Exact Calculation of Expected Values for Splitting Pairs in Blackjack

Abstract: Computer calculations for most exact expected values in blackjack have been available since the 1960's, but exact results for pair splitting and resplitting have previously been too computer intensive. This paper describes a new algorithm for exact pair-splitting. By using dealer probability caching methods and revising the method for recursively generating possible player hands, the estimated calculation time compared to standard methods was reduced by five orders of magnitude. The resulting algorithm was used to calculate the first exact and complete pair splitting results for a single deck game. The exact results were compared to prior approximate theories for resplitting. The prior theories are accurate for many calculations, but inaccurate for resplitting tens. A new approximation method was developed that is accurate for all resplitting calculations.

• The paper introduces an optimized algorithm that reduces CPU time for splitting pair calculations in single-deck Blackjack from thousands of years to 45 days using dealer probability caching and unique hand enumeration.
• The paper demonstrates that commonly used approximations are precise for most pairs but significantly imprecise for resplitting Tens, leading to a new resplitting formula with errors typically under 0.006.
• The study offers practical insights into splitting strategies, showing that while resplitting provides a marginal benefit, splitting Tens remains generally disadvantageous under basic strategy.

This paper tackles the computationally challenging problem of calculating the exact expected values for splitting pairs in single-deck Blackjack, particularly when resplitting is allowed (1909.13710). While methods for calculating expected values for hitting, standing, and doubling down are well-established and fast, splitting requires considering combinations of multiple hands, leading to an exponential increase in complexity ($N^h$, where $N$ is the effective number of single hands and $h$ is the maximum number of split hands). Initial estimates using standard recursive hand generation suggested that calculating exact values for splitting up to 4 hands ($h=4$) would take thousands of years of CPU time.

The core contribution is a significantly optimized algorithm that makes these calculations feasible. The key optimizations are:

1. Dealer Probability Caching: The algorithm avoids recalculating the dealer's final hand probabilities ($\vec D(u,\vec r)$, the probabilities of the dealer finishing with 17, 18, ..., 21, or busting, given their up card $u$ and the set of removed cards $\vec r$) every time. It uses a caching mechanism where the set of removed cards (player hands + dealer up card) serves as a key.
   • Implementation: Removed cards are sorted, and a unique address $K_j$ is calculated using combinatorial formulas based on the number of cards removed ($j$) and their values. This address points to a cache entry storing the pre-calculated dealer probabilities.
   • Formula: The cache address $K_j(x_1,x_2,\dots,x_j)$ for $j$ removed cards $x_1 \ge x_2 \ge \dots \ge x_j$ is calculated efficiently using $$K_j(x_1,x_2,\dots,x_j) = 1+\sum_{i=1}^{j} T_{i}(x_{j-i+1})$$, where $T_i(N) = \binom{N+i-1}{i}$ represents the number of ways to choose $i$ items from $N$ categories with replacement.
   • Impact: This reduced calculation time by up to two orders of magnitude.
2. Pre-enumerated Hands Method: The standard recursive approach generates the same unique hand multiple times via different card orderings (e.g., (2,3,4) vs (2,4,3)). This optimization first generates and stores all unique possible player hands reachable during play. The splitting calculation then iterates over these unique hands instead of recursively exploring card draws.
   • Implementation: A recursive function generates all single hands, using the same hashing/addressing mechanism as the dealer cache to identify and store unique hands along with their frequencies, bet sizes (for doubling down), and whether the second card was a split card (for resplitting logic). The main splitting calculation then uses this pre-built list.
   • Impact: This reduced calculation time by another three orders of magnitude for complex scenarios (high dealer up cards), drastically lowering the effective branching factor $N$.

Combining these methods reduced the total estimated CPU time for calculating all splitting variations (different pairs, up cards, double down rules, resplitting up to $h=4$) from ~11,000 years to ~45 days on 2008-era hardware. The paper notes that parallelization is best achieved by assigning different dealer up cards to separate processors, each maintaining its own caches.

The paper presents the first exact splitting expected value tables for single-deck Blackjack under various rules (resplitting allowed/disallowed, different double down after split options, dealer stands/hits soft 17). Key findings from these results include:

• Comparison with Approximations:
  • Griffin's widely used approximate formula for splitting (without resplitting) is confirmed to be extremely accurate [Griffin:1979].
  • Griffin's approximation for resplitting is accurate for most pairs but significantly inaccurate for resplitting Tens.
  • A new approximate formula for resplitting is developed, refining the probability calculations based on the sequence of split/non-split cards drawn. This new approximation is shown to be highly accurate (errors typically < 0.001, except for Tens where errors are < 0.006) and computationally efficient.
• Practical Implications:
  • The overall impact of allowing resplitting on the game's expected value is small (0.017% to 0.189% benefit to the player).
  • Eliminating splitting entirely penalizes the player by only ~0.36%.
  • Splitting Tens is generally unfavorable in basic strategy. Even for card counters where an initial split might be correct, the threshold count for subsequent resplits increases, making repeated splitting of Tens likely disadvantageous despite anecdotal stories suggesting otherwise.
  • Accurate calculation of overall game expected value requires high precision (at least 5 digits) in the underlying expected value tables for stand/hit/double/split decisions.

Appendices provide pseudocode for the recursive and pre-enumerated hand splitting algorithms (exactSplitExval, handExactSplitExval) and details for calculating probabilities needed in the new approximate resplitting formula. Complete tables and source code are referenced as being available online [Nairn:BJGit, Nairn:BJWeb].