Optimal conditional expectation at the video poker game Jacks or Better

TL;DR

This paper presents a systematic method to derive optimal strategy tables for Jacks or Better video poker by calculating exact conditional expected returns for all initial hands, improving strategy derivation.

Contribution

It introduces a novel approach to compute exact optimal conditional expected returns, enabling the creation of precise hand-rank tables for video poker strategies.

Findings

Identified 1,153 distinct expected return values for initial hands.

Mapped 387 hand categories with optimal expected returns.

Demonstrated the method's applicability to other video poker variants.

Abstract

There are 134,459 distinct initial hands at the video poker game Jacks or Better, taking suit exchangeability into account. A computer program can determine 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 the optimal strategy perfectly, was found for Jacks or Better as early as the mid 1990s. Is there a systematic way to derive such a hand-rank table? We show that there is indeed, and it involves finding the exact optimal conditional expected return, 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 five new cards. We describe the hands corresponding to each of the remaining 387 values of the optimal conditional expected return (sorted from largest to smallest) and show how this leads readily to an optimal strategy hand-rank table for Jacks or Better. Of course, the method applies to other video poker games as well.