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There are 134,459 distinct initial hands at the video poker game Jacks orBetter, taking suit exchangeability into account. A computer program candetermine the optimal strategy (i.e., which cards to hold) for each such hand,but a complete list of these strategies would require a book-length manuscript.Instead, a hand-rank table, which fits on a single page and reproduces theoptimal strategy perfectly, was found for Jacks or Better as early as the mid1990s. Is there a systematic way to derive such a hand-rank table? We show thatthere is indeed, and it involves finding the exact optimal conditional expectedreturn, given the initial hand. In the case of Jacks or Better (paying 800, 50,25, 9, 6, 4, 3, 2, 1, 0), this is a random variable with 1,153 distinct values,of which 766 correspond to garbage hands for which it is optimal to draw fivenew cards. We describe the hands corresponding to each of the remaining 387values of the optimal conditional expected return (sorted from largest tosmallest) and show how this leads readily to an optimal strategy hand-ranktable for Jacks or Better. Of course, the method applies to other video pokergames as well.

Optimal conditional expectation at the video poker game Jacks or Better