What does Perfect Play mean for Video Poker?

Simply put, perfect play means holding cards of a given hand that lead to the best possible expected value. Expected value is the average value of all possible outcomes. In the long-run, perfect holding gives the best average return. A perfect hold is also called an optimal hold. This app always shows perfect holds.

How important is this? Suppose you were dealt the following cards in a Jacks or Better video poker game:

Simply put, perfect play means holding cards of a given hand that lead to the best possible expected value. Expected value is the average value of all possible outcomes. In the long-run, perfect holding gives the best average return. A perfect hold is also called an optimal hold. This app always shows perfect holds.

How important is this? Suppose you were dealt the following cards in a Jacks or Better video poker game:

What should you hold? The best hold depends on the game AND on the payouts for the various types of outcomes. For one common set of Jacks or Better payouts (called the full-pay or 9-6 payouts) the best hold is:

Surprisingly, for another pay table (called the 9-5 pay table) where the only difference is that a Flush pays 5 coins per coin wagered instead of 6, the best hold is:

Had a player chosen this latter hold when playing the 9-6 game, the expected outcome would be 7.13% lower. That adds up! Furthermore, many players choose far worse non-optimal holds. This app hopes to rectify that situation.

If you think this is an isolated case, you would be wrong. Fully 11,568 hands (627 not counting suit permutations of hands) should be played differently between these two pay tables.

If you think this is an isolated case, you would be wrong. Fully 11,568 hands (627 not counting suit permutations of hands) should be played differently between these two pay tables.

What about Multiple Best Plays?

Surprisingly, it may be the case that multiple hold hands can lead to the same expected value of outcomes

Hold Hands with Equal Expected Values

Consider the video poker game Deuces Wild. With certain payouts (i.e., for the pay table labeled 800-200-25-15-9-4-4-3), the following hand has two different holds giving the same optimal expected value (see here how you can display all possible hold hands and their expected values and standard deviations for a given dealt hand.)

One discards the ten of Spades:

and the other discards the Jack of Hearts

Although these both give the same expected payout, the standard deviations - the potential spread of the outcomes - is much higher for the latter case. This is easy to see for this case. If the first hold is held, the worst possible outcome is Four of a Kind (paying 4 units) and the best, Five of a Kind (paying 15 units). If the second option is held, the worst outcome is Three of a Kind (paying 1 unit) and the best a Wild Royal Straight Flush (paying 25 units). The former case has a smaller low to high possible spread of payouts than the latter case.

The technical details are as follow. Both holds give an expected outcome of $4.9362 for every dollar wagered. So, over the long-run, both holds are equivalent. However, the standard deviation for the first hand is $3.0694 and for the latter, $7.1798. Conservative players may prefer the lower spread case where others may want to "go for it" hoping for the higher outcomes - well, this is gambling after all. Of course, they may end up with a lower worse case. In the short run, only luck dictates who would be better-off.

The technical details are as follow. Both holds give an expected outcome of $4.9362 for every dollar wagered. So, over the long-run, both holds are equivalent. However, the standard deviation for the first hand is $3.0694 and for the latter, $7.1798. Conservative players may prefer the lower spread case where others may want to "go for it" hoping for the higher outcomes - well, this is gambling after all. Of course, they may end up with a lower worse case. In the short run, only luck dictates who would be better-off.

This app will recommend the former hold when the default option "break expected value ties using the one with a smaller standard deviation" is set (see Settings). Otherwise the maximum standard deviation hold will be shown. (The player can toggle between the two and examine a table showing all the trade-offs)

Hold Hands with Equal Expected Values and Standard Deviations

In other cases there are equal expected values AND equal standard deviations, so one is really indifferent between which to play. In this case the app breaks ties in its recommendation by choosing the hand with the greater or fewer number of cards according to a user-selected setting. For example, in Jacks or Better, the following hand has two optimal hold hands:

One can hold the four Aces or one can hold all five cards. The app lets you toggle between the two, but uses the recommended hand in the Game Play mode to track optimal play winnings.

Hold Hands with Equal Expected Values, Standard Deviation, and Number of Cards

Unbelievably, perhaps, there may still be equivalent tied, optimal hold hands. For certain Deuces Wild games, the following hand yields two optimal hold hands having the same expected value of outcomes, the same standard deviations and the same number of hold cards:

The two equivalent hold plays are hold the fours or hold the fives. The app breaks these ties using the user selected setting for either choosing the hand having the lower denomination cards (i.e., the pip) or the one with the higher one. The app lets you toggle between the two.

Very rarely - we know of only two cases across all our games and pay tables - there can be a three way tie. For example in Joker Poker (2 Pair) with pay table 100-800-100-100-16-8-5-4, for the hand:

The following shows the three tied hold hands. As one can see, the hands are sorted using the lowest denominations to break ties.

The other hand is: