AlphaBeta is not as good as you think: a simple random games model for a better analysis of deterministic game-solving algorithms

TL;DR

Deterministic game solvers are often evaluated under an independence-based standard model that yields trivially predictable roots; the forward model proposed here injects ancestor dependencies via level-wise conditional sampling with a special inherited child, enabling tunable difficulty while preserving analytical tractability. The authors derive recursive formulations for the average-case complexities of TEST, ALPHA-BETA, and SCOUT under this model, showing that asymptotically the branching factor $r_A$ converges across algorithms, yet finite-depth performance differs, with ALPHABETA incurring larger constants than SCOUT or TEST. Finite-depth numerical analyses across diversified distributions confirm these differences and suggest practical superiority of SCOUT over ALPHA-BETA in realistic deep trees, while open-sourcing the code to reproduce and extend the results. Overall, the forward model provides a richer, yet tractable, framework for rigorously analyzing classical game-solving methods beyond the inadequacies of the standard model, informing both theory and practical benchmarking.

Abstract

Deterministic game-solving algorithms are conventionally analyzed in the light of their average-case complexity against a distribution of random game-trees, where leaf values are independently sampled from a fixed distribution. This simplified model enables uncluttered mathematical analysis, revealing two key properties: root value distributions asymptotically collapse to a single fixed value for finite-valued trees, and all reasonable algorithms achieve global optimality. However, these findings are artifacts of the model's design: its long criticized independence assumption strips games of structural complexity, producing trivial instances where no algorithm faces meaningful challenges. To address this limitation, we introduce a simple probabilistic model that incrementally constructs game-trees using a fixed level-wise conditional distribution. By enforcing ancestor dependencies, a critical structural feature of real-world games, our framework generates problems with adjustable difficulty while retaining some form of analytical tractability. For several algorithms, including AlphaBeta and Scout, we derive recursive formulas characterizing their average-case complexities under this model. These allow us to rigorously compare algorithms on deep game-trees, where Monte-Carlo simulations are no longer feasible. While asymptotically, all algorithms seem to converge to identical branching factor (a result analogous to that of independence-based models), deep finite trees reveal stark differences: AlphaBeta incurs a significantly larger constant multiplicative factor compared to algorithms like Scout, leading to a substantial practical slowdown. Our framework sheds new light on classical game-solving algorithms, offering rigorous evidence and analytical tools to advance the understanding of these methods under a richer, more challenging, and yet tractable model.

Figures (9)

• Figure 1: Illustration of a game-tree of height $h=2$ and branching degree $b=2$. (Left) In the standard model, leaf values are independently sampled from a distribution. (Right) In the forward model, intermediate values are sampled progressively, level by level, until leaf nodes are reached. One random child is chosen to inherit the root value, illustrated by a star symbol, and the remaining ones are sampled according to a fixed distribution, truncated to respect the minimax constraints.
• Figure 2: Finite-depth comparison of average-complexities. Each column corresponds to a parametrization of the forward model, consisting in the choice of the distribution $\mu$. (Top) Probability mass function of $\mu$ (Middle) Difficulty of the generated instances induced by the choice of $\mu$, measured as the dependency of the branching factor $r$ to the branching degree $b$. (Bottom) Relative average-complexity w.r.t. the average performance of the algorithm test, this allows us to visualize the sub-exponential multiplicative factor as a function of the height $h$ (with $b=10$, $n=5$). For each parametrization alpha-beta demonstrates the same convergence rate as test-bruteforce. Conversely, scout and test-bisection exhibit faster convergence, and remarkably scout even seems to incur a smaller multiplicative constant than test-hardest.
• Figure 3: Evolution of the Monte-Carlo mean estimator of the test complexity, as a function of the number of trials, for different settings. Results are averaged over 5 independent random seeds and shaded areas represent bootstrapped 95% confidence interval. The oracle is computed using Equations \ref{['eq:i_test_recurrence']} and \ref{['eq:j_test_recurrence']}.
• Figure 4: Evolution of the Monte-Carlo mean estimator of the alpha-beta complexity, as a function of the number of trials, for different settings. Results are averaged over 5 independent random seeds and shaded areas represent bootstrapped 95% confidence interval. The oracle is computed using Equations \ref{['equation:ab_complexity']} and \ref{['equation:ab_complexity_j']}.
• Figure 5: Evolution of the Monte-Carlo mean estimator of the scout complexity, as a function of the number of trials, for different settings. Results are averaged over 5 independent random seeds and shaded areas represent bootstrapped 95% confidence interval. The oracle is computed using Equations \ref{['eq:i_scout']} and \ref{['eq:j_scout']}.

Theorems & Definitions (8)

• Theorem 4.1
• Theorem 5.1
• Theorem 5.2
• proof
• Proposition A.1
• proof
• proof
• proof