then installs a nominally similar game with another return schedule. How much does the player sacrifice by using the old strategy on the new device?

Consider the two jacks-or-better games with the payoffs represented in the accompanying table. The schedules diverge in that machine 1 pays 9-for-1 on full houses and 6-for-1 on flushes, while machine 2 pays 8-for-1 and 5-for-1 on the respective hands. The probabilities and returns are those which would be obtained for optimum play.

                          Machine 1 (9-6)                         Machine 2 (8-5)	
hand            payoffs    probabilities     returns    payoffs    probabilities    returns
Royal flush         800       0.002476%    1.980661%        800       0.002489%   1.991548%
Straight flush       50       0.010931%    0.546545%         50       0.010766%   0.538321%
Four of a kind       25       0.236255%    5.906364%         25       0.236289%   5.907222%
Full house            9       1.151221%   10.360987%          8       1.151368%   9.210942%
Flush                 6       1.101451%    6.608707%          5       1.090156%   5.450781%
Straight              4       1.122937%    4.491747%          4       1.123512%   4.494050%
Three of a kind       3       7.444870%   22.334610%          3       7.446275%  22.338824%
Two pair              2      12.927890%   25.855780%          2      12.929841%  25.859682%
High pair             1      21.458503%   21.458503%          1      21.507064%  21.507064%
Nothing               0      54.543467%    0.000000%          0      54.502239%   0.000000%
Total                              100%   99.543904%			         97.298434%

The data – based on analyses by Michael Shackleford (the “Wizard of Odds”) – show that when each is played optimally, Machine 1 returns 99.543904 percent of the amount bet while Machine 2 returns 97.298434 percent. The table reveals that the disparity between the alternatives isn’t caused solely by the changes in returns on the full house and flush. The probabilities for corresponding winning hands are unequal up and down the line. Indeed, the 9-6 machine loses rounds slightly more frequently than the 8-5 model – 54.543467 percent as opposed to 54.502239 percent. These phenomena occur because the optimum strategies for the games aren’t the same.

Assume that a player has studied diligently and can execute the optimum strategy for the 8-5 machine flawlessly. One day, this person finds that the bosses – doubtless inadvertently – have installed machines that look like those in the familiar 8-5 game except for that full house pays 9-for-1 and the flush 6-for-1. Following the 8-5 optimum strategy on the 9-6 machine, the probabilities associated with the various hands in the latter will be the same as those in the former because the alternate versions are identical except for the payoffs on the full houses and straights. The theoretical return on the 9-6 machine for this player would be 99.539592 percent. This is marginally less than the 99.543904 percent obtained with the optimum 9-6 strategy.

Conversely, pretend someone whose friendly neighborhood casino has been featuring the 9-6 machines shown in the table, but swaps them out one day for the 8-5 versions indicated. The player knows and uses the optimum 9-6 strategy. The theoretical return in this case is 97.291389 percent. Slightly below the 97.298434 percent achievable with the optimum 8-5 strategy.

If you know the optimum strategy for a game with one set of payoffs, is it worth learning the variations for a modified return schedule? Well, you have to decide for yourself how punctilious you want to get, and why. At $1.25 per round for four hours, with 10 spins per minute, your gross wager would be $3,000. The 8-5 strategy on the 9-6 machine would represent an average sacrifice of $0.13. The 9-6 optimum strategy on the 8-5 machine would sacrifice an average of $0.21. This doesn’t mean that if you learn the best way to play at a jacks-or-better machine and want to switch to deuces wild or some such game only vaguely comparable, you oughtn’t first go back to the books. For, as that masterful muse of the machines, Sumner A Ingmark, muttered:

Similarities incidental
May mask differences fundamental.
Alan Krigman

Alan Krigman was a weekly syndicated newspaper gaming columnist and Editor & Publisher of Winning Ways, a monthly newsletter for casino aficionados. His columns focused on gambling probability and statistics. He passed away in October, 2013.
Alan Krigman
Alan Krigman was a weekly syndicated newspaper gaming columnist and Editor & Publisher of Winning Ways, a monthly newsletter for casino aficionados. His columns focused on gambling probability and statistics. He passed away in October, 2013.