AlphaBeta is not as good as you think: a new probabilistic model to better analyze deterministic game-solving algorithms

2025-06-27

AI Summary

Conventional stochastic game-tree models assume leaf-node values are independent and identically distributed, ignoring structural dependencies inherent in real games—leading to average-case complexity analyses that poorly reflect practical algorithmic behavior. Method: We propose the first progressive probabilistic game-tree model featuring ancestral dependence, where node values are governed by hierarchical conditional distributions that capture structural correlations across levels while preserving mathematical tractability and controllable problem hardness. Contribution/Results: Our theoretical analysis reveals that, on deep finite trees, algorithms such as Alpha-Beta pruning and Scout share the same asymptotic branching factor, yet Alpha-Beta exhibits a significantly larger constant factor—rendering it strictly slower in practice. This is the first rigorous demonstration of performance divergence among classical game-tree search algorithms on nontrivial instances. The model thus establishes a more realistic theoretical foundation for evaluating and comparing adversarial search algorithms.

Technology Category

• Search and Optimization: Adversarial Search
• Reasoning under Uncertainty: Stochastic Optimization
• Game Theory and Economic Paradigms: Adversarial Learning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Web Mining and Content Analysis: Models for Web evolution

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 new probabilistic model that incrementally constructs game-trees using a fixed level-wise conditional distribution. By enforcing ancestor dependency, 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 those 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 more realistic, challenging, and yet tractable model.

Problem

• Analyzes limitations of current deterministic game-solving algorithm models
• Introduces new probabilistic model with ancestor dependency for realistic complexity
• Compares algorithm performance on deep game-trees revealing practical differences

Innovation

• New probabilistic model with ancestor dependency
• Recursive formulas for average-case complexities
• Adjustable difficulty while retaining tractability