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 class of synthetic games generated by a 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.

翻译：确定性博弈求解算法通常通过分析其在随机博弈树分布上的平均情况复杂度进行评估，其中叶子节点的值独立地从固定分布中采样。这一简化模型使得数学分析得以清晰进行，揭示了两个关键性质：对于有限值树，根节点值的分布渐近地坍缩至单一固定值；所有合理的算法都能达到全局最优性。然而，这些发现是该模型设计的人为产物：其长期受到批评的独立性假设剥离了博弈的结构复杂性，产生了没有算法面临有意义挑战的平凡实例。为应对这一局限，我们引入了一类由概率模型生成的合成博弈，该模型使用固定的逐层条件分布增量构建博弈树。通过强制实施祖先依赖性（现实世界博弈的一个关键结构特征），我们的框架生成了具有可调难度同时保留某种形式解析可处理性的问题。对于包括 AlphaBeta 和 Scout 在内的多种算法，我们推导了描述其在该模型下平均情况复杂度的递归公式。这使我们能够在深度博弈树上严格比较算法，而蒙特卡洛模拟在此类情况下已不可行。虽然渐近地，所有算法似乎收敛到相同的分支因子（这一结果类似于基于独立性模型的结果），但深度有限树揭示了显著差异：与 Scout 等算法相比，AlphaBeta 产生了显著更大的常数乘性因子，导致实质性的实际速度下降。我们的框架为经典博弈求解算法提供了新的视角，提供了严格的证据和分析工具，以在更丰富、更具挑战性且仍可处理的模型下推进对这些方法的理解。